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+{"text":"\\section{Introduction}\n\nModeling the behavior of wavelength-sized particles in optical tweezers can \nguide experiments in optical manipulation or assembly.\nDoing so requires calculating the optical forces and torques exerted on particles in a focused laser beam.\nThese calculations are particularly challenging for wavelength-sized particles since \n neither the Rayleigh approximation nor ray optics can be validly used.\nRather, detailed consideration of the electromagnetic fields incident on and scattered by the particle is necessary\n \\cite{jones_optical_2015}.\n$T$-matrices provide a well-established formalism for computing and describing electromagnetic scattering by non-spherical\n particles \\cite{mishchenko_t-matrix_1996, mishchenko_light_1991, mackowski_calculation_1996}.\nTo date, however, the use of $T$-matrices to calculate forces and torques in optical tweezers has mainly been limited\nto rotationally symmetric particles or particles that are smaller than the wavelength\n\\cite{borghese_radiation_2006, borghese_optical_2007, borghese_rotational_2007, \nnieminen_optical_2007, nieminen_t-matrix_2011, cao_equilibrium_2012, \nqi_comparison_2014}.\n\nEven so, being able to compute optical forces and torques is not sufficient for understanding the behavior of\nnon-spherical particles in optical tweezers.\nKey questions include the positions, orientations, and stability of any trapping equilibria that may exist.\nBut as Bui \\emph{et al.}~point out, finding equilibria solely from calculations of forces and torques at fixed positions\nand orientations is difficult for non-spherical particles because of the size of the parameter space \\cite{bui_theory_2017}. \nConsider for example a sphere in a circularly polarized beam.\nRotational symmetry requires that the sphere center lie on\nthe beam axis in equilibrium.\nMoreover, the sphere's orientation is irrelevant.\nSo, to find the equilibrium trapping position, it is only necessary to calculate the axial component of the trapping force \nas a function of the sphere's position along the beam axis.\nNo such simplifications are possible for particles lacking symmetry; mapping the trapping landscape for an asymmetric\nparticle may require considering 3 positional and 3 orientational degrees of freedom \\cite{bui_theory_2017}.\n\nAn alternative strategy for finding equilibria is to perform dynamical simulations that use computations of optical\nforces and torques to model the actual motion of a particle in optical tweezers.\nIf the goal is only to find trapping equilibria, it is not necessary to carefully consider other interactions that a trapped\nparticle may experience but that vanish in static equilibrium, such as hydrodynamic resistance.\nIn contrast, investigating the stability of any trapping equilibria or the dynamics of transitions between multiple\nequilibria requires more care. \nOptically-trapped particles suspended in a fluid \nexperience Brownian motion involving both hydrodynamic interactions and random thermal interactions.\nThese can both be highly anisotropic.\nDetailed consideration of Brownian motion is also needed \nto explore photokinetic effects arising from the angular momentum carried by an elliptically- or circularly-polarized beam\nsince they can involve an interplay between optical and thermal interactions\n\\cite{simpson_first-order_2010, ruffner_optical_2012}.\n\nHere, we report the first Brownian dynamics simulations for wavelength-sized non-spherical particles \nin optical tweezers that \naccount for optical interactions, hydrodynamic interactions, and thermal fluctuations in detail.\nWe combine \\emph{mstm}, a well-established package for calculating $T$-matrices of clusters of spheres \n\\cite{mackowski_calculation_1996, mackowski_multiple_2011}\nwith \\emph{ott}, a package that can calculate the optical forces and torques on a particle in a focused laser beam \ngiven a $T$-matrix \\cite{nieminen_optical_2007, lenton_ilent2ott_2020}.\nFurthermore, we use complete diffusion tensors for the clusters in order to realistically predict\ntheir anisotropic behavior in a fluid.\nOur work not only extends the size regime for which optical forces and torques on sphere clusters have previously\nbeen reported but also predicts their detailed motion in optical tweezers for the first time.\nWhile dynamical simulations of particles in optical tweezers have previously been reported \n\\cite{simpson_first-order_2010, cao_equilibrium_2012, lenton_optical_2018, bui_theory_2017, volpe_simulation_2013,\narmstrong_swimming_2020},\nour simulations incorporate both thermal fluctuations and anisotropic hydrodynamic resistance,\nand we do not linearize the optical forces and torques.\nWe validate our simulations by considering single spheres and then explore the remarkably rich dynamical \neffects that occur for clusters of two spheres that are each comparable to or larger than the wavelength of \nthe trapping laser.\nWe then use our simulations to find multiple trapping equilibria for a highly asymmetric cluster\nof seven spheres.\n\n\n\\section{Simulation methods}\n\n\\subsection{Simulating Brownian motion with external interactions}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=1.75in]{.\/figures\/cartoon\/cartoon}\n\\caption{\\label{fig:cartoon} Coordinate systems used in simulations. In the laboratory coordinate system, a laser beam \nfocused at $(0,0,0)$ propagates in the $+z$ direction. In the particle coordinate system, the spheres of a given \ncluster are located at fixed reference positions. The origin of the particle coordinate system is located at the cluster's\ncenter of mass. The orientation of the particle axes in the laboratory coordinate system describes the \ncluster's orientation.}\n\\end{figure}\n\nWe seek to simulate the trajectory of an arbitrary Brownian particle at temperature $T$ in a fluid of viscosity $\\eta$\nthat is subject to external forces and torques.\nWe assume that our particles are rigid and move in what we call the ``laboratory frame.''\nConsequently, we can describe the position and orientation of our particles by giving the laboratory coordinates of their\ncenter of mass and a rotation matrix.\nSpecifically, for clusters of spheres, we define a reference configuration in which the position of each sphere\nis defined relative to the particle coordinate axes $\\vec{u}_1$, $\\vec{u}_2$, and $\\vec{u}_3$, which we take to be unit vectors.\nWe choose the origin of the particle coordinate axes to lie at the cluster center of mass (CM).\nThe components of the particle coordinate axes in the laboratory frame are given by the columns of a $3\\times3$ rotation matrix.\nFigure \\ref{fig:cartoon} illustrates the laboratory and particle coordinate systems.\n\nIn the overdamped limit in which the particle's inertia is negligible, we model the trajectory of a Brownian particle\nusing a finite-difference approach by computing its generalized displacements during time steps of duration $\\Delta t$\n\\cite{fernandes_brownian_2002, volpe_simulation_2013}.\nFollowing Fernandes and Garc\\'{i}a de la Torre \\cite{fernandes_brownian_2002}, \nwe consider the 6-component generalized displacement vector $\\Delta \\vec{q}$ whose transpose is given by\n\\begin{equation}\n\\Delta \\vec{q}^\\mathrm{tr} = \\left( \\Delta x, \\Delta y, \\Delta z, \\Delta \\phi_1, \\Delta \\phi_2, \\Delta \\phi_3 \\right).\n\\end{equation}\nHere, $\\Delta x$, $\\Delta y$, and $\\Delta z$ are displacements of the center of mass\nand $\\Delta \\phi_1$, $\\Delta \\phi_2$, and $\\Delta \\phi_3$ are infinitesimal rotations about the particle's axes $\\vec{u}_1$, $\\vec{u}_2$,\nand $\\vec{u}_3$.\nThe generalized displacement $\\Delta \\vec{q}$ is computed in the particle coordinate system and subsequently needs\nto be transformed to laboratory coordinates.\nTo find $\\Delta \\vec{q}$ for a given time step, we separately consider two\ncontributions:\n\\begin{equation} \\label{eq:Delta_q_decomp}\n\\Delta \\vec{q} = \\Delta \\vec{q}^{B} + \\Delta \\vec{q}^\\mathrm{ext}. \n\\end{equation}\nThe first term is due to the particle's random Brownian motion, while the second term is due to the external interactions.\nThe Brownian displacement $\\Delta \\vec{q}^B$ with components $\\Delta q^B_i$ is simulated by generating \nnormally distributed random numbers with covariance \n\\begin{equation} \\label{eq:Delta_q_b}\n\\langle \\Delta q^B_i \\Delta q^B_j \\rangle = 2D_{ij} \\Delta t,\n\\end{equation}\nwhere the $D_{ij}$ are the elements of the particle's $6\\times6$ diffusion tensor $\\mathbf{D}$ \\cite{fernandes_brownian_2002}.\nThe displacement $\\Delta \\vec{q}^\\mathrm{ext}$ due to an external force $\\vec{F}_\\mathrm{ext}$ and torque $\\vec{n}_\\mathrm{ext}$\nis given by\n\\begin{equation} \\label{eq:Delta_q_ext}\n\\Delta \\vec{q}^\\mathrm{ext} = \\left( \\frac{\\Delta t}{k_BT} \\right) \\mathbf{D}\\cdot\\mathcal{F}, \n\\end{equation}\nwhere $\\vec{F}_\\mathrm{ext}$ and $\\vec{n}_\\mathrm{ext}$ have been combined into the 6-component vector $\\mathcal{F}$\n\\cite{fernandes_brownian_2002}.\nOur main computational challenges \nare computing the external forces and torques $\\mathcal{F}$ and finding the diffusion tensor $\\mathbf{D}$.\n\n\\subsection{Calculating optical forces and torques with $T$-matrices}\nWhile the approach in Eqs.~(\\ref{eq:Delta_q_b}) and (\\ref{eq:Delta_q_ext}) is relevant to any external interaction, we now focus\non the optical forces and torques exerted by optical tweezers.\nWe consider a laser beam of vacuum wavelength $\\lambda_0$ in a medium of refractive index $n_\\mathrm{med}$ that is focused \nby an objective lens with a given numerical aperture (NA).\n\nCalculating the optical forces and torques is fundamentally a light scattering problem.\nAs we briefly summarize here, we expand the position-dependent\nelectric field of the incident beam $\\vec{E}_\\mathrm{inc}(k\\vec{r})$ in vector spherical wave functions (VSWFs) \n\\cite{nieminen_optical_2014}:\n\\begin{equation}\n\\vec{E}_\\mathrm{inc}(k\\vec{r}) =  \\sum_{n=1}^\\infty \\sum_{m=-n}^{n} a_{nm}\\mathrm{Rg}\\vec{M}_{nm}(k\\vec{r}) + b_{nm} \\mathrm{Rg}\\vec{N}_{nm}(k\\vec{r}).\n\\end{equation}\nHere, $\\mathrm{Rg}\\vec{M}_{nm}(k\\vec{r})$ and $\\mathrm{Rg}\\vec{N}_{nm}(k\\vec{r})$ are solutions of the vector\nHelmholtz equation that are finite at the origin and $k$ is the beam's wavenumber.\n(See Nieminen \\emph{et al.} and references therein for details \\cite{nieminen_optical_2014}.)\nWe use \\emph{ott} to truncate the expansion and calculate the expansion coefficients $a_{nm}$ and $b_{nm}$ \nfor a focused Gaussian beam using\nan overdetermined least squares fit \\cite{nieminen_multipole_2003, nieminen_optical_2007}.\nThe fitting procedure is necessary because a Gaussian beam is not a solution to the vector Helmholtz equation.\n\nWe similarly expand the scattered field $\\vec{E}_\\mathrm{sca}(k\\vec{r})$ in VSWFs \\cite{nieminen_optical_2014}:\n\\begin{equation}\n\\vec{E}_\\mathrm{sca}(k\\vec{r}) =  \\sum_{n=1}^\\infty \\sum_{m=-n}^{n} p_{nm}\\vec{M}^{(1)}_{nm}(k\\vec{r}) + q_{nm} \\vec{N}^{(1)}_{nm}(k\\vec{r}).\n\\end{equation}\nThe $\\vec{M}^{(1)}_{nm}$ and $\\vec{N}^{(1)}_{nm}$ VSWFs asymptotically behave as outgoing spherical waves.\nThe details of the scattering process are described by the $T$-matrix $\\mathbf{T}$, which\nrelates the scattered coefficients to the incident coefficients:\n\\begin{equation} \\label{eq:t_matrix_mult}\n\\begin{pmatrix}\np_{nm} \\\\ q_{nm}\n\\end{pmatrix}\n=\n\\mathbf{T}\n\\begin{pmatrix}\na_{nm} \\\\ b_{nm}\n\\end{pmatrix},\n\\end{equation}\nwhere we combine the incident and scattered field coefficients into single column vectors.\n\\emph{ott} can natively calculate $T$-matrices for homogenous spheres, whose elements are the Lorenz-Mie coefficients.\nHowever, \\emph{ott} cannot calculate $T$-matrices for sphere clusters.\nInstead, we use the multisphere superposition code \\emph{mstm} developed by Mackowski and Mishchenko \n\\cite{mackowski_multiple_2011}.\nOur code calls \\emph{mstm}, reads the $T$-matrix it generates, and then uses \\emph{ott} to perform the matrix multiplication in \nEq.~(\\ref{eq:t_matrix_mult}) to find the scattered field expansion coefficients $p_{mn}$ and $q_{mn}$.\nSubsequently, \\emph{ott} determines the optical forces and torques from the \nfield expansion coefficients \n$a_{nm}$, $b_{nm}$, $p_{nm}$, and $q_{nm}$ \\cite{nieminen_optical_2007}. \nIn this approach, the computationally challenging light scattering problem only needs to be solved once in a general way when\ncomputing the $T$-matrix; subsequent force and torque calculations are fast.\n\n\n\\subsection{Finding diffusion tensors}\nBrownian dynamics simulations require knowing the diffusion tensor $\\mathbf{D}$ of \na particle.\nFinding $\\mathbf{D}$ requires solving the Stokes equations for creeping flow around the particle.\nFor spheres of radius $a$, $\\mathbf{D}$ is diagonal with two unique, nonzero elements:\nthe translational diffusion constant  $D_t = k_BT \/ (6\\pi\\eta a)$ and \nthe rotational diffusion constant $D_r = k_BT \/ (8\\pi\\eta a^3)$.\nFor clusters of two identical spheres, or dimers, $\\mathbf{D}$ is also diagonal and has four unique elements.\n$D_{t,\\parallel}$ describes translational diffusion along the dimer's long axis while $D_{t,\\perp}$ describes translational\ndiffusion perpendicular to the long axis. Similarly, $D_{r,\\parallel}$ describes rotational diffusion about the long axis \nand $D_{r,\\perp}$ describes rotational diffusion about the other two axes.\nWe use the analytic solution of Nir \\& Acrivos to find $\\mathbf{D}$ for dimers \\cite{nir_creeping_1973}. \nHowever, for more complex sphere clusters, numerical methods are needed.\nWe use the Fortran program BEST, which implements a boundary element solution to the Stokes equations \nover an arbitrary triangulated surface \\cite{aragon_precise_2004}.\nAs recommended, we perform repeated BEST calculations using increasing numbers of triangles and \nextrapolate to the limit of an infinite number of triangles.\nBy comparing the results of BEST for single spheres and dimers to the analytic results, we estimate that using\nBEST results in uncertainties in the diffusion tensor elements of no more than 1\\%.\n\n\n\\subsection{Implementation}\nWe implement our Brownian dynamics algorithms in the package \\emph{brownian\\_ot}, which is freely available on Github\n\\cite{jerome_jeromefungbrownian_ot_nodate}.\nOur code integrates both a Matlab package (\\emph{ott}) and a Fortran 90 package (\\emph{mstm}) in a single interface.\nTo make this interfacing straightforward, we implement \\emph{brownian\\_ot} in Python.\n\nThe initial steps in simulating a given particle in a beam of a specified wavelength are determining the particle's diffusion\ntensor, determining the particle's $T$-matrix for scattering of light of that wavelength, and determining the beam's\nVSWF expansion coefficients. These calculations only need to be done once.\nThereafter, we simulate the trajectory of a particle beginning from an arbitrary initial position and orientation as follows:\n\\begin{enumerate}\n \\item Use \\emph{ott} to compute the optical force and torque $\\mathcal{F}$ on the particle in the laboratory coordinate system.\n \\item Correct the optical torque to be relative to the particle's center of diffusion \\cite{harvey_coordinate_1980}\n and transform all elements of $\\mathcal{F}$ to the particle coordinate system.\n \\item Use Eq.~(\\ref{eq:Delta_q_ext}) to calculate $\\Delta \\vec{q}^\\mathrm{ext}$ in the particle coordinate system.\n \\item Use Eq.~(\\ref{eq:Delta_q_b}) to calculate $\\Delta \\vec{q}^B$ in the particle coordinate system.\n \\item Compute $\\Delta \\vec{q} = \\Delta \\vec{q}^\\mathrm{ext} + \\Delta \\vec{q}^B$ [Eq.~(\\ref{eq:Delta_q_decomp})].\n \\item Transform the CM displacement (the spatial components of $\\Delta \\vec{q}$) to the laboratory coordinate system and update\n the CM position.\n \\item Calculate an unbiased rotation corresponding to the angular components of $\\Delta \\vec{q}$\n\\cite{beard_unbiased_2003}\n and update the particle's orientation by rotation composition.\n \\item Repeat.\n\\end{enumerate}\nInternally, our orientation calculations use unit quaternions due to their compact storage and numerical stability.\n\nWhile our code runs on personal computers under Macintosh, Windows, or Linux operating systems, \nwe typically run longer simulations using the Comet cluster\nat the San Diego Supercomputing Center, which is part of the Extreme Science and Engineering Discovery Environment (XSEDE)\n\\cite{towns_xsede_2014}.\nThe primary reason to use Comet is that we can run tens of independent simulations simultaneously, \neither to build up statistics or to explore parameter space.\nRun times depend primarily on the number of time steps to be simulated but also on the particle size\nsince larger particles generally require a larger maximum order $n_\\mathrm{max}$ in the truncation of the VSWF expansions for \nthe incident and scattered fields.\nOn Comet, simulating $3\\times10^6$ steps for a 1-\\si{\\micro\\meter}-diameter sphere takes approximately \\SI{8e4}{\\second},\nand simulating a trajectory of the same length for a cluster of seven $0.8$-\\si{\\micro\\meter}-diameter spheres takes approximately \n\\SI{1e5}{\\second}.\n\n\\subsection{Simulation parameters}\nThroughout what follows, we consider a 1064-nm-wavelength beam focused by a 1.2 NA objective lens.\nExcept when noted, the beam has a power of \\SI{5}{\\milli\\watt} and is left-circularly-polarized (LCP) with Jones vector\n$(1, i)$.\nWe also assume that the beam propagates in an isotropic medium with $n_\\mathrm{med}=1.33$, which corresponds to water.\nWe assume that the medium has viscosity $\\eta = \\SI{1e-3}{\\pascal\\second}$\nand that the particles are at $T=\\SI{295}{\\kelvin}$.\nWe consider non-absorbing particles with two different refractive indices: silica (Si; $n=1.45$) and polystyrene (PS; \n$n=1.58$).\nHowever, our code can be easily applied to particles with other refractive indices, including absorbing particles \nsuch as metallic nanospheres.\n\nWe choose a simulation time step $\\Delta t = \\SI{1e-5}{\\second}$ throughout our simulations. \nBecause optical forces and torques are strongly dependent on particle position and orientation, we choose the time step \nto be small enough that none of our particles translates a significant fraction of the wavelength or rotates through a significant angle\nduring a single step.\nAt the same time, $\\Delta t$ is large enough that ballistic motion is negligible \\cite{volpe_simulation_2013, huang_direct_2011}. \nFinally, except when noted, all simulations begin with the particle centers of mass at the origin and the particle reference\naxes aligned with the laboratory frame axes.\n\n\n\\section{Results and discussion}\n\n\\subsection{Spheres}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/sphere_exemplar\/example_sphere}\n\\caption{\\label{fig:sphere} (a) Optical forces on a 1-\\si{\\micro\\meter}-diameter PS sphere in water in a 5 mW LCP beam as a \nfunction of particle position. $F_z$ is calculated as a function of $z$ at $x=y=0$. $F_x$ is calculated as a function of $x$\nat $y=0$ and $z=z_\\mathrm{eq}$ (at which $F_z = 0$). Dashed lines indicate linear approximation to optical forces near equilibrium\ngiving stiffnesses $\\kappa_x$ and $\\kappa_z$.\n(b) First \\SI{2}{\\second} of a 30-s-long simulated trajectory. \n(c) Laboratory-frame MSD and $\\vec{u}_1$ MSAD averaged from 5 independent trajectories with\nidentical initial conditions. \nShaded regions indicate standard deviations. Dotted lines are the predictions of Eqs.~(\\ref{eq:MSD}) and (\\ref{eq:MSAD}).}\n\\end{figure}\n\nTo validate our simulations, we consider the behavior of a 1-\\si{\\micro\\meter}-diameter PS sphere.\nThe optical forces as a function of position\nare approximately linear near equilibrium (Fig.~\\ref{fig:sphere}(a)):\n\\begin{gather} \\label{eq:linear_restoring_force}\nF_x = -\\kappa_x x, \\nonumber \\\\\nF_z = -\\kappa_z (z - z_\\mathrm{eq}),\n\\end{gather}\nwhere $\\kappa_x$ and $\\kappa_z$ are the stiffnesses in the $x$ and $z$ directions, respectively.\nBecause of radiation pressure, $F_z = 0$ at $z=z_\\mathrm{eq}$ with $z_\\mathrm{eq}>0$ rather than at $z =0$.\nThe first \\SI{2}{\\second} of a simulated sphere trajectory shows that the sphere exhibits Brownian fluctuations about equilibrium\n(Fig.~\\ref{fig:sphere}(b)).\n\nMean-squared displacements (MSDs) calculated from the simulated trajectories demonstrate that the simulations perform\nas expected. The MSD for the coordinate $x_i$ is defined as $\\mathrm{MSD}=\\langle \\left[ x_i(t+\\tau) - x_i(t) \\right]^2\\rangle$.\nWe plot the $x$ and $z$ MSDs calculated from our simulations in Fig.~\\ref{fig:sphere}(c).\nSince we know the stiffnesses $\\kappa_x$ and $\\kappa_z$ from the force calculations in Fig.~\\ref{fig:sphere}(a) and also know\nthe sphere radius, we can also predict the MSDs \\emph{a priori}.\nSolving the Langevin equation shows that a sphere of radius $a$ moving in one dimension subject to a linear restoring force with \nstiffness $\\kappa$ has the MSD \\cite{jones_optical_2015}\n\\begin{equation} \\label{eq:MSD}\n\\mathrm{MSD} = \\frac{2k_BT}{\\kappa}\\left[  1 - \\exp\\left( -\\frac{\\kappa \\tau}{\\gamma}\\right)  \\right],\n\\end{equation}\nwhere the friction coefficient is given by $\\gamma = 6\\pi\\eta a$.\nThe dotted lines in the upper panel of Fig.~\\ref{fig:sphere}(c) show that the predictions of Eq.~(\\ref{eq:MSD})\nagree well with the values calculated from the trajectories.\n\nIn addition, because there are no significant optical torques, the sphere should undergo free rotational\ndiffusion. We can similarly characterize the rotational dynamics of the sphere by calculating the mean-squared angular\ndisplacement (MSAD) of any of the reference axes in the laboratory frame. The MSAD for axis $\\vec{u}_i$\nis defined as $\\langle (\\Delta\\vec{u}_i)^2\\rangle = \\langle \\left|\\vec{u}_i(t+\\tau) - \\vec{u}_i(t)\\right|^2 \\rangle$. \nThe lower panel of Fig.~\\ref{fig:sphere}(c) shows the MSAD for $\\vec{u}_1$ since, for a sphere, all three axes are equivalent.\nA sphere with rotational diffusion constant $D_r$ has an MSAD given by \n\\cite{doi_theory_1988, fung_holographic_2013}\n\\begin{equation} \\label{eq:MSAD}\n\\langle (\\Delta \\vec{u}_i )^2\\rangle = 2\\left[ 1 - \\exp(-2D_r\\tau  )  \\right].\n\\end{equation}\nThe agreement between the prediction of Eq.~(\\ref{eq:MSAD}) (the dotted line in the lower panel of Fig.~\\ref{fig:sphere}(c))\nand the values determined from the simulated trajectories shows that the sphere indeed undergoes free rotational diffusion as\nexpected.\n\n\n\\subsection{Dimers}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/dimer_forces\/dimer_forces_final}\n\\caption{\\label{fig:dimer_forces} Optical forces and torques on small, intermediate, and large dimers in a 5 mW LCP beam.\nAll force calculations are shown with the dimer long axis aligned with the $z$ axis.\n$F_x$ and $n_y$ are calculated at $z=z_\\mathrm{eq}$.\nIn (c), the equilibrium position with $z_{eq}>0$ is chosen.\nInset in (a) shows definition of $\\theta$. \nDotted lines in force graphs indicate linear approximations near equilibrium. \nThe dotted line in the torque graph for (a) shows a small-angle fit to Eq.~(\\ref{eq:restoring_torque}).}\n\\end{figure}\n\nWe next consider the simplest multi-sphere cluster: a dimer consisting of two identical spheres.\nFigure \\ref{fig:cartoon} shows the reference orientation for a dimer; we choose the long axis to be the\n$\\vec{u}_3$ axis.\n\nIn order to verify that the dimers we consider can indeed be optically trapped, we compute the optical forces and torques on \nseveral characteristic dimers (Fig.~\\ref{fig:dimer_forces}).\nWe perform all force calculations with the dimer long axis aligned with the direction of beam propagation.\nWe observe three regimes.\nFirst, for a dimer whose constituent spheres are smaller than the wavelength of light, which we term ``small,'' both \n$F_x$ and $F_z$ resemble the corresponding calculations for spheres (Fig.~\\ref{fig:dimer_forces}(a)).\nIn particular, there is a linear regime near equilibrium and an equilibrium height $z_\\mathrm{eq}>0$.\nIn addition, the lateral stiffness $\\kappa_x$ is larger than the axial stiffness $\\kappa_z$.\nThese results are similar to those reported by Borghese \\emph{et al}.~for 220-nm-diameter PS spheres \\cite{borghese_optical_2007}.\nSecond, we consider an ``intermediate'' dimer whose spheres are comparable in size to the wavelength of light \nin the surrounding medium.\nFor the dimer of 0.8-\\si{\\micro\\meter}-diameter spheres shown in Fig.~\\ref{fig:dimer_forces}(b),\nwe observe a qualitatively different $F_z$ graph with two maxima.\nAlso, in contrast to the small dimer, the equilibrium trap stiffnesses $\\kappa_x$ and $\\kappa_z$ are comparable.\nThird, we consider a ``large'' dimer consisting of spheres that are larger than the wavelength,\nsuch as the 1.6-\\si{\\micro\\meter}-diameter silica spheres in Fig.~\\ref{fig:dimer_forces}(c).\nHere, there are two heights $z$ where $F_z=0$. In addition,  \nthe trap is stiffer axially than laterally: $\\kappa_z>\\kappa_x$.\n\nThe optical torques experienced by small, intermediate, and large dimers also differ.\nSince the incident beam is axisymmetric, we consider without loss of generality\nrotating the dimer by $\\theta$ about the $y$ axis and compute the $y$ component of the optical torque $n_y$.\nIn all three cases, there is a restoring torque resulting in a stable angular equilibrium at $\\theta=0$.\nNotably, for the small dimer in Fig.~\\ref{fig:dimer_forces}(a), the torque is approximately proportional to $\\sin(2\\theta)$:\n\\begin{equation} \\label{eq:restoring_torque}\nn_y = -\\kappa_r\\sin(2\\theta),\n\\end{equation}\nwhere $\\kappa_r$ is a rotational stiffness. \nHowever, Eq.~(\\ref{eq:restoring_torque}) poorly describes the torques on intermediate and large dimers.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/dimer_trajectory\/dimer_trajectory}\n\\caption{\\label{fig:small_traj} First \\SI{2}{s} of the CM trajectory and rotation matrix elements for a small \ndimer composed of 0.4-\\si{\\micro\\meter}-diameter silica spheres in a 5 mW LCP trap. \nThe complete 30-s-long trajectory is shown in \\textcolor{urlblue}{Visualization 1}.}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{.\/figures\/visualization\/frame0123}\n\\caption{\\label{fig:movie} Rendering of an optically-trapped dimer composed of 0.4-\\si{\\micro\\meter}-diameter silica spheres from \n\\textcolor{urlblue}{Visualization 1}, which shows the trajectory of Fig.~\\ref{fig:small_traj}.\nThe stripes are drawn to enable visualizing rotations about all axes.\nLeft: front view. The laser propagates upwards and is focused at the center of the box, \nwhose sides are \\SI{6}{\\micro\\meter} long.\nRight: perspective view. The axes are \\SI{3.5}{\\micro\\meter} long. \nAll subsequent visualizations have the same scale, and all visualizations are slowed $5\\times$ from real time.}\n\\end{figure}\n\n\nSmall, intermediate, and large dimers exhibit qualitatively different dynamical behavior in optical tweezers.\nSmall dimers exhibit positional fluctuations similar to those for a trapped sphere and \norientational fluctuations that would be expected for a particle experiencing a restoring torque.\nThe beginning of a simulation trajectory for a small dimer is shown in Fig.~\\ref{fig:small_traj}, where\nwe plot the CM coordinates and the laboratory-frame coordinates of all 3 particle-frame axes.\n\\textcolor{urlblue}{Visualization 1} shows a rendering of the complete trajectory, and Fig.~\\ref{fig:movie} is a still frame from the rendering.\nOnce again, the CM position fluctuates about $x=y=0$ and $z=z_\\mathrm{eq}$.\nUnlike spheres, however, the restoring torque keeps the dimer axis $\\vec{u}_3$ pointing near the laboratory $+z$ axis:\n$u_{3x}$ and $u_{3y}$ fluctuate near 0 and $u_{3z}$ fluctuates near 1.\nThe horizontal components of $\\vec{u}_{1}$ and $\\vec{u}_2$ range fully between -1 and +1, indicating that the dimer is free to\nrotate about its long axis. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/dimer_example_msds\/dimer_example_msds}\n\\caption{\\label{fig:dimer_msds}\nCluster-frame MSDs and $\\vec{u}_3$ MSAD for dimer of 0.4-\\si{\\micro\\meter}-diameter silica spheres averaged from 5 \nindependent trajectories. \nPredictions of Eq.~(\\ref{eq:MSD}): dotted lines in (a). Prediction of Eq.~(\\ref{eq:msad_limit}) for MSAD \nlimiting value: dotted line in (b).\nPrediction of Eq.~(\\ref{eq:MSAD}) for short-time behavior of MSAD: dashed line in (b).}\n\\end{figure}\n\nWe analyze the behavior shown in the trajectory by computing MSDs and MSADs (Fig.~\\ref{fig:dimer_msds}).\nBecause the diffusion tensor is anisotropic, we compute MSDs in the particle frame by resolving laboratory-frame displacements\ninto components along the particle axes \\cite{han_brownian_2006, fung_holographic_2013}. \nWe determine trap stiffnesses $\\kappa_x$ and $\\kappa_z$ from the linear behavior of $F_x$ and $F_z$ near equilibrium shown in \nFig.~\\ref{fig:dimer_forces}(a).\nWe also extract translational drag coefficients $\\gamma_\\parallel = k_BT\/D_{t,\\parallel}$ and $\\gamma_\\perp = k_BT\/D_{t,\\perp}$ \nfrom the diffusion tensor.\nWe can therefore compare the predictions of Eq.~(\\ref{eq:MSD}) to the MSDs calculated from the trajectories.\n\nWe also calculate the MSAD for the long axis $\\vec{u}_3$ since \nthe orientation of this axis can be observed in experiments using techniques such as holographic microscopy \n\\cite{fung_measuring_2011}. \nAt short lag times $\\tau$, the optical torques do not affect the dimer: the MSAD is diffusive and is well-described by \nEq.~(\\ref{eq:MSAD}) with $D_r = D_{r,\\perp}$ (Fig.~\\ref{fig:dimer_msds}(b)).\nAt longer times, however, the effects of the optical tweezers become significant.\nEven approximating the torque to be proportional to $\\sin(2\\theta)$ [Eq.~(\\ref{eq:restoring_torque})], we cannot analytically \nsolve the Einstein-Smoluchowski equation in order to determine the $\\vec{u}_3$ MSAD for arbitrary lag times. \nBut assuming that the particle is in thermal equilibrium, we can compute the limiting value of the MSAD as $\\tau\\rightarrow\\infty$.\n(See Supplemental Document for details.)\nThe limiting value is given by\n\\begin{equation} \\label{eq:msad_limit}\n\\lim_{\\tau \\rightarrow \\infty} \\langle( \\Delta \\vec{u}_3)^2\\rangle = 2\\left[ 1 - \\frac{1}{4\\beta\\kappa_r} \n\\left( \\frac{\\exp(\\beta\\kappa_r) - 1}{\\exp(\\beta \\kappa_r)F(\\sqrt{\\beta\\kappa_r})}\\right)^2    \\right],\n\\end{equation}\nwhere $\\beta = 1\/(k_BT)$ and $F(x)$ is Dawson's integral.\nSince we can determine $\\kappa_r$ by fitting the torque calculations shown in Fig.~\\ref{fig:dimer_forces}(a),\nwe can plot the predictions of Eq.~(\\ref{eq:msad_limit}) in Fig.~\\ref{fig:dimer_msds}(b).\n\nThe results in Fig.~\\ref{fig:dimer_msds} all differ from the analytical predictions by several percent.\nThe discrepancies suggest that the translational and orientational degrees of freedom cannot be considered independently\nof each other.\nIn particular, the optical forces on the dimer depend not only on position but also on orientation.\nSimilarly, the optical torques depend not only on orientation but also on position.\nMoreover, the optical torque is only approximately described by Eq.~(\\ref{eq:restoring_torque}).\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/medium_dimer_traj\/medium_dimer_trajectory}\n\\caption{\\label{fig:medium_traj} First \\SI{2}{s} of the CM trajectory and rotation matrix elements for an intermediate dimer \ncomposed of 0.8-\\si{\\micro\\meter}-diameter PS spheres in a 5 mW LCP trap. The complete 30-s-long trajectory is shown in \n\\textcolor{urlblue}{Visualization 2}. The dimer exhibits deterministic rotation about its long axis.}\n\\end{figure}\n\nThe trajectory of the intermediate dimer in Fig.~\\ref{fig:medium_traj} shows a marked difference from that of the small dimer:\nthe dimer rotates deterministically about its long axis.\nThis can be clearly seen in the behavior of $\\vec{u}_1$ and $\\vec{u}_2$ and in \\textcolor{urlblue}{Visualization 2}.\nThe rotation is polarization-dependent: the dimer does not rotate when the incident beam is linearly polarized.\nMoreover, the direction of rotation reverses when the beam is right-circularly polarized (\\textcolor{urlblue}{Visualization 3}).\nWe attribute this behavior to photokinetic spin-curl effects \\cite{ruffner_optical_2012, yevick_photokinetic_2017}.\nWhile their analysis is only strictly valid in the Rayleigh limit, Ruffner and Grier predicted theoretically \nand demonstrated experimentally that the rotation frequency for particles experiencing spin-curl torques is proportional to \n$S_3\/S_0$, where $(S_0, S_1, S_2, S_3)$ is the Stokes vector describing the incident polarization \\cite{ruffner_optical_2012}. \nWe thus perform additional simulations with elliptically-polarized beams and \ndetermine the periods of rotation, averaged over 5 trajectories, by computing the $\\vec{u}_1$ MSAD and\nlocating the first minimum (Fig.~\\ref{fig:dimer_rotation}(a)).\nThe rotation frequencies $\\Omega$ we observe are indeed proportional to $S_3\/S_0$ (Fig.~\\ref{fig:dimer_rotation}(b)).\nHere, a positive $\\Omega$ corresponds to a counterclockwise rotation and a negative $\\Omega$ corresponds to a counterclockwise\nrotation.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/dimer_rotation\/dimer_rotation_final}\n\\caption{ \\label{fig:dimer_rotation}\n(a) $\\vec{u}_1$ MSADs for dimer of 0.8-\\si{\\micro\\meter}-diameter PS spheres in 5 mW beams with left circular polarization, elliptical polarization with Jones vector $(1\/\\sqrt{2}, \\exp(i\\pi\/6)\/\\sqrt{2})$, and linear polarization with Jones vector \n$(1\/\\sqrt{2}, 1\/\\sqrt{2})$. MSADs are averaged from 5 trajectories.\nThe period of the dimer's deterministic rotation is determined from the first minimum of the MSADs (arrows).\nNo rotation is observed for the linearly polarized beam.\n(b) Rotation frequency $\\Omega$ as a function of the ratio of Stokes vector elements $S_3\/S_0$. \nThe blue and orange points correspond to the MSADs shown in (a). \nThe sign of $\\Omega$ corresponds to the direction of rotation. Dashed line: linear fit. }\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/large_dimer_traj\/large_dimer_trajectory}\n\\caption{\\label{fig:large_traj} First \\SI{2}{s} of the CM trajectory and rotation matrix elements for a large dimer \ncomposed of 1.6-\\si{\\micro\\meter}-diameter silica spheres in a 5 mW LCP trap. The complete 30-s-long trajectory is shown in \n\\textcolor{urlblue}{Visualization 4}. The dimer exhibits both a slower deterministic rotation about its long axis ($\\vec{u}_3$) \nas well as a faster wobble of its long axis.}\n\\end{figure}\n\nFor the large dimer, we observe a similar deterministic rotation about the dimer axis\n(Fig.~\\ref{fig:large_traj} and \\textcolor{urlblue}{Visualization 4}).\nHowever, there is an additional effect: the particle undergoes a wobble that is faster than the axial rotation.\nThe effects of the wobble can be seen in the $z$ component of $\\vec{u}_1$ and $\\vec{u}_2$, \nin the lateral components of $\\vec{u}_3$, and in the lateral position of the center of mass.\nBoth the axial rotation and the wobble reverse direction when the beam is \nright-circularly polarized (\\textcolor{urlblue}{Visualization 5}).\nWhile experimentally detecting the axial rotation described in Fig.~\\ref{fig:dimer_rotation} for an optically homogeneous\nmedium dimer would be challenging, it would be straightforward to detect the wobble we predict here.\nWe plan to explore these effects both computationally and experimentally in the future.\n\n\n\\subsection{Chiral 7-sphere cluster}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/chiral7_reference\/reference_final}\n\\caption{\\label{fig:chiral7_ref} Reference orientation for chiral 7-sphere cluster. \n(a) and (b): Space-filling models viewed from two perspectives. \n(c): Ball-and-stick model viewed from same perspective as (b). The red sphere causes the cluster to be chiral.}\n\\end{figure}\n\nFinally, our Brownian dynamics simulations reveal the existence of multiple trapping equilibria for a highly asymmetric\nsphere cluster.\nThe 7-sphere cluster shown in Fig.~\\ref{fig:chiral7_ref} has no axes of rotational symmetry\nor planes of mirror symmetry, but it is the smallest rigid sphere packing that exhibits chirality \\cite{arkus_deriving_2011}.\nWe simulate clusters of 0.8-\\si{\\micro\\meter}-diameter silica spheres held in a beam that is \nlinearly polarized in the $x$ direction.\nAs in the previous simulations, the cluster begins with its center of mass at the origin and in its reference orientation.\n\nRather than fluctuating or exhibiting deterministic motion about a \\emph{single} equilibrium, the simulated trajectory \nfluctuates about two different orientations (Fig.~\\ref{fig:chiral7_ab} and \\textcolor{urlblue}{Visualization 6}). \nThis is most clearly visible in the behavior of the $x$ and $y$ components of the cluster-frame axes, \nalthough the two orientations also have different values of the CM position.\n\nIn order to determine whether the observed plateaus in the trajectory (e.g., between $t=14$ and \\SI{19}{\\second}) \nin fact correspond to equilibria, we estimate candidate equilibrium positions and orientations from the plateaus\nand perform additional athermal simulations starting from the candidate equilibria.\nTo estimate the equilibrium position, we average each CM coordinate in a plateau.\nTo estimate the equilibrium orientation, we use the quaternion averaging algorithm\nof Markley and co-workers \\cite{markley_averaging_2007}.\nWe then perform additional simulations with the cluster starting from the candidate equilibria but without thermal fluctuations,\nand we confirm that the resulting trajectories approach asymptotic values.\nThe refined equilibria obtained from the limiting values\nare shown in the horizontal lines of Fig.~\\ref{fig:chiral7_ab} and in Fig.~\\ref{fig:chiral7_equilibria}(a)-(b).\nThe two equilibria here correspond to a rotation of approximately $180^\\circ$ about the $z$ axis.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/chiral7_ab\/chiral7_ab.pdf}\n\\caption{\\label{fig:chiral7_ab} CM and rotation matrix for a chiral 7-sphere cluster of\n0.8-\\si{\\micro\\meter}-diameter silica spheres trapped in a \\SI{5}{\\milli\\watt} horizontally-polarized beam.\n(See \\textcolor{urlblue}{Visualization 6}.)\nDashed lines indicate equilibria shown in Fig.~\\ref{fig:chiral7_equilibria}(a)-(b).}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/chiral7_equilibria\/equilibria.pdf}\n\\caption{\\label{fig:chiral7_equilibria}\nEquilibrium orientations indicated by dashed lines in Figs.~\\ref{fig:chiral7_ab} and \\ref{fig:chiral7_cd}. In orientations (a) and (b), \nwhich are rotated by approximately $180^\\circ$ about the $z$ axis, the trap focus is on the sphere \nhighlighted in blue. In orientations (c) and (d), also rotated by $180^\\circ$ about the $z$ axis, the trap focus is on the sphere\nhighlighted in red.}\n\\end{figure}\n\nRemarkably, in a different simulation with the \\emph{same} initial conditions, we observe a different set of candidate equilibria\n(Fig.~\\ref{fig:chiral7_cd} and \\textcolor{urlblue}{Visualization 7}).\nWe again perform additional simulations without thermal fluctuations and show the refined equilibria in Figs.~\\ref{fig:chiral7_cd}\nand \\ref{fig:chiral7_equilibria}(c)-(d).\nOnce again, the two equilibria in this trajectory correspond to a $180^\\circ$ rotation about the $z$ axis. \nHowever, the equilibria of Fig.~\\ref{fig:chiral7_equilibria}(c)-(d) involve a different sphere lying at the beam focus than in \nFig.~\\ref{fig:chiral7_equilibria}(a)-(b).\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/chiral7_ab\/chiral7_cd.pdf}\n\\caption{\\label{fig:chiral7_cd} As in Fig.~\\ref{fig:chiral7_ab}, but for a different simulation. \n (See \\textcolor{urlblue}{Visualization 7}.) \n Dashed lines indicate equilibria shown in Fig.~\\ref{fig:chiral7_equilibria}(c)-(d).} \n\\end{figure}\n\nThese results demonstrate the importance of considering thermal fluctuations in simulations of particles in optical tweezers.\nHere, thermal fluctuations allow the cluster to explore its trapping landscape and reveal multiple equilibria that we did not\nexpect to find \\emph{a priori}. \nThere may be additional trapping equilibria besides the ones we have reported.\nWe intend to explore the trapping of this cluster in more detail as well as the transitions between\nequilibria.\n\n\n\\section{Conclusions}\nCombining $T$-matrix-based calculations of optical interactions with a model of anisotropic Brownian motion\nhas resulted in dynamical simulations that realistically capture the behavior of complex, wavelength-sized sphere \nclusters in optical tweezers.\nWe have observed rich photokinetic effects even for the simplest case, clusters of two identical isotropic spheres, when\nthe individual spheres are comparable to or larger than the wavelength.\nOur simulations have also demonstrated that multiple trapping equilibria exist for a wavelength-sized 7-sphere cluster \nwith no symmetry.\nFinding this cluster's trapping equilibria solely from calculations of optical interactions at fixed positions and orientations\nwould have been computationally challenging.\nMoreover, our incorporation of thermal fluctuations allowed us to more easily explore the trapping landscape.\nThe fully general, on-demand computations of optical interactions needed for these simulations were only possible because of the\nspeed of the $T$-matrix-based calculations.\n\nOur work suggests two intriguing directions for future inquiry.\nFirst, systematically investigating effects such as the photokinetic axial rotation and wobble\nwe have observed for intermediate and large dimers would be worthwhile.\nOur simulations allow continuous variation of parameters such as the particle size and refractive\nindex and could usefully complement experimental investigations in which the particle properties \ncannot be as easily tuned.\nMoreover, since we can in principle determine the electromagnetic fields everywhere \nfrom the VSWF expansion of the incident beam and the $T$-matrix, more detailed investigation into \nthe physical mechanism of these effects might be possible.\nThis may be worthwhile since theoretical analyses based on the \nRayleigh approximation cannot be strictly valid for particles of these sizes \\cite{yevick_photokinetic_2017}.\n\nSecond, it is also possible to extend this work to other systems.\nUsing \\emph{ott}, we could investigate particle dynamics in beams that are not Gaussian, \nincluding Bessel beams \\cite{arlt_optical_2001}\nand Laguerre-Gaussian beams \\cite{simpson_mechanical_1997, \nsimpson_optical_2009}, both of which can carry orbital angular momentum.\nIn addition, the generality of the $T$-matrix approach and the modularity of our code\ncould allow us to consider other particles.\n\\emph{ott} natively supports $T$-matrix computations for small spheroids, and\nother packages reliably calculate the $T$-matrix for spheroids \\cite{somerville_smarties_2016}\nor other axisymmetric particles \\cite{mishchenko_capabilities_1998}.\nThe discrete dipole approximation, which is implemented in \\emph{ott},\n could also enable $T$-matrices to be computed for arbitrarily-shaped\nparticles \\cite{mackowski_discrete_2002, loke_discrete-dipole_2011}.\nThe broad generality as well as detail of the Brownian dynamics simulations we have introduced here \nmay make them useful both for guiding or interpreting experiments on optically-trapped \nparticles as well as for gaining further insight into the underlying physics of optical trapping and manipulation.\n\n\\section*{Funding}\nThis work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National \nScience Foundation grant number ACI-1548562.\nThis work used the XSEDE resource Comet at the San Diego Supercomputing Center through allocation PHY190049.\nO.~L. was supported by the Summer Scholars program of the Ithaca College School of Humanities \\& Sciences.\n\n\\section*{Acknowledgments}\nWe thank Isaac Lenton, Miranda Holmes-Cerfon, and Sergio Aragon for helpful discussions.\n\n\\section*{Disclosures}\n\nThe authors declare no conflicts of interest.\n\n\\vspace{1em}\n\\noindent\nSee Supplement 1 for supporting content.\n\n\n\n\n\n\\section{Analytical model for MSAD limit for small, axisymmetric particles}\n\nHere, we calculate the limiting value as the delay time $\\tau\\rightarrow\\infty$ for the mean-squared angular \ndisplacement (MSAD) of the symmetry axis of an axisymmetric particle (such as a dimer) strongly trapped \nin optical tweezers.\nLet $\\vec{u}_3$ be the symmetry axis of \nan axisymmetric particle at temperature $T$ that is fixed at the origin but undergoes rotational diffusion subject to an external\nrestoring torque given in spherical polar coordinates by\n\\begin{equation} \\label{eq:torque}\n\\vec{n} = -\\kappa_r\\sin(2\\theta) \\hat{\\theta},\n\\end{equation}\nwhere $\\kappa_r$ is a rotational stiffness.\nThe MSAD for $\\vec{u}_3$ is defined as \n\\begin{equation} \\label{eq:msad_defn}\n\\langle (\\Delta \\vec{u}_3)^2\\rangle =  \\langle \\left| \\vec{u}_3(t+\\tau) - \\vec{u}_3(t) \\right|^2 \\rangle.\n\\end{equation}\nAs $\\tau\\rightarrow\\infty$, $\\vec{u}_3(t+\\tau)$ and $\\vec{u}_3(t)$ become statistically independent, and\nthe limiting value for $\\langle (\\Delta \\vec{u}_3)^2 \\rangle$\nmay be found using equilibrium statistical mechanics.\n \nWe focus on calculating the axis autocorrelation $\\langle\\vec{u}_3(t+\\tau) \\cdot \\vec{u}_3(t) \\rangle$ since\nexpansion of $\\langle (\\Delta \\vec{u}_3)^2 \\rangle$ (Eq.~\\ref{eq:msad_defn}) leads to \n\\begin{equation} \\label{eq:msad_expansion}\n\\langle (\\Delta \\vec{u}_3)^2\\rangle = 2\\left[1 - \\langle \\vec{u}_3(t+\\tau) \\cdot \\vec{u}_3(t) \\rangle \\right].\n\\end{equation}\nSince $\\vec{u}_3$ is a unit vector,\nwe hereafter refer to the axis autocorrelation as $\\langle \\cos\\psi \\rangle$, \nwhere $\\psi$ is the angle subtended between $\\vec{u}_3$ at $t$ and $\\vec{u}_3$ at $t+\\tau$.\nLet the coordinates of $\\vec{u}_3$ on the unit sphere be $(\\theta, \\phi)$ at time $t$ and\n$(\\theta', \\phi')$ at time $t+\\tau$.\nExpressing the Cartesian components of $\\vec{u}_3$ in spherical coordinates and computing the dot product leads to\n\\begin{equation} \\label{eq:cos_psi}\n\\cos\\psi = \\sin\\theta\\sin\\theta'\\left( \\cos\\phi\\cos\\phi' + \\sin\\phi\\sin\\phi'  \\right) + \\cos\\theta\\cos\\theta'.\n\\end{equation}\n\nIn thermal equilibrium, the system is governed by Boltzmann statistics.\nTaking the system to have potential energy $U=0$ at $\\theta=0$,\nintegration of the external torque (Eq.~\\ref{eq:torque}) leads to  \n\\begin{equation} \\label{eq:potential_energy}\nU = \\kappa_r \\sin^2\\theta.\n\\end{equation}\nSo, at time $t$, the probability of $\\vec{u}_3$ lying \nwithin solid angle $\\mathrm{d}\\Omega$ near $(\\theta, \\phi)$ is given by\n\\begin{equation} \\label{eq:prob}\nP(\\theta, \\phi) \\mathrm{d}\\Omega = \\frac{\\exp\\left( -\\beta \\kappa_r \\sin^2\\theta \\right)\\,\\mathrm{d}\\Omega}{\\iint \\exp\\left( -\\beta \\kappa_r \\sin^2\\theta \\right) \\, \n\\mathrm{d}\\Omega},\n\\end{equation}\nwhere $\\beta \\equiv 1\/k_BT$, $\\mathrm{d}\\Omega = \\sin\\theta\\,\\mathrm{d}\\phi\\,\\mathrm{d}\\theta$,\nand the integrals in the denominator run over the entire unit sphere.\nIf $\\tau$ is sufficiently large that the orientation of $\\vec{u}_3$ at time $t+\\tau$ is independent of its orientation at time $t$, \nthen the probability of $\\vec{u}_3$ lying near $(\\theta',\\phi')$ at $t+\\tau$ is similarly given by  \n\\begin{equation} \\label{eq:prob'}\nP(\\theta', \\phi') \\mathrm{d}\\Omega' = \\frac{\\exp\\left( -\\beta \\kappa_r \\sin^2\\theta '\\right)\\,\\mathrm{d}\\Omega'}{\\iint \\exp\\left( -\\beta \\kappa_r \\sin^2\\theta' \\right) \\, \n\\mathrm{d}\\Omega'}.\n\\end{equation}\n(To calculate $\\langle \\cos \\psi \\rangle$ at an arbitrary value of $\\tau$, we would instead need the transition probability of finding \n$\\vec{u}_3$ at $(\\theta', \\phi')$ after a time interval $\\tau$ given $\\vec{u}_3$ initially at $(\\theta,\\phi)$.)\nIt follows that\n\\begin{equation} \\label{eq:large_integral}\n\\lim_{\\tau\\rightarrow\\infty} \\langle \\cos\\psi\\rangle = \\iint \\cos \\psi P(\\theta, \\phi) P(\\theta', \\phi')\\,\\mathrm{d}\\Omega\\,\\mathrm{d}\\Omega'.\n\\end{equation}\n\nEquation (\\ref{eq:large_integral}) simplifies considerably because the probabilities (Eqs.~\\ref{eq:prob}, \\ref{eq:prob'}) \nhave no explicit $\\phi$ or $\\phi'$ dependence. \nSince the first two terms of Eq.~(\\ref{eq:cos_psi}) are proportional to $\\cos\\phi$ or \n$\\sin\\phi$, they vanish upon integration over $\\phi$.\nThus, only the last term of Eq.~(\\ref{eq:cos_psi}) contributes to $\\langle \\cos\\psi\\rangle$. \nIntegrating over $\\phi$ and $\\phi'$,\n\\begin{equation} \\label{eq:big_integrals}\n\\lim_{\\tau\\rightarrow\\infty}  \\langle\\cos\\psi\\rangle = \n\\frac{\\left(\\int \\cos\\theta \\exp\\left( -\\beta \\kappa \\sin^2\\theta \\right) \\sin\\theta\\,\\mathrm{d}\\theta \\right)\n\\left(\\int \\cos\\theta' \\exp\\left( -\\beta \\kappa \\sin^2\\theta' \\right) \\sin\\theta'\\,\\mathrm{d}\\theta' \\right) }{ \\left( \n\\int \\exp\\left( -\\beta \\kappa \\sin^2\\theta \\right) \\sin\\theta\\,\\mathrm{d}\\theta \\right) \\left(\\int  \\exp\\left( -\\beta \\kappa \\sin^2\\theta' \\right) \\sin\\theta'\\,\\mathrm{d}\\theta' \\right)} .\n\\end{equation}\n\nThe limits of integration now require some attention. The integrals in the numerator of \nEq.~(\\ref{eq:big_integrals}) are proportional to $\\langle \\cos \\theta \\rangle$.\nBut the potential energy in Eq.~(\\ref{eq:potential_energy}) implies that \n$\\vec{u}_3$ is equally likely to be at $\\theta$ or at  $\\pi-\\theta$.\nConsequently, $\\langle \\theta \\rangle = \\pi\/2$ and $\\langle \\cos\\theta\\rangle = 0$.\nInstead, we note that in our simulations, the dimers are strongly trapped: $\\vec{u}_3$ always lies near\n$\\theta = 0$ and never flips to lie near $\\theta = \\pi$.\nAssuming that $\\vec{u}_3$ is confined to the upper hemisphere, we can restrict \nthe upper limits of integration to $\\theta$ or $\\theta' = \\pi\/2$.\nSince the integrals over $\\theta$ and $\\theta'$ are independent, we then have\n\\begin{equation}\n\\lim_{\\tau\\rightarrow\\infty} \\langle \\cos \\psi\\rangle  = \\left( \\frac{I}{Z} \\right)^2,\n\\end{equation}\nwhere\n\\begin{equation} \nI \\equiv \\int_0^{\\pi\/2} \\sin\\theta\\cos\\theta \\exp\\left( -\\beta \\kappa_r \\sin^2\\theta \\right) \\,\\mathrm{d}\\theta\n\\end{equation}\nand\n\\begin{equation} \nZ \\equiv \\int_0^{\\pi\/2} \\sin\\theta \\exp\\left( -\\beta \\kappa_r \\sin^2\\theta \\right) \\,\\mathrm{d}\\theta.\n\\end{equation}\n\nEvaluating $I$ by substitution leads to \n\\begin{equation} \\label{eq:I1_result}\nI = \\frac{1}{2\\beta\\kappa_r}\\left[ 1 - \\exp(-\\beta\\kappa_r)  \\right].\n\\end{equation}\nEvaluating $Z$ leads to \\cite{zwillinger_table_2014}\n\\begin{equation} \\label{eq:Z_result}\nZ = \\frac{1}{2}\\sqrt{\\frac{\\pi}{\\beta\\kappa_r}}\\exp(-\\beta\\kappa_r) \\mathrm{erfi}(\\sqrt{\\beta\\kappa_r}),\n\\end{equation}\nwhere $\\mathrm{erfi}(x)$ is the imaginary error function.\nSo, \n\\begin{equation} \n\\lim_{\\tau\\rightarrow\\infty} \\langle \\cos\\psi \\rangle = \\frac{1}{\\pi\\beta\\kappa_r}  \\left( \\frac{\\exp(\\beta\\kappa_r) -1 }{\\mathrm{erfi} \n(\\sqrt{\\beta\\kappa_r})}  \\right)^2.\n\\end{equation}\nFor numerical calculations, it is convenient to express the result in terms of Dawson's integral\n\\begin{equation}\nF(x) \\equiv \\exp(-x^2) \\int_0^x \\exp(t^2)\\,\\mathrm{d}t,\n\\end{equation}\nwhere $\\mathrm{erfi}(x) = \\frac{2}{\\sqrt{\\pi}}\\exp(x^2)F(x)$.\nThus, \n\\begin{equation} \\label{eq:main_result}\n\\lim_{\\tau\\rightarrow\\infty} \\langle \\cos\\psi \\rangle = \\frac{1}{4\\beta\\kappa_r}  \\left( \\frac{\\exp(\\beta\\kappa_r) -1 }{\\exp(\\beta\\kappa_r) F(\\sqrt{\\beta\\kappa_r}) }  \\right)^2.\n\\end{equation}\nThe dependence of Eq.~(\\ref{eq:main_result}) on $\\beta\\kappa_r$ is shown in Fig.~\\ref{fig:betakappa}.\nNote that the limiting value approaches 1 as $\\beta\\kappa_r\\rightarrow\\infty$:\nas the trap becomes increasingly stiff, $\\vec{u}_3$ lies increasingly close to the $+z$ axis.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/plateau_autocorr\/beta_kappa_dependence}\n\\caption{\\label{fig:betakappa}\nDependence of limiting value of $\\langle \\cos\\psi\\rangle$ [Eq.~(\\ref{eq:main_result})]\non $\\beta\\kappa_r$. In the limit of increasing rotational stiffness, \n$\\langle \\cos \\psi\\rangle \\rightarrow 1$.\n}\n\\end{figure}\n\nFinally, substitution of Eq.~(\\ref{eq:main_result}) into Eq.~(\\ref{eq:msad_expansion}) leads to\n\\begin{equation} \\label{eq:msad_limit}\n\\lim_{\\tau\\rightarrow\\infty} \\langle( \\Delta \\vec{u}_3)^2\\rangle = 2\\left[ 1 - \\frac{1}{4\\beta\\kappa_r} \n\\left( \\frac{\\exp(\\beta\\kappa_r) - 1}{\\exp(\\beta \\kappa_r)F(\\sqrt{\\beta\\kappa_r})}\\right)^2    \\right],\n\\end{equation}\nwhich is \nEq.~(12) in the main text.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics{.\/figures\/sinusoidal_torque\/sinusoidal_plateau}\n\\caption{\\label{fig:sinusoid} $\\vec{u}_3$ MSADs calculated from simulations of a particle \nat $T=\\SI{295}{\\kelvin}$ with \n$D_{r,\\perp} = \\SI{5}{\\second^{-1}}$\nand the labeled values of $\\kappa_r$. Each MSAD is averaged from 5 trajectories. Dashed lines indicate predictions\nof Eq.~(\\ref{eq:msad_limit}), and the dotted line indicates the predictions of Eq.~(\\ref{eq:diffusive_msad}).}\n\\end{figure}\n\nTo validate this result, we perform Brownian dynamics simulations in which the particle is fixed in place (with all elements of the\ntranslational diffusion tensor identically 0) and experiences the external torque given by Eq.~\\ref{eq:torque} \n(Fig.~\\ref{fig:sinusoid}). The limiting values of the MSADs for $\\vec{u}_3$ calculated from the simulated trajectories \nagree with the predictions of Eq.~(\\ref{eq:msad_limit}).\nThe simulations also show that the short-time dynamics are governed by \n\\begin{equation} \\label{eq:diffusive_msad}\n\\lim_{\\tau\\rightarrow 0} \\langle (\\Delta \\vec{u}_3)^2 \\rangle = 2\\left[1 - \\exp(-2D_{r,\\perp}\\tau)\\right].\n\\end{equation}\nwhere $D_{r,\\perp}$ is the element of the rotational diffusion tensor for rotations about $\\vec{u}_1$ and\n$\\vec{u}_2$.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nIn the 60s and 70s, various results were obtained providing rather precise control over arbitrary definable sets in\nthe fields ${\\mathbb Q}_p$ of $p$-adic numbers and in fields $k((t))$ of formal Laurent series with coefficients in a field $k$ of characteristic $0$ \\cite{AK1,AK2,AK3,Cohen,Ersov,Mac}.\nBased on those, Denef \\cite{Denef} solved a question of Serre \\cite{Serre} on the rationality of certain Poincar\\'e series and in this way established a strong connection between number theory, arithmetic geometry, and model theory of valued fields. This connection became a driving motivation for the further development of that model theory \\cite{Pas,Basarab}, and in the late 90s, it became the fundament for motivic integration \\cite{DLinvent,DL},\nan integration theory on arc spaces of varieties measuring definable subsets using the model theory of $k((t))$,\nan approach which turned out to be useful in particular for the study of (invariants of) singularities \\cite{DLBarc}.\nThose connections of model theory of valued fields to number theory, arithmetic geometry and singularity theory are the motivation for our present work. Our goal was to understand what really lies behind tameness of definable sets in valued fields and to describe this axiomatically,\nthereby providing a fundament for further research in that direction.\n\nIn real closed fields, model theoretic tameness is very well captured by the notion of o-minimality, an axiomatic condition about first order structures on such fields \\cite{DriesTarski, vdD, KnightPillSt, PillaySteinhornI}. It became a central tool in real algebraic geometry and its generalizations, on the one hand because of its extreme simplicity and on the other hand because of its vast consequences on geometry of definable sets in such structures and its applications to the Andr\u00e9-Oort Conjecture \\cite{Pila}.\n\nSoon after, people started to seek for similar notions in valued fields, and indeed, several such notions have been described, partly in close analogy to o-minimality (like P-minimality \\cite{Haskell}, C-minimality \\cite{Macpherson,HM}), and later more motivated by applications (like V-minimality \\cite{HK}, b-minimality \\cite{CLb}). However, none of those notions has all the desirable properties. In the present paper, we introduce a new notion we call ``Hensel minimality'', with three goals in mind: easy verifiability, broad applicability, and strong consequences.\nHensel minimality is easier to axiomatize and to verify than, say, ``finite b-minimality with centers and Jacobian Property'', it applies more broadly than P-minimality, C-minimality or V-minimality (which only apply to $p$-adically closed or algebraically closed valued fields), and it has stronger consequences than P-minimality and C-minimality.\n\nWhile our original hope had been to come up with one single ``most natural'' notion, we ended up with several variants of Hensel minimality, abbreviated by $\\ell$-h-minimality with $\\ell$ taking values in ${\\mathbb N}\\cup\\{\\omega\\}$. Our favorite notions are with $\\ell=1$ and $\\ell=\\omega$:\nWe develop most of our theory under the (weaker) assumption of $1$-h-minimality, but most examples of Hensel minimal theories we are aware of are\n$\\omega$-h-minimal, which is the more robust notion. (In particular, it is preserved under coarsenings of the valuation.)\nWe continue to use the term ``Hensel minimality'' to talk about any of those minimality notions in an imprecise way.\n\nThe notions of Hensel minimality can be introduced in any valued field of characteristic zero. However, it does become more technical if the characteristic of the residue field is positive, so to keep things more readable, we restrict most of the present paper to the case where the residue field also has characteristic $0$. Nevertheless, we obtain\na mixed characteristic version of $\\omega$-h-minimality almost for free (including many of its consequences), by passing through a coarsening of the valuation.\nWe illustrate how this works on some examples, leaving a more general and more complete treatment of $\\ell$-h-minimality in mixed characteristic for the future.\n\n\\medskip\n\nOne can consider o-minimality as requiring that every unary definable set is controlled by a finite set, namely its set of boundary points; in this sense, Hensel minimality is very similar to o-minimality. To make this more precise,\nthe definition of an o-minimal field $R$ can be phrased as the following condition on definable subsets $X$ of $R$: There exists a finite subset $C$ of $R$ such that, for any $x\\in R$, whether $x$ lies in $X$ depends only on the tuple $(\\sgn(x-c))_{c\\in C}$, where $\\sgn$ stands for the sign, which can be $-$, $0$, or $+$. The definition of Hensel minimality is similar,\nwhere the sign function is replaced by a suitable function adapted to the valuation, namely the leading term map $\\operatorname{rv}$. However,\nwhereas in the o-minimal world, $C$ is automatically definable over the same parameters as $X$, in the valued field setting, we need to impose precise conditions on the parameters over which $C$ can be defined.\nThis is also where the subtle differences between the different variants of Hensel minimality (for different values of $\\ell$) arise.\n\nThe bulk of this paper consists in proving that Hensel minimality implies geometric properties similar to those obtained from o-minimality, in particular\ncell decomposition, dimension theory,\nthe ``Jacobian Property'' (which plays a key role to obtain motivic integration, and which can be considered as an analogue of the Monotonicity Theorem from the o-minimal context where $\\sgn$ is replaced by $\\operatorname{rv}$),\nand versions of the Jacobian Property of higher order and in higher dimension,\nstating that definable functions have good approximations by their Taylor polynomials.\n\nBased on those properties, various recent results in the model theory of Henselian valued fields readily generalize to arbitrary Hensel minimal valued fields, like on Lipschitz continuity \\cite{CCLLip} and t-stratifications (which were introduced in \\cite{Halup} and studied further in \\cite{Gar.cones,Gar.powbd,HalYin}).\nMoreover, our Taylor approximation results lay the ground for further generalizations of motivic integration (both, the one from \\cite{CLoes, CLexp} and the one from \\cite{HK}) and its applications\n(e.g. in \\cite{Bilu-Th,ChaL,CCL,CGH5,CLexp,ForeyDens,HKPoiss,HMartin,Kien:rational,Nicaise-semi,Yin-int,Yimu-t-conv}).\nThey also lay the ground for $C^r$ parameterizations and for bounds on the number of rational points as in \\cite{CCL-PW,CFL}, analogous to results by Yomdin \\cite{YY2} and by Pila--Wilkie \\cite{PW}. We leave these generalizations to the future.\n\nIn another direction, ideas of non-standard analysis can be used to deduce results in ${\\mathbb R}$ and ${\\mathbb C}$ from corresponding results in valued fields.\nBuilding on \\cite{DL.Tcon1,Dri.Tcon2} and on a variant of the above-mentioned Jacobian property, this approach has been used in \\cite{HalYin} to deduce the existence of Mostowksi's Lipschitz stratifications \\cite{Mos.biLip} in arbitrary power-bounded o-minimal real closed fields. As an example application of Hensel minimality, we use our higher order Jacobian Property result to deduce a uniform Taylor approximation result in power-bounded o-minimal real closed fields, which might be useful for strengthenings of the results from \\cite{HalYin}.\n\nOne of the original goals of this work was to deduce the Jacobian Property starting from abstract conditions on unary sets only.\nTo our knowledge, previous proofs of the Jacobian Property either use piecewise analyticity arguments, even in the semi-algebraic case (like in \\cite{CLip}), or they are restricted to rather specific situations (like \\cite{HK} for V-minimal valued fields and \\cite{Yin.tcon} for power-bounded $T$-convex valued fields). Our new proof includes arbitrary Henselian valued fields of characteristic $0$, and (non-first order) analyticity arguments are completely absent.\n\nAs extra upshot, Hensel minimality has resplendency properties in the spirit of resplendent quantifier elimination, i.e., it is preserved by different kinds\nof expansions of the language. In particular, it should be considered as a notion of tameness ``relative to the leading term structure $\\mathrm{RV}$''.\n\n\\subsection{The main results in some more detail}\n\\label{sec:main:res}\n\nWe start by giving the precise definition of Hensel minimality. To this end, we quickly introduce the necessary notation; more details are given in the following two subsections.\n\nLet $K$ be a non-trivially valued field of equi-characteristic $0$,\nconsidered as an ${\\mathcal L}$-structure for some language ${\\mathcal L}$ expanding the language ${\\mathcal L}_\\val = \\{+,\\cdot,{\\mathcal O}_K\\}$ of valued fields.\nWe use the multiplicative notation for valuations and\nwe denote the value group by $\\Gamma^\\times_K$ and the valuation map by\n$$\n|\\cdot|\\colon K\\to \\Gamma_K :=  \\Gamma_K^\\times \\cup\\{0\\};\n$$\nsee Section \\ref{sec:not:val} for more detailed definitions.\n\nThe analogue to the sign map from the o-minimal context will be the ``leading term map'' $\\operatorname{rv}\\colon K \\to \\mathrm{RV}$, defined as follows:\n\n\\begin{defn}[Leading term structure $\\mathrm{RV}_\\lambda$]\\label{defn:RVI}\nLet $\\lambda \\le 1$ be an element of $\\Gamma_K^\\times$ and set $I := \\{x \\in K \\mid |x| < \\lambda\\}$. We define $\\mathrm{RV}_\\lambda^\\times$ to be the quotient of multiplicative groups $K^\\times\/(1 + I)$, and we let\n\\[\n\\operatorname{rv}_\\lambda\\colon K \\to \\mathrm{RV}_\\lambda := \\mathrm{RV}_\\lambda^\\times \\cup \\{0\\}\n\\]\nbe the map extending the projection map $K^\\times\\to \\mathrm{RV}_\\lambda^\\times$ by sending $0$ to $0$. We abbreviate $\\mathrm{RV}_1$ and $\\operatorname{rv}_1$ by $\\mathrm{RV}$ and $\\operatorname{rv}$, respectively. If several valued fields are around, we may also write $\\mathrm{RV}_K$ and $\\mathrm{RV}_{K,\\lambda}$.\n\\end{defn}\n\nWe can now make precise in which sense a set $X\\subset K$ can be controlled by a finite set $C$.\n\n\\begin{defn}[Prepared sets]\\label{defn:lambda-prepared}\nLet $\\lambda \\le 1$ be in $\\Gamma_K^\\times$, let $C$ be a finite non-empty subset of $K$ and let $X \\subset K$ be an arbitrary subset.\nWe say that $C$ \\emph{$\\lambda$-prepares} $X$ if whether some $x\\in K$ lies in $X$ depends only on the tuple $(\\operatorname{rv}_\\lambda(x-c))_{c\\in C}$. In other words, if $x, x' \\in K$ satisfy\n\\begin{equation}\\label{eq:xyIball}\n\\operatorname{rv}_\\lambda(x-c) = \\operatorname{rv}_\\lambda(x'-c)  \\mbox{ for each } c\\in C,\n\\end{equation}\nthen one either has $x\\in X$ and $x'\\in X$, or, one has $x\\not\\in X$ and $x'\\not\\in X$.\n\\end{defn}\n(The condition that $C$ is non-empty is essentially irrelevant -- one could always add $0$ to $C$ --, but it will sometimes avoid pathologies.)\n\n\\begin{example}\nA subset of $K$ is $\\lambda$-prepared by the set $C = \\{0\\}$ if and only if it is the preimage under $\\operatorname{rv}_\\lambda$ of a subset of $\\mathrm{RV}_\\lambda$.\n\\end{example}\n\nA first approximation to the notion of Hensel minimality is: Any $A$-definable set $X \\subset K$ (for $A \\subset K$) can be $1$-prepared by a finite $A$-definable set $C$. This however is not yet strong enough: We need some strong control of parameters from $\\mathrm{RV}_\\lambda$. This is where we obtain different variants of Hensel minimality, the difference consisting only in some number $\\ell$ of allowed parameters:\n\n\\begin{defn}[$\\ell$-h-minimality]\\label{defn:hmin:intro}\nLet $\\ell\\geq 0$ be either an integer or $\\omega$, and let ${\\mathcal T}$ be a complete theory\nof valued fields of equi-characteristic $0$, in a language ${\\mathcal L}$ expanding the language ${\\mathcal L}_\\val$ of valued fields. We say that ${\\mathcal T}$ is \\emph{$\\ell$-h-minimal} if every model $K \\models {\\mathcal T}$ has the following property:\n\\begin{condition}\\label{cond:l-h-min}\nFor every $\\lambda \\le 1$ in $\\Gamma_K^\\times$, for every set $A\\subset K$\nand for every set $A' \\subset \\mathrm{RV}_\\lambda$ of cardinality $\\#A' \\le \\ell$, every $(A \\cup \\mathrm{RV} \\cup A')$-definable set $X \\subset K$ can be $\\lambda$-prepared by a finite $A$-definable set $C \\subset K$.\n\\end{condition}\n\\end{defn}\n\n\\begin{remark}\nIn this definition, the parameters from $\\mathrm{RV}$ and $\\mathrm{RV}_\\lambda$ are imaginary elements, i.e., either one works in $K^{\\mathrm{eq}}$, or (equivalently)\none interprets ${\\mathcal L}(A \\cup \\mathrm{RV} \\cup A')$\nas the language ${\\mathcal L}$ expanded by the constants form $A$ and by predicates $P_\\xi$ for the preimages in $K$ of\nthe elements $\\xi \\in \\mathrm{RV} \\cup A'$; see Section~\\ref{sec:not:mod} for more details.\n\\end{remark}\n\n\n\\begin{remark}\nIn the case $\\ell = 0$, $\\lambda$ plays no role and the definition simplifies to: Any $(A \\cup \\mathrm{RV})$-definable set (for $A \\subset K$) can be $1$-prepared by a finite $A$-definable set.\n\\end{remark}\n\n\\begin{remark}\nIn the case $\\ell = \\omega$, we can more generally allow $X$ to use parameters $\\xi_i \\in \\mathrm{RV}_{\\lambda_i}$ for different $\\lambda_i$ (using that we have surjections $\\mathrm{RV}_\\lambda \\to \\mathrm{RV}_{\\lambda'}$ for $\\lambda' > \\lambda$);\nin that case, $C$ is required to $\\lambda$-prepare $X$ for $\\lambda := \\min_i \\lambda_i$.\n\\end{remark}\n\n\\begin{remark}\nThe case $\\ell = 1$ seems to play an important role: Most of the results in this paper follow from 1-h-minimality, and we obtain a rather different (but equivalent) characterization of 1-h-minimality in terms of definable functions (see Theorem~\\ref{thm:tame2vf}).\nWe do not have a similar characterization for integers $\\ell \\ge 2$, and we do not know whether those separate cases are of particular interest. On the other hand, the case $\\ell = \\omega$ is more robust and is useful for the mixed characteristic case.\n\\end{remark}\n\n\n\\begin{remark}\nThe assumption that $C$ is definable using only the parameters from $K$ will enable us\nto simultaneously prepare families of sets parametrized by $\\mathrm{RV}$ (and $\\mathrm{RV}_\\lambda$).\nThis plays a central role to make Hensel minimality independent of the structure on $\\mathrm{RV}$.\n\\end{remark}\n\n\n\\medskip\n\n\nMany languages where model theory is known to behave well are\nHensel minimal. In particular, in Section~\\ref{sec:examples}, we give the following examples:\n\n\\begin{enumerate}\n \\item\n    Any Henselian valued field of equi-characteristic $0$ in the pure valued field language ${\\mathcal L}_\\val$ has an $\\omega$-h-minimal theory.\n \\item\n   The theory stays $\\omega$-h-minimal if we work in an expansion ${\\mathcal L}$ of ${\\mathcal L}_\\val$ by analytic functions (forming an analytic structure as in \\cite{CLip} or \\cite{CLips}): For example,\n   in the case $K = k((t))$ with $\\operatorname{char} k = 0$, we can add all analytic functions ${\\mathcal O}_K^n \\to K$ for all $n$ to the language, where analytic here means that the function is given by evaluating a $t$-adically converging power series in $x\\in {\\mathcal O}_K^n$ (see Section 3.4 of \\cite{CLips}).\n\n   More generally, we can expand ${\\mathcal L}_\\val$ with function symbols for the elements of any separated analytic structures\n   in the sense of \\cite{CLip}, or of any strictly converging analytic structure in the sense of \\cite{CLips}.\n\\item\\label{it:tcon}\n   If ${\\mathcal T}_\\omin$ is the theory of a power-bounded o-minimal expansion of a real closed field, then the theory of ${\\mathcal T}_\\omin$-convex valued fields in the sense of \\cite{DL.Tcon1} is $1$-h-minimal.\n   Recall that a ${\\mathcal T}_\\omin$-convex valued fields is obtained by taking a (sufficiently big) model $K$ of ${\\mathcal T}_\\omin$ and turning it into a valued field using the convex closure of an elementary substructure $K_0 \\prec K$ as valuation ring; see Section~\\ref{sec:Tcon} for details.\n\\end{enumerate}\n\nIn addition, Hensel minimality is preserved under several modifications of the language:\n\\begin{enumerate}\\setcounter{enumi}{4}\n \\item\\label{it:resp} Any expansion of the language by predicates on Cartesian powers of $\\mathrm{RV}$ preserves $0$-, $1$- and $\\omega$-h-minimality (Theorem~\\ref{thm:resp:h}).\n  \\item\\label{it:coars-rob} In addition, $\\omega$-h-minimality is also preserved under coarsening of the valuation (Corollary~\\ref{cor:coarse}).\n\\end{enumerate}\nMotivated by the robustness of $\\omega$-h-minimality as in (\\ref{it:coars-rob}), we make the passage to mixed characteristic:\n\\begin{enumerate}\\setcounter{enumi}{6}\n\\item\n   In Subsection~\\ref{ssec:mixed}, we give a condition on valued fields of mixed characteristic (using a coarsening) which implies that many results from this paper carry over with only small modifications. That condition is satisfied by the pure valued field language and by analytic structures as in (2) above.\n\\end{enumerate}\n\nAs a converse to the results of Section~\\ref{sec:examples}, we show that a valued field which is Hensel minimal is automatically Henselian, see Theorem \\ref{thm:hens}.\n\n\\medskip\n\nWe now list some of the consequences of 1-h-minimality (partly in simpler but slightly weaker forms than the versions in the main part of the paper).\nThe first result is the ``Jacobian Property'' which plays a crucial role e.g.\\ in motivic integration, both in \\cite{CLoes}, \\cite{CLexp} and \\cite{HK}.\n\n\n\\begin{thm}[Jacobian Property; see Corollary~\\ref{cor:JP}]\\label{thm:JP:intro}\nLet $f\\colon K \\to K$ be definable. Then there exists a finite set $C \\subset K$ such that the following holds: For every fiber $B$ of the map sending $x \\in K$ to the tuple $(\\operatorname{rv}(x - c))_{c \\in C}$, there exists a $\\xi_B \\in \\mathrm{RV}$ such that\n\\begin{equation}\\label{eq:JP:intro}\n\\operatorname{rv}\\left(\\frac{f(x_1) - f(x_2)}{x_1 - x_2}\\right) = \\xi_B\n\\end{equation}\nfor all $x_1, x_2 \\in B$, $x_1 \\ne x_2$.\n\\end{thm}\n\n\\begin{remark}\nIf, in this formulation of the Jacobian Property, one replaces all occurrences of $\\operatorname{rv}$ by $\\sgn$ and if one assumes continuity on the fibers, one gets exactly the Monotonicity Theorem from o-minimality.\n\\end{remark}\n\nCorollary~\\ref{cor:JP} also includes the following strengthenings of Theorem~\\ref{thm:JP:intro}:\n\\begin{itemize}\n \\item\nThe theorem still holds with $\\operatorname{rv}$ replaced by $\\operatorname{rv}_\\lambda$ for any $\\lambda \\le 1$ in $\\Gamma_K^\\times$ (still only assuming 1-h-minimality), and one can moreover choose a single finite set $C$ that works for all such $\\lambda$.\n \\item If $f$ is definable with parameters from $A \\cup \\mathrm{RV}$ with $A \\subset K$, then $C$ can taken to be $A$-definable. (Corollary~\\ref{cor:JP} only speaks about $\\emptyset$-definable sets $f$; Remark~\\ref{rem:acl} explains how to deduce this more general version.)\n\\end{itemize}\n\nAnother point of view of the Jacobian Property is that on each fiber $B$ (using notation from the Theorem), $f$ has a good approximation by its first order Taylor series.\nOne of the deepest results of this paper is a similar result for higher order Taylor approximations:\n\n\\begin{thm}[Taylor approximations; see Theorem~\\ref{thm:high-ord}]\\label{thm:high-ord:intro}\nLet $f\\colon K \\to K$ be a definable function and let $r \\in {\\mathbb N}$ be given. Then there exists a finite set $C$ such that for every\nfiber $B$ of the map sending $x \\in K$ to the tuple $(\\operatorname{rv}(x - c))_{c \\in C}$,\n$f$ is $(r+1)$-fold differentiable on $B$ and we have\n\\begin{equation}\\label{eq:t-higher:intro}\n \\left|f(x) -  \\sum_{i = 0}^r \\frac{f^{(i)}(x_0)}{i!}(x - x_0)^i \\right| \\leq |f^{(r+1)}(x_0)\\cdot (x-x_0)^{r+1}|\n\\end{equation}\nfor every $x_0, x \\in B$.\n\\end{thm}\nAs in Theorem~\\ref{thm:JP:intro}, if $f$ is $(A \\cup \\mathrm{RV})$-definable for $A \\subset K$, then $C$ can be taken $A$-definable (again using Remark~\\ref{rem:acl}).\n\nUsing the results about Hensel minimality ${\\mathcal T}_\\omin$-convex valued fields mentioned under (\\ref{it:tcon}) above (on p.~\\pageref{it:tcon}), this Taylor approximation result implies\na uniform Taylor approximation result in power-bounded o-minimal real closed fields; see Corollary~\\ref{cor:arch}.\n\n\n\\medskip\n\nAs in the o-minimal context, the preparation of unary sets and the Jacobian Property lend themselves well (by logical compactness) to obtain results about higher dimensional objects (namely, definable subsets of $K^n$ and definable functions on $K^n$). Indeed, this can be pursued, \\emph{mutatis mutandis}, in the style of how cell decomposition and dimension theory are built up from b-minimality in \\cite{CLb}. In particular, we obtain results about\n\\begin{itemize}\n \\item\nalmost everywhere differentiability (Subsection~\\ref{sec:cont}), \\item\ncell decomposition in two variants (Subsections~\\ref{sec:cd} and ~\\ref{sec:cd:classical}),\n\\item\ndimension theory (Subsection~\\ref{sec:dim}), and\n\\item higher dimensional versions\nof the Jacobian Property and of Taylor approximations (Subsections~\\ref{sec:sjp} and \\ref{sec:taylor-box}).\n\\end{itemize}\nFor cell decomposition, Subsection~\\ref{sec:cd} provides a new approach specific to Hensel minimality: Usually, the notion of cells in valued fields is a lot more technical than in o-minimal structures, partly due to the lack of (certain) Skolem functions. Item~(\\ref{it:resp}) above (on p.~\\pageref{it:resp}) allows us to add the missing Skolem functions to the language without destroying Hensel minimality, and there are also some tools enabling us to get back to the original language afterwards (Subsection~\\ref{sec:alg:skol}).\nTherefore, the notion of cell introduced in Subsection~\\ref{sec:cd} is much less technical than most previous ones in valued fields.\n\n\n\n\\medskip\n\n\nThe paper is organized as follows: In the remainder of Section~\\ref{sec:intro}, we mainly fix notation and terminology.\nIn Section~\\ref{sec:first-properties}, we develop many tools that are useful for proofs in Hensel minimal theories,\nand we obtain first elementary results like a key ingredient to dimension theory (Lemma \\ref{lem:fin-inf}) and\na weak version of the Jacobian Property (Lemma~\\ref{lem:gammaLin}). We also show that those two results are essentially equivalent to\n$1$-h-minimality (Theorem~\\ref{thm:tame2vf}).\n\nThe deepest results of this paper are contained in Section~\\ref{sec:derJac} about definable functions from $K$ to $K$,\nnamely almost everywhere differentiability and Taylor approximation.\n\nSection~\\ref{sec:respl} is devoted to understanding in which sense Hensel minimality is a notion ``relative to $\\mathrm{RV}$'' and to proving that\nvarious modifications of the language preserve Hensel minimality.\n\nThe geometric results in $K^n$ (like cell decomposition, dimension theory) are collected in\nSection~\\ref{sec:compactn}. That section also contains our main application, of t-stratifications, and our higher dimensional Taylor approximation results.\n\nIn Section~\\ref{sec:mixed}, we explain how (and under which assumptions) our results\nin equi-characteristic $0$ imply similar results in mixed characteristic. We also take the opportunity to indicate how to work with non-complete theories.\n\nFinally, in Section~\\ref{sec:examples}, we give many examples of Hensel minimal structures (including examples of the mixed characteristic version from Section~\\ref{sec:mixed}), we compare our notion to V-minimality and we list some open questions. Moreover, we show, in Subsection~\\ref{sec:Tcon} how Hensel minimality results yield corresponding results in power-bounded real closed fields.\n\n\n\\subsection{Notation and terminology for valued fields and balls}\\label{sec:not:val}\n\n\nIn the entire paper, $K$ will denote a non-trivially valued field of characteristic zero, with valuation ring ${\\mathcal O}_K \\subset K$ and maximal ideal ${\\mathcal M}_K \\subset {\\mathcal O}_K$, where non-trivially means ${\\mathcal O}_K \\ne K$. Moreover, apart from parts of Sections \\ref{sec:mixed} and \\ref{sec:examples}, $K$ will always be of equi-characteristic $0$, meaning that both $K$ and also the residue field ${\\mathcal O}_K\/{\\mathcal M}_K$ have characteristic $0$.\n\nBy ``valued field'', from now on, we mean non-trivially valued field. Note that we allow Krull-valuations (and thus valuations of arbitrary rank), that is,  we allow ${\\mathcal O}_K$ to be an arbitrary (proper, by the non-triviality) subring of $K$ such that for every element $x\\in K^\\times$, at least one of $x$ or $x^{-1}$ belongs to ${\\mathcal O}_K$.\nThe value group is then defined to be the quotient $\\Gamma_K^\\times := K^\\times \/ {\\mathcal O}_K^\\times$ of multiplicative groups.\n\n\nWe denote the valuation map by $|\\cdot|\\colon K^\\times \\to \\Gamma_K^\\times$ and use multiplicative notation for the value group. We write $\\Gamma_K$ for the disjoint union $\\Gamma_K^\\times \\cup \\{0\\}$, we extend the valuation map to $|\\cdot|\\colon K \\to \\Gamma_K$ by setting $|0| := 0$,\nand we define the order on $\\Gamma_K$ in such a way that ${\\mathcal O}_K = \\{x \\in K \\mid |x| \\le 1\\}$ and $|x|<|y|$ whenever  $x\/y\\in {\\mathcal M}_K$  for  $x$ and nonzero $y$ in $K$.\n\nFor $x = (x_1, \\dots, x_n) \\in K^n$, we set $|x| := \\max_i |x_i|$.\n\n\nWe use the (generalized) leading term structures $\\mathrm{RV}_\\lambda$ (for $\\lambda \\le 1$ in $\\Gamma_K^\\times$) that have already been introduced in Definition~\\ref{defn:RVI},\nand we denote the natural map $\\mathrm{RV}_\\lambda \\to \\Gamma_K$ by $|\\cdot|$.\nNote that $\\mathrm{RV}_\\lambda$ is a semi-group for multiplication.\n\n\\begin{remark}\nRecall that one has a natural short exact sequence of multiplicative groups $({\\mathcal O}_K\/{\\mathcal M}_K)^\\times \\to \\mathrm{RV}^\\times \\to \\Gamma^\\times_K$. (So\n$\\mathrm{RV}$ combines information from the \\underline{r}esidue field and \\underline{v}alue group.)\n\\end{remark}\n\n\n\\begin{example}\nIn the case $K = k((t))$, the above short exact sequence naturally splits, giving an isomorphism $\\mathrm{RV}^\\times \\to  ({\\mathcal O}_K\/{\\mathcal M}_K)^\\times \\times \\Gamma^\\times_K$,\nwhich, for $a = \\sum_{i=N}^\\infty a_i t^i \\in K^\\times$ with $a_N \\ne 0$, sends $\\operatorname{rv}(a)$ to $(a_N, N)$.\n\\end{example}\n\n\\begin{remark}\nFor $a, a' \\in K$, one has $\\operatorname{rv}_\\lambda(a) = \\operatorname{rv}_\\lambda(a')$ if and only if either $a = a' = 0$, or $|a - a'| < \\lambda\\cdot |a|$.\n\\end{remark}\n\n\n\nWe consider several kinds of balls:\n\n\\begin{defn}[Balls]\\label{defn:balls}\n\\begin{enumerate}\n \\item\nWe call a subset $B \\subset K^n$ a \\emph{ball} if $B$ is infinite, $B \\ne K^n$, and for all $x,x'\\in B$ and all $y\\in K^n$ with $|x-y|\\leq |x-x'|$ one has $y\\in B$.\n\\item\nBy an \\emph{open ball}, we mean a set of the form\n$$\nB = B_{<\\gamma}(a) := \\{x\\in K^n\\mid | x - a | < \\gamma \\}\n$$\nfor some $a\\in K^n$ and some $\\gamma \\in \\Gamma_K^\\times$.\n\\item\nBy a \\emph{closed ball}, we mean a set of the form\n$$\nB = B_{\\le\\gamma}(a) := \\{x\\in K^n\\mid | x - a | \\leq  \\gamma \\}\n$$\nfor some $a\\in K^n$ and some $\\gamma \\in \\Gamma_K^\\times$.\n\\item\nFor $B$ as in (2) or (3), we call $\\gamma$ the \\emph{radius} of the open (resp.~closed) ball and denote it by $\\rad_{\\mathrm{op}}(B)$ (resp.~$\\rad_{\\mathrm{cl}}(B)$).\n\\end{enumerate}\n\\end{defn}\n\nWe call the valuation on $K$ discrete if there is a uniformizing element $\\varpi$ in ${\\mathcal O}_K$, namely satisfying $\\varpi{\\mathcal O}_K={\\mathcal M}_K$.\n\n\\begin{remark}\\label{rem:rad-op}\nWhen $\\Gamma_K^\\times$ is discrete, a ball $B$ can at the same time be open and closed, but with $\\rad_{\\mathrm{op}}(B) > \\rad_{\\mathrm{cl}}(B)$.\n\\end{remark}\n\nNote also that for any $\\lambda \\le 1$ in $\\Gamma_K^\\times$ and for any $\\xi \\in \\mathrm{RV}_\\lambda \\setminus \\{0\\}$, the preimage $\\operatorname{rv}_{\\lambda}^{-1}(\\xi)$ is an open ball satisfying\n$\\rad_{\\mathrm{op}}(\\operatorname{rv}_{\\lambda}^{-1}(\\xi)) = |\\xi| \\cdot \\lambda$.\n\n\\begin{defn}[$\\lambda$-next balls]\\label{defn:next}\nFix $\\lambda \\le 1$ in $\\Gamma^\\times_K$.\n\\begin{enumerate}\n \\item\nWe say that a ball $B$ is $\\lambda$-next to an element $c \\in K$\nif\n$$\nB= \\{x\\in K\\mid \\operatorname{rv}_\\lambda(x-c) = \\xi \\}\n$$\nfor some (nonzero) element $\\xi$ of $\\mathrm{RV}_\\lambda$.\n\\item\nWe say that a ball $B$ is $\\lambda$-next to a finite non-empty set $C\\subset K$\nif $B$ equals $\\bigcap_{c\\in C} B_c$\nwith $B_c$ a ball $\\lambda$-next to $c$ for each $c\\in C$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{remark}\nUsing that the intersection of the finitely many balls $B_c$ is either empty or equal to one of the $B_c$, one deduces that\nevery ball $B$ which is $\\lambda$-next to $C$ is in particular $\\lambda$-next to one element $c \\in C$.\n\\end{remark}\n\n\n\n\\begin{remark}\nGiven a finite non-empty set $C \\subset K$,\nthe fibers of the map $x \\mapsto (\\operatorname{rv}_\\lambda(x -  c))_{c \\in C}$ are exactly the singletons consisting of one element of $C$ and the balls $\\lambda$-next to $C$. In particular, the balls $\\lambda$-next to $C$ form a partition of $K \\setminus C$, and a subset $X$ of $K$\nis $\\lambda$-prepared by $C$ (as in Definition~\\ref{defn:lambda-prepared}) if and only if every ball $B$ which is $\\lambda$-next to $C$ is either contained in $X$ or disjoint from $X$.\n\\end{remark}\n\n\n\\begin{example}\\label{ex:1prep}\nThe balls $1$-next to an element $c \\in K$ are exactly the maximal balls in $K$ not containing $c$. From this, one deduces that\na ball $1$-next to a finite set $C$ is exactly a maximal ball disjoint from $C$. This means that a set $X \\subset K$ is $1$-prepared by $C$ if and only if every ball disjoint from $C$ is either contained in $X$ or disjoint from $X$. Note again how closely ``every definable set can be $1$-prepared'' resembles o-minimality (where ``balls disjoint from $C$'' becomes ``intervals disjoint from $C$'').\n\\end{example}\n\n\nGiven a subset $A$ of a Cartesian product $B\\times C$ and an element $b\\in B$, we write $A_b$ for the fiber $\\{c\\in C\\mid (b,c)\\in A\\}$. Also, for a function $g$ on $A$, we write $g(b,\\cdot)$ for the function on $A_b$ sending $c$ to $g(b,c)$.\n\n\n\n\n\\subsection{Model theoretic notations and conventions}\n\\label{sec:not:mod}\n\nAs already stated, in the entire paper, $K$ is a non-trivially valued field of characteristic zero, and outside of Sections \\ref{sec:mixed} and \\ref{sec:examples}, $K$ is moreover of equi-characteristic $0$.\nIn the entire paper, we fix a language ${\\mathcal L}$ expanding the language ${\\mathcal L}_\\val$ of valued fields, and we consider the valued field $K$ as an ${\\mathcal L}$-structure.\nAs ``language of valued fields'', it suffices for us to take ${\\mathcal L}_\\val = \\{+,\\cdot, {\\mathcal O}_K\\}$; in any case, we only care about which sets are definable (and not how they are definable). For that reason, we will often specify languages only up to interdefinability.\n\nIf not specified otherwise, ``definable'' always refer to the fixed language ${\\mathcal L}$. As usual, ``${\\mathcal L}$-definable'' means definable (in ${\\mathcal L}$) without additional parameters, ``$A$-definable'' means ${\\mathcal L}(A)$-definable, and\n``definable'' means ${\\mathcal L}(K)$-definable (i.e., with arbitrary parameters).\n\nSometimes, we will also consider $K$ as a structure in other languages (e.g.\\ ${\\mathcal L}'$); in that case, we may specify the language as an index, writing e.g.\\ $\\operatorname{Th}_{{\\mathcal L}'}(K)$ for the theory of $K$ considered as an ${\\mathcal L}'$-structure.\n\n\nIn almost the entire paper (more precisely, everywhere except in parts of Section~\\ref{sec:fixI}), ${\\mathcal L}$ will be a one-sorted language. Nevertheless, we  often\nwork with imaginary sorts of $K$, i.e., quotients $K^n\/\\mathord{\\sim}$ where $\\sim$ is a $\\emptyset$-definable equivalence relation. In particular, we consider imaginary definable sets and imaginary elements.\nAs usual, this can be made formal either by working in the ${\\mathcal L}^{\\mathrm{eq}}$-structure $K^{\\mathrm{eq}}$ (see e.g.\\ \\cite[Proposition~8.4.5]{TentZiegler}), or, equivalently, by ``simulating'' imaginary objects in ${\\mathcal L}$, namely as follows:\n\\begin{itemize}\n \\item By a ``definable subset $X$ of $K^n\/\\mathord{\\sim}$'', we really mean its preimage in $K^n$, i.e., a definable set $\\tilde X \\subset K^n$ which is a union of $\\sim$-equivalence classes.\n \\item If, in a formula $\\phi(x, \\dots)$, the variable $x$ runs over an imaginary sort $K^n\/\\mathord{\\sim}$, this means that we really have an $n$-tuple $\\tilde{x}$ of variables\n (running over $K^n$) and that the truth value of $\\phi(\\tilde{x}, \\dots)$\n only depends on the equivalence class of $\\tilde x$ modulo $\\sim$.\n \\item If $A$ is a set of potentially imaginary elements, then by ${\\mathcal L}(A)$, we mean the expansion of ${\\mathcal L}$ by predicates for the equivalence classes (in $K^n$ for some $n$) corresponding to the imaginary elements $a \\in A$. By ``$A$-definable'', we mean ${\\mathcal L}(A)$-definable in that sense.\n\\item A (potentially imaginary) element $b$ is said to be in the definable closure of a set $A$ (of potentially imaginary elements) if the equivalence class in $K^n$ corresponding to $b$\n  is ${\\mathcal L}(A)$-definable. If $X$ is any imaginary sort (or even more generally an arbitrary set of imaginary elements), we write $\\dcl_X(A)$ for the set of elements from $X$ that are\n  in the definable closure of $A$. Being in the algebraic closure, with notation $\\acl_X(A)$, is defined accordingly.\n\\end{itemize}\n\n\nThe value group $\\Gamma_K$ is of course an imaginary sort. In general, $\\mathrm{RV}_\\lambda$ (for $\\lambda \\le 1$ in $\\Gamma^\\times_K$) is, by itself, not an imaginary sort, since the equivalence relation used to define it may not be $\\emptyset$-definable. However, the disjoint union\n\\[\n\\mathrm{RV}_\\bullet := \\bigcup_{\\lambda \\le 1} \\mathrm{RV}_\\lambda\n\\]\nis an imaginary sort and $\\mathrm{RV}_\\lambda$ is a definable subset of\n$\\mathrm{RV}_\\bullet$; in particular, it makes sense to use elements from $\\mathrm{RV}_\\lambda$ as parameters.\n\n\n\\subsection{Basic model theoretic properties of Hensel minimality}\n\nLet us now verify that the notion of Hensel minimality (in all its variants) has some basic\nproperties one would expect from any good model theoretic notion.\n\nFirst of all, note that it is preserved under expansions of the language by constants; more precisely:\n\n\\begin{lem}[Adding constants]\\label{lem:addconst}\nLet $\\ell\\geq 0$ be an integer or $\\omega$,\nsuppose that $\\operatorname{Th}(K)$ is $\\ell$-h-minimal, and let $A$ be any subset of $K \\cup \\mathrm{RV}^{\\mathrm{eq}}$. Then $\\operatorname{Th}_{{\\mathcal L}(A)}(K)$ is also $\\ell$-h-minimal.\n\\end{lem}\n\nHere, by $\\mathrm{RV}^{\\mathrm{eq}}$, we mean imaginary sorts of the form $\\mathrm{RV}^n\/\\mathord{\\sim}$, for some $n$ and some $\\emptyset$-definable equivalence relations $\\sim$. In particular, we can add constants from $\\Gamma_K$.\nNote however that adding parameters from other sorts than $K$ and $\\mathrm{RV}^{\\mathrm{eq}}$ may destroy Hensel minimality.\n\nThe lemma should be clear in the case $A \\subset K \\cup \\mathrm{RV}$; we mainly give the following proof to show that parameters from $\\mathrm{RV}^{\\mathrm{eq}}$ are not a problem either.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:addconst}]\nWe verify Definition~\\ref{defn:hmin:intro}:\nLet $K'\\models \\operatorname{Th}_{{\\mathcal L}(A)}(K)$ and let $X\\subset K'$ be ${\\mathcal L}(A\\cup A'\\cup\\mathrm{RV}_{K'}\\cup A'')$-definable, for some\n$A' \\subset K'$ and some $A'' \\subset \\mathrm{RV}_{K',\\lambda}$ satisfying $\\#A'' \\le \\ell$, with $\\lambda \\le 1$ in $\\Gamma_{K'}$.\nChoose $\\tilde A \\subset \\mathrm{RV}_{K'}$ such that every element of $A \\cap \\mathrm{RV}_{K'}^{\\mathrm{eq}}$ is ${\\mathcal L}(\\tilde A)$-definable.\nThen $X$ is ${\\mathcal L}((A \\cap K) \\cup \\tilde{A}\\cup A'\\cup\\mathrm{RV}_{K'}\\cup A'')$-definable,\nso by $\\ell$-h-minimality of $\\operatorname{Th}_{{\\mathcal L}}(K')$, $X$ can be $\\lambda$-prepared by a finite ${\\mathcal L}((A \\cap K) \\cup A')$-definable set $C$.\nIn particular, \\(C\\) is ${\\mathcal L}(A\\cup A')$-definable, as desired.\n\\end{proof}\n\nWith this lemma in mind, many results in this paper are formulated for $\\emptyset$-definable sets; those results then automatically also hold for $A$-definable sets, when $A \\subset K \\cup \\mathrm{RV}^{\\mathrm{eq}}$,\nand using a compactness argument, one then often obtains family versions of the results.\n\n\nAs so often in model theory, it is sufficient to verify Hensel minimality in sufficiently saturated models. To see this, we first prove that preparation is a first order property in the following sense:\n\n\n\\begin{lem}[Preparation is first order]\\label{lem:prep:def}\nLet $X_q$ and $C_q$ be $\\emptyset$-definable families of subsets of $K$, where $q$ runs over a $\\emptyset$-definable subset $Q$ of an arbitrary imaginary sort. Suppose that $C_q$ is finite for every $q \\in Q$. Then the\nset of pairs $(q,\\lambda) \\in Q \\times \\Gamma^\\times_K$ with\n$\\lambda \\le 1$ such that $C_q$ $\\lambda$-prepares $X_q$ is $\\emptyset$-definable.\n\\end{lem}\n\n\\begin{proof}\nIf $\\phi(x, q)$ defines $X_q$ and $\\psi(z, q)$ defines $C_q$, then\nthe above set of pairs $(q,\\lambda)$ is defined by the following ${\\mathcal L}$-formula:\n\\[\n\\forall x,x'\\colon \\big(\\underbrace{\\forall z\\colon (\\psi(z, q) \\to \\operatorname{rv}_\\lambda(x - z) = \\operatorname{rv}_\\lambda(x' - z))}_{\\text{i.e., }(\\operatorname{rv}_\\lambda(x'-c))_{c\\in C_q} = (\\operatorname{rv}_\\lambda(x'-c))_{c\\in C_q}} \\to (\\phi(x, q) \\leftrightarrow \\phi(x', q))\\big)\n\\]\n\\end{proof}\n\n\n\\begin{lem}[Saturated models are enough]\\label{lem:emin-sat} Let $\\ell\\geq 0$ be either an integer or $\\omega$ and suppose that $K$ is $\\aleph_0$-saturated. Then the theory $\\operatorname{Th}(K)$ of $K$ is $\\ell$-h-minimal if and only if $K$ satisfies Condition~(\\ref{cond:l-h-min}) from Definition~\\ref{defn:hmin:intro}.\n\\end{lem}\n\n\\begin{proof}\nWe need to show that if $K$ satisfies (\\ref{cond:l-h-min}), then so does any\nother model $K'$ of $\\operatorname{Th}(K)$.\n\nSuppose for contradiction that $K'$ is a model not satisfying (\\ref{cond:l-h-min}), i.e., there exist a $\\lambda \\le 1$ in $\\Gamma^\\times_{K'}$, tuples $a \\in (K')^n$, $\\zeta \\in \\mathrm{RV}_{K'}^{n'}$, $\\xi \\in \\mathrm{RV}_{K',\\lambda}^{n''}$ with $n, n'$ arbitrary and $n'' \\le \\ell$, and an $(a, \\zeta, \\xi)$-definable set\n$X = \\phi(K', a, \\zeta, \\xi) \\subset K'$ such that no finite non-empty $a$-definable set $C \\subset K$ $\\lambda$-prepares $X$.\n\nFor fixed $\\phi$, the non-existence of $C$ can be expressed by an infinite conjunction of ${\\mathcal L}$-formulas\nin $(\\lambda, a, \\zeta, \\xi)$. Indeed,\nfor every formula $\\psi(z, a)$ that could potentially define $C$ and for every integer $k \\ge 1$, there is (by Lemma~\\ref{lem:prep:def}) an ${\\mathcal L}$-formula $\\chi_\\psi(\\lambda, a, \\zeta, \\xi)$ expressing\n``$\\psi(K', a)$ has cardinality $k$ and $\\psi(K', a)$ does not $\\lambda$-prepare $\\phi(K', a, \\zeta, \\xi)$.''\n\nThe fact that this partial type $\\{\\chi_\\psi \\mid \\psi\\text{ as above}\\}$ is realized in $K'$ implies that it is also realized in $K$, so that Condition~(\\ref{cond:l-h-min}) fails in $K$.\n\\end{proof}\n\nWhether a structure is o-minimal can also be characterized via its 1-types. For $0$-h-minimality, we have a similar characterization:\n\n\n\\begin{lem}[$0$-h-minimality in terms of types]\\label{lem:type-0-h-min}\nSuppose that $K$ is $\\aleph_0$-saturated.\nThe theory $\\operatorname{Th}(K)$ is $0$-h-minimal\nif and only if, for every parameter set $A \\subset K$ and for every ball $B \\subset K \\setminus \\acl_K(A)$, all elements of $B$ have the same type over $A \\cup \\mathrm{RV}$.\n\\end{lem}\n\n\n\\begin{remark}\nOne could also formulate similar conditions for $1$-h-minimality and $\\omega$-h-minimality, but those would be more technical.\n\\end{remark}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:type-0-h-min}]\n``$\\Rightarrow$'': Suppose for contradiction that $B$ contains two elements $x, x'$ with $\\tp(x\/A \\cup \\mathrm{RV}) \\ne \\tp(x'\/A \\cup \\mathrm{RV})$. This means that there exists an $(A \\cup \\mathrm{RV})$-definable set $X$ containing $x$ but not $x'$. By $0$-h-minimality, there exists a finite $A$-definable set $C$ $1$-preparing $X$. In particular, $C \\subset \\acl_K(A)$ and hence $C \\cap B = \\emptyset$.\nHowever, by Example~\\ref{ex:1prep}, this implies that $B$ is either contained in $X$ or disjoint from $X$, contradicting the properties of $x$ and $x'$.\n\n``$\\Leftarrow$'': Let $X$ be $(A \\cup \\mathrm{RV})$-definable, for some parameter set $A \\subset K$ which we may assume to be finite, and suppose that no finite $A$-definable $C \\subset K$ $1$-prepares $X$.\nThis means (using Example~\\ref{ex:1prep} again) that for every finite $A$-definable $C \\subset K$, there exists an open ball $B \\subset K$ that is disjoint from $C$ and such that neither $B \\subset X$ nor $B \\subset K \\setminus X$.\nTaking all those conditions on $B$ together (for all finite $A$-definable $C$), we obtain a (finitely satisfiable) type in an imaginary variable running over the open balls in $K$. A realization of this type is a ball $B$ that is disjoint from $\\acl_K(A)$ on the one hand, but on the other hand contains elements $x$ and $x'$ that satisfy $x \\in X$ and $x' \\notin X$ and hence have different types over $A \\cup \\mathrm{RV}$.\n\\end{proof}\n\nYet another way to characterize o-minimality is: Every unary definable set is already quantifier free definable in the language $\\{<\\}$.\nUsing Lemma~\\ref{lem:prep eq qf}, one obtains a similar kind of characterization of $0$-h-minimality and of $\\omega$-h-minimality.\n\n\n\n\n\n\n\\section{First properties of h-minimal theories}\\label{sec:first-properties}\n\nRecall that in the entire paper, $K$ is a non-trivially valued field of characteristic zero, considered as a structure in a language ${\\mathcal L} \\supset {\\mathcal L}_\\val = \\{+, \\cdot,{\\mathcal O}_K\\}$. Outside of Sections \\ref{sec:mixed} and \\ref{sec:examples}, $K$ is moreover of equi-characteristic $0$.\nMost of the time, we will assume $\\operatorname{Th}(K)$ to be $1$-h-minimal, but we start with some results that hold more generally.\n\n\n\n\n\\subsection{Basic properties under weaker assumptions}\n\\label{sec:weak-h}\n\nIn this and the following subsection, we assume $\\operatorname{Th}(K)$ to be ``Hensel minimal without control of parameters'', namely:\n\\begin{assumption}\\label{ass:no-ctrl}\nFor every $K' \\equiv K$, every definable (with any parameters) subset of $K'$ can be $1$-prepared by a finite set $C$. (We do not impose definability conditions on $C$.)\n\\end{assumption}\n\nNote that this assumption is preserved under adding constants to ${\\mathcal L}$ (even from arbitrary imaginary sorts), so below, every occurrence of ``$\\emptyset$-definable'' can also be replaced by ``$A$-definable''.\n\n\n\\begin{lem}[$\\exists^\\infty$-elimination]\\label{lem:finite}\nUnder Assumption~\\ref{ass:no-ctrl},\nevery infinite definable set $X \\subset K$ contains an (open) ball. In particular, if $\\{X_q\\mid q\\in Q\\}$ is a $\\emptyset$-definable family of subsets $X_q$ of $K$, for some $\\emptyset$-definable set $Q$ in an arbitrary imaginary sort, then\nthe set $Q' := \\{q\\in Q \\mid X_q\\text{ is finite}\\}$ is a $\\emptyset$-definable set, and there exists a uniform bound $N \\in {\\mathbb N}$ on the cardinality of $X_q$ for all $q \\in Q'$.\n\\end{lem}\n\n\\begin{proof}\nIf $X$ is infinite, then it is not contained in the finite set $C$ preparing it, which implies that it contains a ball.\nThe definability of $Q'$ then follows since the condition that a set contains an open ball can be expressed by a formula. The existence of the bound $N$ then follows by compactness: If no bound would exist, then in a sufficiently saturated model, we would find a $q \\in Q'$ with $\\#X_q > N$ for every $N$.\n\\end{proof}\n\n\n\n\\begin{lem}[Finite sets are $\\mathrm{RV}$-parametrized]\\label{lem:average}\nUnder Assumption~\\ref{ass:no-ctrl}, let $C_q\\subset K$ be a $\\emptyset$-definable family of finite sets, where $q$ runs over some $\\emptyset$-definable set $Q$ in an arbitrary imaginary sort.\nThen there exists a $\\emptyset$-definable family of injective maps $f_q\\colon C_q \\to \\mathrm{RV}^k$ (for some $k$).\n\\end{lem}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:average}]\nWe do an induction over the maximal cardinality $\\max\\{\\#C_q \\mid q \\in Q\\}$. Note that by Lemma~\\ref{lem:finite}, this maximum exists.\n\nIf $C_q$ is always a singleton or empty, we can define $f_q$ to always be constant. Otherwise,\nthe lemma is obtained by repeatedly taking averages of the elements of $C_q$ and subtracting. More precisely, setting $a_q := \\frac1{\\#C_q}\\sum_{x \\in C_q} x$, we get that the map $\\hat f_q\\colon C_q\\to \\mathrm{RV}, x \\mapsto \\operatorname{rv}(x - a_q)$ is not constant on $C_q$.\nTherefore, each fiber $\\hat f^{-1}_q(\\xi)$ of $\\hat f$ (for $\\xi \\in \\hat f_q(C_q)$) has cardinality less than $C_q$, so\nby induction, we obtain a definable family of\ninjective maps $g_{q,\\xi}\\colon \\hat f^{-1}_q(\\xi) \\to \\mathrm{RV}^k$. Now set $f_q(x) := (\\hat f_q(x), g_{q,\\hat f_q(x)}(x))$.\n\\end{proof}\n\n\nThe family of balls $1$-next to some finite set $C \\subset K$ can be parameterized by $\\mathrm{RV}$-variables; more precisely (and more generally), we have the following:\n\n\\begin{lem}[$\\lambda$-next balls as fibers]\\label{lem:InextFam}\nUnder Assumption~\\ref{ass:no-ctrl}, let $\\lambda \\le 1$ be an element of $\\Gamma^\\times_K$,\nlet $A$ be any set of possibly imaginary parameters containing $\\lambda$, and\nlet $C\\subset K$ be a finite non-empty $A$-definable set.\nThen there exists an $A$-definable map $f\\colon K \\to \\mathrm{RV}^k \\times \\mathrm{RV}_\\lambda$ (for some $k$) such that each fiber of $f$ is either a singleton contained in $C$ or a ball $\\lambda$-next to $C$.\n\\end{lem}\n\n\n\\begin{proof}\nGiven $x \\in K$, let $\\mu(x) := \\min\\{|x-c| \\mid c \\in C\\}$ be the minimal distance to elements of $C$, let $C(x) := \\{c \\in C \\mid |x-c| = \\mu(x)\\}$ be the set $c \\in C$ realizing that distance,\nand let $a(x) := \\frac1{\\#C(x)}\\sum_{c \\in C(x)} c$ be the average of those elements.\nNote that the map $a\\colon K \\to K$ has finite image.\nUsing Lemma~\\ref{lem:average}, we find an injective map $\\alpha$ from the image of $a$ to $\\mathrm{RV}^k$.\nNow we define\n\\[\nf(x) := \\begin{cases}\n(\\alpha(a(x)), \\operatorname{rv}_\\lambda(x - a(x))) & \\text{if } |x - a(x)| \\ge \\mu(x)\\cdot\\lambda\\\\\n(\\alpha(a(x)), \\operatorname{rv}_\\lambda(0))        & \\text{if } |x - a(x)| < \\mu(x)\\cdot\\lambda.\n        \\end{cases}\n\\]\nTo see that this works, first note that if $x, x' \\in K$ lie in the same ball $\\lambda$-next to $C$, then\n$C(x) = C(x')$ and hence $a(x) = a(x')$, so it remains to check that the restriction of this $f$ to $\\{x \\in K \\mid a(x) = a_0\\}$\nhas the right fibers, for each fixed $a_0$ in the image of $a$. This is a straight-forward computation after noting that for $x_1, x_2$ as in the first case of the definition of $f$ and satisfying $a(x_1) = a(x_2) = a_0$, we have $\\operatorname{rv}_\\lambda(x_1 - a(x_1)) = \\operatorname{rv}_\\lambda(x_2 - a(x_2))$ if and only if $\\operatorname{rv}_\\lambda(x_1 - c) = \\operatorname{rv}_\\lambda(x_2 - c)$ for each $c\\in C$.\n\\end{proof}\n\n\\begin{remark}\\label{rem:InextFam}\nIn Lemma~\\ref{lem:InextFam}, we can also find a map $f$ with codomain $\\mathrm{RV}_\\lambda^{k+1}$ instead of $\\mathrm{RV}^k \\times \\mathrm{RV}_\\lambda$; indeed, in Lemma~\\ref{lem:average} and its proof, $\\mathrm{RV}$ can be replaced by $\\mathrm{RV}_\\lambda$ everywhere.\n\\end{remark}\n\n\n\\subsection{Henselianity of the valued field $K$}\\label{sec:hens}\n\nAs an analogue of o-minimal fields being real closed, in this subsection, we prove that any equi-characteristic zero valued field satisfying the weak form of Hensel minimality from the previous subsection (in any language expanding ${\\mathcal L}_\\val$) is Henselian. This is one reason why we call our notion ``Hensel minimality''.\n\n\nA collection of balls is called nested if for any two balls in the collection, one is contained in the other.\n\n\\begin{lem}[Definable spherical completeness]\\label{lem:intersection}\nUnder Assumption~\\ref{ass:no-ctrl}, let $\\{B_q \\mid q\\in Q\\}$ be a $\\emptyset$-definable family of nested balls in $K$,\nfor some non-empty definable set $Q$ in an arbitrary imaginary sort. Then the intersection $\\bigcap_{q \\in Q} B_q$ is non-empty.\n\\end{lem}\n\\begin{proof}\nFirst we suppose that $Q \\subset \\Gamma^\\times_K$ and that each $B_q$ is an open ball of radius $q$.\nBy Corollary~\\ref{cor:prep} (and Remark~\\ref{rem:prep}), there exists a finite set $C$ preparing the family of balls $B_q$.\nNow one checks that at least one of the following two situations occurs.\nFirstly: For each $q\\in Q$, the intersection of $C$ with $B_q$ is non-empty. Secondly: The set $Q$ has a minimum $q_0\\in Q$. (Indeed, suppose that the intersection of $C$ with $B_{q_0}$ is empty for some $q_0\\in Q$; then $Q$ contains no $q < q_0$, since $B_q$ would not be $1$-prepared by $C$.) In both situations, the lemma follows.\n\nFinally, we reduce the general case to the case that $Q\\subset \\Gamma^\\times_K$ and that each $B_q$ is an open ball of radius $q$. To this end, for $\\gamma\\in\\Gamma^\\times_K$, let $B'_{\\gamma}$ be the (necessarily unique) open ball of radius $\\gamma$ containing some $B_q$ ($q \\in Q$) if such a ball exists, and let it be the empty set otherwise. Then it is clear that the non-empty $B'_{\\gamma}$ form a nested definable family of open balls. Moreover, the intersection of the non-empty $B'_\\gamma$ equals the intersection of the $B_q$ (since each $B_q$ is equal to the intersection of all open balls containing $B_q$).\n\\end{proof}\n\n\n\n\n\\begin{thm}[Hensel minimality implies Henselian]\\label{thm:hens}\nSuppose that $K$ is a valued field of equi-characteristic $0$, which, when considered as a structure in a language ${\\mathcal L} \\supset {\\mathcal L}_\\val$, satisfies\nAssumption~\\ref{ass:no-ctrl}. Then $K$ is Henselian.\n\\end{thm}\nIf ${\\mathcal L}$ is the pure valued field language, Corollary~\\ref{cor:ex:equi} implies the converse. Combining, we have, for $K$ of equi-characteristic $0$:\n$K$ is Henselian if and only if $K$ satisfies Assumption~\\ref{ass:no-ctrl}, if and only if\n$\\operatorname{Th}_{{\\mathcal L}_\\val}(K)$ is $\\omega$-h-minimal.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:hens}]\nLet $P\\in{\\mathcal O}_K[X]$ be a polynomial such that $P(0)\\in{\\mathcal M}_K$ and $P'(0)\\in{\\mathcal O}^{\\times}_K$; we need to prove that $P$ has a root in ${\\mathcal M}_K$. (Note that uniqueness of such a root then follows automatically.)\nThe idea is to use ``Newton approximation'' as in the usual proof of Hensel's lemma for complete discretely valued fields, but where complete and discretely valued is replaced by definably spherically complete.\n\nTo make this formal, we suppose that $P$ has no root in ${\\mathcal M}_K$ and we set $B_x := B_{\\le |P(x)|}(x)$ for $x \\in {\\mathcal M}_K$. We will prove that (a) all these balls form a chain under inclusion and (b) that an element in the intersection of all those balls (which is non-empty by Lemma~\\ref{lem:intersection}) is, after all, a root of $P$.\n\n(a) Let $x_1, x_2 \\in {\\mathcal M}_K$ be given and set $\\varepsilon := x_2 - x_1$. To see that the balls $B_{x_1}$ and $B_{x_2}$ are not disjoint, we verify that $|\\epsilon| \\le \\max\\{|P(x_1)|, |P(x_2)|\\}$. Taylor expanding $P$ around $x_1$ yields\n\\begin{equation}\\label{eq:tay}\n|P(x_1+\\varepsilon) - P(x_1) - \\varepsilon P'(x_1)| \\le |\\varepsilon^2|,\n\\end{equation}\nwhich, together with $|P'(x_1)| = 1$, implies $|\\epsilon| \\le \\max\\{|P(x_1)|, |P(x_1 + \\epsilon)|\\}$.\n\n(b) Let $x_1$ be in the intersection $\\bigcap_{x \\in {\\mathcal M}_K} B_x$ and suppose that $P(x_1) \\ne 0$. Then (\\ref{eq:tay}) with $\\varepsilon := -  \\frac{P(x_1)}{P'(x_1)}$ implies $|P(x_1 + \\varepsilon)| \\le |\\varepsilon^2| < |\\varepsilon|$ and hence $x_1 \\notin B_{x_1 + \\varepsilon}$, contradicting our choice of $x_1$. \\end{proof}\n\n\n\\begin{remark}\nLemma~\\ref{lem:intersection} implies a ``definable Banach Fixed Point Theorem'' (exactly in the form of \\cite[Lemma~2.32]{Halup}, and with the same proof).\nThe above proof of Theorem~\\ref{thm:hens} can be considered as applying that Fixed Point Theorem to\nthe map ${\\mathcal M}_K \\to {\\mathcal M}_K, x \\mapsto x - P(x)\/P'(x)$.\n\\end{remark}\n\n\\private{This subsection directly works in finitely ramified mixed char. With infinite ramification, I'm unsure.}\n\n\\subsection{Preparing families}\n\\label{sec:Prepfam}\n\nIn this subsection, we fix an integer $\\ell \\ge 0$, and we will mostly assume that $\\operatorname{Th}(K)$ is $\\ell$-h-minimal. (If $\\operatorname{Th}(K)$ is $\\omega$-h-minimal, then the results hold for every $\\ell$.)\nWe will see that $\\ell$-h-minimality does not only imply that we can\nprepare definable subsets of $K$ (by finite sets $C$), but also\nvarious other kinds of definable objects that ``live in $K \\times \\mathrm{RV}^k \\times \\mathrm{RV}_\\lambda^\\ell$''.\n\nWe write $\\mathrm{RV}_\\bullet$ for the disjoint union $\\bigcup_{\\lambda \\le 1} \\mathrm{RV}_{\\lambda}$; recall that $\\mathrm{RV}_\\bullet$ is an imaginary sort.\n\n\\begin{def-prop}[Preparing families]\\label{prop:uniform\nAssume that $\\operatorname{Th}(K)$ is $\\ell$-h-minimal and that $A$ is a subset of $K$.\nFor any integer $k>0$\nand any $(A \\cup \\mathrm{RV})$-definable set\n$$\nW\\subset K\\times \\mathrm{RV}^k\\times \\mathrm{RV}_{\\bullet}^\\ell,\n$$\nthere exists a finite non-empty $A$-definable set $C$ such that for every $\\lambda \\le 1$ in $\\Gamma^\\times_K$ and every ball $B$ $\\lambda$-next to $C$, the fiber $W_{x,\\lambda} := \\{\\xi \\in \\mathrm{RV}^k\\times \\mathrm{RV}_{\\lambda}^\\ell \\mid (x,\\xi)\\in W\\}$\ndoes not depends on $x$ when $x$ runs over $B$. In other words, for all $x,x'\\in B$,\none has $W_{x,\\lambda} = W_{x',\\lambda}$.\n\nWe say that $C$ \\emph{uniformly prepares} $W$. We sometimes also say that $C$ uniformly prepares the family $W_\\xi \\subset K$, $\\xi \\in \\mathrm{RV}^k\\times \\mathrm{RV}_{\\bullet}^\\ell$; indeed, note that the above statement is equivalent to: the set $C$ $\\lambda$-prepares $W_\\xi$ for each $\\xi \\in \\mathrm{RV}^k\\times \\mathrm{RV}_{\\lambda}^\\ell$ and for each $\\lambda$.\n\\end{def-prop}\n\n\\begin{proof}\nFor each $\\lambda$ and each $\\xi\\in \\mathrm{RV}^k\\times \\mathrm{RV}_{\\lambda}^\\ell$, let $C_\\xi$ be a finite $A$-definable set $\\lambda$-preparing $W_\\xi$. By a usual compactness argument (see Remark~\\ref{rem:compact} below), we may suppose that $C := \\bigcup_\\xi C_\\xi$ is finite and  $A$-definable. It prepares each $W_\\xi$ and hence also $W$.\n\\end{proof}\n\n\\begin{remark}\\label{rem:compact}\nIn the above proof, we used a compactness argument which we will be using (in variants) many times in this paper. We give some details once: First recall that by Lemma~\\ref{lem:prep:def}, ``preparing'' is a definable condition. In particular, the set\n\\[\n\\Xi_{\\xi} := \\bigcup_{\\lambda \\le 1}\\{\\xi' \\in  \\mathrm{RV}^k\\times \\mathrm{RV}_{\\lambda}^\\ell \\mid\nC_\\xi\\text{ $\\lambda$-prepares $W_{\\xi'}$}\n\\}\\subseteq \\mathrm{RV}^k\\times \\mathrm{RV}_\\bullet^\\ell\n\\]\nis $A$-definable.\nSince $\\xi \\in \\Xi_{\\xi}$, the union of all $\\Xi_{\\xi}$ is equal to $\\mathrm{RV}^k\\times \\mathrm{RV}_{\\bullet}^\\ell$, and then compactness implies that finitely many sets $\\Xi_{\\xi_i}$ suffice to cover everything. Now \\(C := \\bigcup_i C_{\\xi_i}\\) is a finite \\(A\\)-definable set which prepares every \\(W_\\xi\\).\n\\end{remark}\n\n\\begin{remark}\\label{rem:finiterange:set}\nIn Proposition-Definition~\\ref{prop:uniform}, we could also replace $\\mathrm{RV}^k\\times \\mathrm{RV}_{\\bullet}^\\ell$ by $\\bigcup_{\\lambda \\le 1}\\big((\\mathrm{RV}^k\\times \\mathrm{RV}_{\\lambda}^\\ell)\/\\mathord{\\sim_\\lambda}\\big)$, for any $\\emptyset$-definable family of equivalence relations $(\\sim_\\lambda)_{\\lambda\\le 1}$.\nIndeed, in that case, just apply the original version of the proposition to the preimage of $W$ in $K\\times \\mathrm{RV}^k\\times \\mathrm{RV}_{\\bullet}^\\ell$.\nIn particular, $W$ can additionally use (arbitrarily many) $\\Gamma_K$-coordinates, since $\\Gamma_K$ is a quotient of $\\mathrm{RV}$.\n\\end{remark}\n\n\nNote that the proposition allows us to ``prepare any definable object involving one $K$-coordinate, at most $\\ell$ $\\mathrm{RV}_\\bullet$-coordinates and arbitrarily many $\\mathrm{RV}^{\\mathrm{eq}}$-coordinates''. For example,\nwe can $1$-prepare functions $\\mathrm{RV}^k \\to K$ or functions $K \\to \\mathrm{RV}^k$, meaning that we $1$-prepare their graphs.\n\nWe formulate some useful special cases of Proposition-Definition~\\ref{prop:uniform} as corollaries:\n\n\\begin{cor}[Preparing families]\\label{cor:prep}\nAssume that $\\operatorname{Th}(K)$ is $\\ell$-h-minimal, that $A$ is a subset of $K$ and that $\\lambda \\le 1$ is an element of $\\Gamma^\\times_K$. For any $k>0$ and any $(A \\cup \\mathrm{RV})$-definable set\n$$\nW\\subset K\\times \\mathrm{RV}^k\\times \\mathrm{RV}_{\\lambda}^\\ell,\n$$\nthere exists a finite non-empty $A$-definable set $C$\n$\\lambda$-preparing $W$ in the following sense: For any ball $B$ $\\lambda$-next to $C$, the fiber $W_x \\subset  \\mathrm{RV}^k\\times \\mathrm{RV}_{\\lambda}^\\ell$\ndoes not depends on $x$ when $x$ runs over $B$.\n\\end{cor}\n\n\\begin{remark}\nNote that the condition on $C$ is equivalent to: $C$\n$\\lambda$-prepares $W_\\xi \\subset K$ for every $\\xi \\in \\mathrm{RV}^k \\times \\mathrm{RV}^\\ell_\\lambda$. In the sequel, we will use both points of view of what $\\lambda$-preparing $W$ means.\n\\end{remark}\n\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:prep}]\nClear.\n\\end{proof}\n\n\n\\begin{cor}[$\\mathrm{RV}$-unions stay finite]\\label{cor:finiterange:set} Assume that $\\operatorname{Th}(K)$ is $\\ell$-h-minimal.\nFor any $k \\ge 0$ and any $\\emptyset$-definable set $W\\subset \\mathrm{RV}^k \\times \\mathrm{RV}_\\bullet^\\ell \\times K$ such that the fiber $W_\\xi\\subset K$ is finite for each $\\xi\\in \\mathrm{RV}^k \\times \\mathrm{RV}_\\bullet^\\ell$, the union $\\bigcup_{\\xi} W_\\xi$ is also finite.\n\\end{cor}\n\n\\begin{proof}\nBy Proposition-Definition \\ref{prop:uniform}, we find \\(C\\) such that, for all \\(\\lambda\\) and \\(\\xi\\in \\mathrm{RV}^k\\times\\mathrm{RV}_\\lambda^\\ell\\), $W_\\xi$ is $\\lambda$-prepared by \\(C\\). Since \\(W_\\xi\\) is finite, \\(W_\\xi\\subset C\\) and hence \\(\\bigcup_\\xi W_\\xi \\subset C\\) is finite.\n\\end{proof}\n\n\\begin{cor}[Finite image in $K$]\\label{cor:finiterange}\nAssume that $\\operatorname{Th}(K)$ is $\\ell$-h-minimal and let $\\lambda \\le 1$ be in $\\Gamma^\\times_K$. The image of any definable (with parameters) function $f\\colon \\mathrm{RV}^k \\times \\mathrm{RV}_\\bullet^\\ell \\to K$ for any $k \\ge 0$ is finite.\n\\end{cor}\n\n\\begin{proof}\nApply Corollary \\ref{cor:finiterange:set} to the graph of $f$.\n\\end{proof}\n\n\\begin{remark}\nLater, we will develop dimension theory for $1$-h-minimal theories, which then implies that definable functions $f\\colon \\mathrm{RV}^k \\times \\mathrm{RV}_\\lambda^\\ell \\to K$ have finite image for arbitrary $\\ell$, see Section~\\ref{sec:dim}.\nHowever, for the moment, we content ourselves with this version.\n\\end{remark}\n\n\\begin{remark}\\label{rem:prep}\nRemark~\\ref{rem:finiterange:set} also applies to Corollaries \\ref{cor:prep} to \\ref{cor:finiterange}. In particular, we can additionally allow (arbitrarily many) $\\Gamma_K$-coordinates in $W$\n(in \\ref{cor:prep}, \\ref{cor:finiterange:set}) and in the range of $f$ (in \\ref{cor:finiterange}).\n\\end{remark}\n\n\\begin{cor}[Removing $\\mathrm{RV}$-parameters]\\label{cor:acl}\nAssume that $\\operatorname{Th}(K)$ is $0$-h-minimal. For any $A \\subset K$ and any finite $(A \\cup \\mathrm{RV}^{\\mathrm{eq}})$-definable set $C \\subset K$, there exists a finite $A$-definable set $C' \\subset K$ containing $C$. In other words, $\\acl_K(A \\cup \\mathrm{RV}^{\\mathrm{eq}}) = \\acl_K(A)$.\n\\end{cor}\n\n\\begin{proof}\nAdd constants for $A$ to the language.\nWe have $C = W_ {\\xi_0}$ for some $\\emptyset$-definable $W \\subset K \\times \\mathrm{RV}^k$ and some $\\xi_0 \\in \\mathrm{RV}^k$. We may assume  that all fibers $W_{\\xi}$ have cardinality at most the cardinality of $C$. Let $C'$ be their union, which is finite by Corollary~\\ref{cor:finiterange:set}.\n\\end{proof}\n\n\\begin{remark}\\label{rem:acl}\nMany results in this paper are stated in the form: For some $\\emptyset$-definable object $X$, there exists a finite $\\emptyset$-definable set $C \\subset K$ which in some sense ``prepares'' $X$. Using Lemma~\\ref{lem:addconst}, we can always allow $X$ to be $(A \\cup \\mathrm{RV})$-definable, for $A \\subset K$, and get an $(A \\cup \\mathrm{RV})$-definable $C$. By applying Corollary~\\ref{cor:acl}, we then even get a finite $A$-definable $C$ ``preparing'' $X$. (In this reasoning, the only assumption about ``preparation'' is that if $C$ prepares $X$, then so does any set containing $C$; this will generally be the case.)\n\\end{remark}\n\nWe end this subsection by noting that $\\mathrm{RV}$ is stably embedded in a strong sense (namely with the $\\mathrm{RV}$-parameters being in the definable closure of the original parameters):\n\n\\begin{prop}[Stable embeddedness of $\\mathrm{RV}$]\\label{prop:stab}\nAssume that $\\operatorname{Th}(K)$ is $0$-h-minimal. Then\n$\\mathrm{RV}$ is stably embedded in the following strong sense: Given any $A \\subset K$, every $A$-definable set $X \\subset \\mathrm{RV}^n$\nis already $\\dcl_{\\mathrm{RV}}(A)$-definable.\n\\end{prop}\n\n\\begin{proof\nWe may assume that $A$ is finite; we do an induction on the cardinality of $A$.\n\nLet $A = \\hat A \\cup \\{a\\}$. Then we have an $\\hat A$-definable set $Y \\subset K \\times \\mathrm{RV}^n$ such that\n$X$ is equal to the fiber $Y_a \\subset \\mathrm{RV}^n$.\nBy applying Corollary~\\ref{cor:prep} to $Y$, we find a finite $\\hat A$-definable set $C \\subset K$ such that\neither $a \\in C$, or for every $a' \\in K$ in the same ball $1$-next to $C$ as $a$, we have $Y_{a'} = Y_a$.\nUsing Lemma~\\ref{lem:InextFam}, we find an $\\hat A$-definable map $f\\colon K \\to \\mathrm{RV}^k$ (for some $k$)\nwhose fibers are exactly the elements of $C$ and the balls $1$-next to $C$. In particular, the set $X = Y_a$ is definable\nusing $\\hat A$ and $f(a)$ as parameters. Thus we have $X = Z_{f(a)}$ for some $\\hat A$-definable set\n$Z \\subset \\mathrm{RV}^k \\times \\mathrm{RV}^n$. By induction, $Z$ is $\\dcl_{\\mathrm{RV}}(\\hat A)$-definable, so $X$ is $\\dcl_{\\mathrm{RV}}(\\hat A)\\cup \\{f(a)\\}$-definable and hence $\\dcl_{\\mathrm{RV}}(A)$-definable.\n\\end{proof}\n\n\n\\begin{remark}\\label{rem:stab}\nIf $\\operatorname{Th}(K)$ is $\\omega$-h-minimal, there are also various variants of Proposition~\\ref{prop:stab} involving $\\mathrm{RV}_\\lambda$, with similar proofs. For example, building on Remark~\\ref{rem:InextFam} instead of Lemma~\\ref{lem:InextFam}, one obtains that\nany $A$-definable subset of $\\prod_i \\mathrm{RV}_{\\lambda_i}$ (for $A \\subset K$) is\n$\\dcl_{\\mathrm{RV}_{\\lambda}}(A)$-definable with $\\lambda = \\min_i \\lambda_i$.\n\\end{remark}\n\n\n\n\n\n\n\\subsection{Definable functions\n\\label{sec:fctn}\n\nWe continue assuming that $K$ is an equi-characteristic $0$ valued field, and we now assume that $\\operatorname{Th}(K)$ is $1$-h-minimal (unless specified otherwise). Under those assumptions, we now prove first basic properties of definable functions in one variable; in particular, we already obtain a weak version of the Jacobian Property  (Lemma~\\ref{lem:gammaLin}) and simultaneous domain and image preparation (Proposition~\\ref{prop:range}).\n\n\\private{In case we want to check again how much 1-h-min we really need: In this section, 1-h-min is used only as follows: (1) in \\ref{lem:fin-inf}; Use \\ref{cor:finiterange:set};\n(2) in \\ref{lem:balltosmall}: use \\ref{cor:prep}.}\n\n\nThe first result is key to dimension theory (though in our proofs of dimension theory,\nthis will only be used indirectly, namely in the proof of Proposition \\ref{prop:b-min}).\n\\begin{lem}[Basic preservation of dimension]\\label{lem:fin-inf}\nAssume (as convened for the whole Section \\ref{sec:fctn}) that  $\\operatorname{Th}(K)$ is $1$-h-minimal.\nLet $f\\colon K \\to K$ be a definable function.\nThen there are only finitely many function values which are taken infinitely many times.\n\\end{lem}\n\n\n\\begin{proof\nWe may assume $f$ to be $\\emptyset$-definable (say, after adding enough parameters from $K$ to the language).\nSuppose $f$ takes infinitely many values $y$ infinitely many times.\nThen for each such $y$,\n$f^{-1}(y)$ contains a ball (by Lemma~\\ref{lem:finite}). Thus, letting $X \\subset K$ be the set of points where $f$ is locally constant, $f(X)$ is still infinite.\n\nLet $W\\subset K\\times \\Gamma^\\times_K$ consist of those $(x,\\lambda)$ such that $f$ is constant on $B_{<\\lambda}(x)$. This set $W$ is $\\emptyset$-definable, so we find a finite set $C$ $1$-preparing $W$ (in the sense of Corollary \\ref{cor:prep}).\nBy enlarging $C$, we may moreover assume that $C$ also $1$-prepares $X$.\n\nFrom the fact that $f(X)$ is infinite, we can deduce that there exists a ball $B_0 \\subset X$ $1$-next to $C$ such that $f(B_0)$ is still infinite. Indeed, letting $g\\colon K \\to \\mathrm{RV}^k$ be a map whose fibers are the balls $1$-next to $C$ (using Lemma~\\ref{lem:InextFam}), if $f(g^{-1}(\\xi))$ would be finite for every $\\xi \\in g(X)$, then so would be $f(X)$ (by Corollary~\\ref{cor:finiterange:set}).\n\nChoose $x \\in B_0$ and $\\lambda_0 \\in \\Gamma^\\times_K$ such that $f$ is constant on $B_{<\\lambda_0}(x)$. Since $C$ 1-prepares $W$, $f$ is constant on $B_{<\\lambda_0}(x')$ for every $x' \\in B_0$.\n\nSet $\\lambda_1 := \\rad_{\\mathrm{op}}(B_0)\/\\lambda_0$. Then the family $F_1$ of open balls of radius $\\lambda_0$ contained in $B_0$ can be definably parametrized by a subset of $\\mathrm{RV}_{\\lambda_1}$\n(using some parameters). Indeed, if we fix $c \\in K$ such that $B_0$ is $1$-next to $c$, then each member of $F_1$ is of the form $c + \\operatorname{rv}_{\\lambda_1}^{-1}(\\xi)$ for some\n$\\xi \\in \\mathrm{RV}_{\\lambda_1}$. We define $F_2$ to be the family of $f(B)$, for $B$ in $F_1$. Then each family member of $F_2$ is a singleton, yet their union is infinite, contradicting Corollary \\ref{cor:finiterange:set}.\n\\end{proof}\n\n\n\nUsing this, we obtain that definable functions are (in a strong sense) locally constant or injective:\n\n\\begin{lem}[Piecewise constant or injective]\\label{lem:loc-inj}\nFor every $\\emptyset$-definable map $f\\colon K \\to K$, there exists a finite $\\emptyset$-definable set $C$ such that\nfor every ball $B$ $1$-next to $C$, $f$ is either constant or injective on $B$.\n\\end{lem}\n\n\\begin{proof}\nFirst, consider the set $Y_\\infty$ of $y \\in K$ such that $f^{-1}(y)$ is infinite. By Lemma~\\ref{lem:fin-inf}, this set $Y_\\infty$ is finite, so we can find\na finite $\\emptyset$-definable set $C$ $1$-preparing $f^{-1}(y)$ for each $y \\in Y_\\infty$. Indeed, for each $y \\in Y_\\infty$, one finds a finite $y$-definable\nset $C_y$ $1$-preparing $f^{-1}(y)$ (uniformly in $y$), and one lets $C$ be the union of the sets $C_y$.\n\nFor each $y \\in K \\setminus Y_\\infty$, the set $f^{-1}(y)$ is finite, so by Lemma~\\ref{lem:average},\nthere exists a $\\emptyset$-definable\nfamily of injective functions $g_y\\colon f^{-1}(y) \\to \\mathrm{RV}^k$ for some $k \\ge 1$. For convenience, we set $g_y(x) := 0$ if $y \\in Y_\\infty$ and $x \\in f^{-1}(y)$,\nso that we can define a function $h\\colon K \\to \\mathrm{RV}^k$ by $h(x) := g_{f(x)}(x)$.\nWe then enlarge our above set $C$ (using  Corollary~\\ref{cor:prep}) so that it also $1$-prepares (the graph of) $h$.\n\nWe claim that this set $C$ is as desired, so let $B$ be a ball $1$-next to $C$.\nIf $f(B) \\cap Y_\\infty \\ne \\emptyset$, then $B$ is contained in one of the sets $f^{-1}(y)$, for $y \\in Y_\\infty$, and hence $f$ is constant on $B$.\nOtherwise, we use that $h$ is constant on $B$ to deduce that $f$ is injective on $B$. Indeed, $f(x_1) = f(x_2) = y$ for some $y \\in K \\setminus Y_\\infty$\nimplies $h(x_1) = g_{y}(x_1) = g_{y}(x_2) = h(x_2)$,\nso injectivity of $h$ on $f^{-1}(y)$ implies $x_1 = x_2$.\n\\end{proof}\n\nThe next lemma says that a definable function sends most open balls either to points or to open balls.\n\n\\begin{lem}[Images of balls are balls]\\label{lem:open-to-open} Let $f\\colon K \\to K$ be a $\\emptyset$-definable function.\nThere exists\na $\\emptyset$-definable finite set $C \\subset K$ such that for every open ball $B$ disjoint from $C$,\n$f(B)$ is either a point or an open ball.\n\\end{lem}\n\n\\begin{proof}\nDefine $W \\subset K \\times \\Gamma^\\times_K$ to consist of those $(x, \\lambda)$ for which the open ball $B_{<\\lambda}(x)$ ``has a good image'', i.e., $f(B_{<\\lambda}(x))$ is a singleton or an open ball.\nBy  Corollary \\ref{cor:prep}, we find a finite $\\emptyset$-definable subset $C_0$ of $K$ such that for any ball $B$ $1$-next to $C_0$, the fiber $W_x \\subset \\Gamma^\\times_K$ does not depend on $x \\in B$.\n\nFix a ball $B_0$ $1$-next to $C_0$. We first prove that any open ball $B$ strictly contained in $B_0$ has a good image.\nSuppose otherwise, i.e., that $B_1$ is an open ball strictly contained in $B_0$ with bad image.\nThen the fact that $W_x$ does not depend on $x \\in B_0$ implies that every translate of $B_1$ contained in $B_0$ also has bad image.\nWe can find an infinite definable (with parameters) family $F_1$ consisting of such translates of $B_1$ and parameterized by\na subset of $\\mathrm{RV}$. Indeed, the sets $x_0 + \\operatorname{rv}^{-1}(\\xi)$ form such a family for a suitable $x_0 \\in K$ and when $\\xi$ runs over a suitable subset of $\\mathrm{RV}$.\n\nConsider the family $F_2$ of the sets $f(B)$ for $B$ in $F_1$,\nuse Corollary \\ref{cor:prep} to find a finite set $D \\subset K$ $1$-preparing the family $F_2$, and let $g\\colon K \\to \\mathrm{RV}^k$ be a function whose fibers are the balls $1$-next to $D$ and the individual points of $D$ (as obtained using Lemma~\\ref{lem:InextFam}).\nSince none of the balls $B$ in $F_1$ are good, none of the sets $f(B)$ in $F_2$ are exactly a point or an open ball, so each $f(B)$ consists of several fibers of $g$; in other words, $g \\circ f\\colon K \\to \\mathrm{RV}^k$ is non-constant on every $B$ in $F_1$.\n\nNow we get a contradiction by applying Corollary \\ref{cor:prep} to the graph of $g \\circ f$. Indeed, any set $C'$ $1$-preparing\nthat graph would have to contain at least one point in each ball $B$ from $F_1$ (since $g \\circ f$ is not constant on $B$), so $C'$ cannot be finite.\nThis finishes the proof that balls strictly contained in $B_0$ have good image.\n\nThe only problematic open balls left (i.e., which are disjoint from $C_0$ and might have bad image) are the ones $1$-next to $C_0$. To get hold of those, we run a similar argument as above: We let $F_1$ be the family of balls $1$-next to $C_0$; we find a finite set $D$ $1$-preparing $f(B)$ for each\n$B$ in $F_1$, and we define $g\\colon K \\to \\mathrm{RV}^k$ as before, so that in particular\nif $g \\circ f$ is constant on a ball $B$ $1$-next to $C_0$, then that ball $B$ has good image.\n\nNow we find a finite set $C'$ $1$-preparing the graph of $g \\circ f$ and we set $C := C_0 \\cup C'$. In this way, among the balls $1$-next to $C_0$, all those which have bad image are not disjoint from $C$.\nNote that since this time, $F_1$ is $\\emptyset$-definable, and hence so are $D$, $g$ and $C'$.\n\\end{proof}\n\n\n\nThe following is a key technical lemma, which serves later in the proof of the Jacobian Property. The main point of the statement is that a definable map cannot scale all small balls by one factor and all big balls by a different factor.\n\n\\begin{lem}[Preservation of scaling factor]\\label{lem:balltosmall}\nLet $B$ be either ${\\mathcal O}_K$ or ${\\mathcal M}_K$ and\nlet $f\\colon B\\to K$ be a $\\emptyset$-definable function.\nSuppose that there are $\\alpha$ and $\\beta$ in $\\Gamma^\\times_K$\nwith\n$\\alpha<1$\nsuch that for every open ball $B' \\subset B$ of radius $\\alpha$, $f(B')$ is contained in an open ball of radius $\\beta$. Then the following hold.\n\\begin{enumerate}\n \\item\nThe image $f(B)$ is contained in a finite union of closed balls of radius $\\beta\/\\alpha$.\n\\item\nIf we moreover assume that $B = {\\mathcal M}_K$ and that $f(B)$ is an open ball, then $\\rad_{\\mathrm{op}} f(B) \\le \\beta\/\\alpha$.\n\\end{enumerate}\n\\end{lem}\n\n\n\\begin{proof}\n(1)\nLet $B_\\xi$ be the family of open balls of radius $\\alpha$ contained in $B$, (definably) parametrized by $\\xi$ running over a set $\\Lambda\\subset \\mathrm{RV}_\\alpha$.\nBy Corollary \\ref{cor:prep}, there is a finite set $C$ such that for each $\\xi\\in\\Lambda$, the set $f(B_\\xi)$ is $\\alpha$-prepared by $C$.\nConsider any open ball $B'$ which\nis $\\alpha$-next to $C$. Then, since $C$ prepares $f(B_\\xi)$ for every $\\xi\\in\\Lambda$, one has either\n$$\nB'\\subset f(B_\\xi), \\mbox{ or, } B'\\cap  f(B_\\xi)=\\emptyset.\n$$\nHence, if the radius of the open ball $B'$ is larger than $\\beta$, then $B'\\cap  f(B_\\xi)$ is empty for all $\\xi\\in\\Lambda$. (Indeed, by assumption, $f(B_\\xi)$ is contained in an open ball of radius $\\beta$, so $f(B_\\xi)$ cannot contain the larger ball $B'$.)\n\nThus, we find that $f(B)$ is contained in the union of $C$ with those open balls $\\alpha$-next to $C$ that have radius at most $\\beta$. This union\nequals the union over $c\\in C$ of the closed balls $B_{\\le \\beta\/\\alpha}(c)$. This proves (1).\n\n(2)\nIf the value group is dense, then (2) follows from (1) using that the largest open balls contained in a finite\nunion of closed balls of radius $\\beta\/\\alpha$ have radius $\\beta\/\\alpha$.\n\nIf the value group is discrete (see just above Remark \\ref{rem:rad-op}), we apply Part (1) to $g(x) := f(\\varpi x)\\colon {\\mathcal O}_K \\to K$, where $\\varpi \\in K$ is a\nuniformizing element. Since $g$ sends open balls of radius $|\\varpi|^{-1} \\cdot \\alpha$ to open balls of radius $\\beta$,\nwe obtain that\n$f({\\mathcal M}_K) = g({\\mathcal O}_K)$ is contained in a finite union of closed balls of radius $|\\varpi| \\cdot \\beta\/\\alpha$, which\nis the same as a finite union of open balls of radius $\\beta\/\\alpha$. Now the claim follows from the assumption that\n$f({\\mathcal M}_K)$ is itself an open ball.\n\\end{proof}\n\nBy combining various previous lemmas, we already obtain a weak form of the Jacobian Property.\n\n\\begin{lem}[Valuative Jacobian Property]\\label{lem:gammaLin}\nFor every $\\emptyset$-definable function $f\\colon K \\to K$, there exists a finite $\\emptyset$-definable set $C \\subset K$ such that\nfor every ball $B$ $1$-next to $C$, we have the following:\nEither $f$ is constant on $B$, or there exists a $\\mu_B \\in \\Gamma^\\times_K$ such that\n\\begin{enumerate}\n \\item\nfor every open ball $B' \\subset B$, $f(B')$ is an open ball of radius $\\mu_B \\cdot\\rad_{\\mathrm{op}}(B')$;\n \\item for every $x_1, x_2 \\in B$, we have $|f(x_1) - f(x_2)| = \\mu_B\\cdot |x_1 - x_2|$.\n\\end{enumerate}\n\\end{lem}\n\n\n\\begin{proof}\nChoose $C$ $\\emptyset$-definable such that for every ball $B$ $1$-next to $C$, we have:\n\\begin{itemize}\n \\item $f$ is constant or injective on $B$ (using Lemma~\\ref{lem:loc-inj});\n \\item $f(B')$ is a point or an open ball for every open ball $B' \\subset B$ (using Lemma~\\ref{lem:open-to-open}).\n\\end{itemize}\nMoreover, we assume (using Corollary~\\ref{cor:prep}) that $C$ $1$-prepares the graph of the function $r\\colon K \\times \\Gamma^\\times_K \\to \\Gamma_K$\ndefined by\n\\[r(x,\\lambda) = \\begin{cases}\n                  \\rad_{\\mathrm{op}}(f(B_{<\\lambda}(x))) & \\text{if } f(B_{<\\lambda}(x)) \\text{ is an open ball}\\\\\n                  0 & \\text{otherwise}.\n                 \\end{cases}\n\\]\nWe claim that this $C$ is as desired, so fix a ball $B$ $1$-next to $C$ for the remainder of the proof.\nNote that for $x$ running over $B$, the function $r(x,\\cdot)$ is independent of $x$,\nso from now on we write $r(\\lambda)$ instead of $r(x, \\lambda)$.\n\nIf $f$ is constant on $B$, there is nothing to do, so we may assume that $f$ is injective on $B$.\nWe claim that the lemma holds with $\\mu_B := \\rad_{\\mathrm{op}}(f(B))\/\\rad_{\\mathrm{op}}(B)$. (Note that indeed, $f(B)$ is an open ball.)\n\n(1) Fix $\\alpha \\in \\Gamma_K$ with $0 < \\alpha < \\rad_{\\mathrm{op}}(B)$.\nAny open ball $B' \\subset B$ of radius $\\alpha$ is sent to an open ball $f(B')$ of radius $r(\\alpha)$.\nBy applying Lemma~\\ref{lem:balltosmall} (to a suitably rescaled function), we deduce that the radius of the open ball $f(B)$ is at most\n$\\rad_{\\mathrm{op}}(B)\\cdot r(\\alpha)\/\\alpha$; this implies $r(\\alpha)\/\\alpha \\ge \\mu_B$.\n\nTo get the other inequality, namely $r(\\alpha)\/\\alpha \\le \\mu_B$, we apply the same argument to the inverse function\n$f^{-1}\\colon f(B) \\to B$. This inverse is well-defined since $f$ is injective on $B$, so it remains\nto verify that $f^{-1}$ sends open balls $B'' \\subset f(B)$ of radius $r(\\alpha)$ to open balls of radius $\\alpha$.\nIndeed, choose any $x \\in f^{-1}(B'')$; then $f(B_{< \\alpha}(x)) = B''$\nand hence $f^{-1}(B'') = B_{< \\alpha}(x)$.\n\n(2) For every $x_1, x_2 \\in B$, (1) implies $|f(x_1) - f(x_2)| \\le \\mu_B\\cdot |x_1 - x_2|$. Indeed,\napplying (1) to a ball of the form $B_{<\\alpha}(x_1)$ with $|x_1 - x_2| < \\alpha \\le \\rad_{\\mathrm{op}}(B)$\nyields $|f(x_1) - f(x_2)| < \\mu_B\\cdot \\alpha$ for every $\\alpha > |x_1 - x_2|$, and hence\n$|f(x_1) - f(x_2)| \\le \\mu_B\\cdot |x_1 - x_2|$.\nThe same argument with $f$ replaced by $f^{-1}\\colon f(B) \\to B$ yields the other inequality.\n\\end{proof}\n\nUsing Lemma~\\ref{lem:gammaLin}, we deduce that the domain and the image of a definable function can be prepared simultaneously and in a compatible way.\n\n\n\n\\begin{prop}[Domain and range preparation]\\label{prop:range}\nLet $f\\colon K \\to K$ be a $\\emptyset$-definable function\nand let $C_0 \\subset K$ be a finite, $\\emptyset$-definable set.\nThen there exist finite, $\\emptyset$-definable sets $C, D \\subset K$ with $C_0 \\subset C$ such that $f(C) \\subset D$, and for every $\\lambda \\le 1$ in $\\Gamma^\\times_K$ and every ball $B \\subset K$ that is $\\lambda$-next to $C$, the image $f(B)$ is either a singleton in $D$ or a ball $\\lambda$-next to $D$.\n\\end{prop}\n\nThe role of the $C_0$ is to make it possible to combine this proposition with other preparation results for functions (notably Theorem~\\ref{thm:high-ord}): First apply the other results to get a set $C_0$; then apply Proposition~\\ref{prop:range} to enlarge $C_0$ to $C$ and to obtain $D$.\n\nNote however that the proposition cannot be combined very well with itself:\nGiven $f_1, f_2\\colon K \\to K$ as in the proposition, there are, in general, no $C, D_1, D_2$ such that $(f_i, C, D_i)$ are as in the proposition for both $i = 1, 2$, as the following example shows.\n\n\\begin{example}\nFix $r \\in K$ of negative valuation. For $x \\in {\\mathcal O}_K$, we set $f_1(x) = f_2(x) = f_1(x + r) = x$ and $f_2(x + r) = x+1$, with the exception that $f_2(0) = r$. Extend $f_1$ and $f_2$ by $0$ outside of ${\\mathcal O}_K \\cup ({\\mathcal O}_K + r)$. Then one successively deduces: $0 \\in C \\Rightarrow 0 \\in D_1 \\Rightarrow r \\in C \\Rightarrow 1 \\in D_2 \\Rightarrow 1 \\in C \\Rightarrow 1 \\in D_1 \\Rightarrow \\dots$; this shows that $C$ cannot be finite.\n\\end{example}\n\nIn Addendum \\ref{add:cd:alg:range} to the cell decomposition Theorem~\\ref{thm:cd:alg:skol}, we will state a version of simultaneous preparation of the domain and image which avoids this problem (and thus works for several functions simultaneously), by working piecewise.\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:range}]\nFirst, enlarge $C_0$ to a set $C_1$ using Lemma~\\ref{lem:gammaLin}, so that on each ball $B$ $1$-next to $C_1$, $f$ sends open balls to open balls and the quotient\n\\begin{equation}\\label{eq:quoco}\n|f(x_1) - f(x_2)|\/|x_1 - x_2|\n\\end{equation}\nis constant.\nThen let $D$ be a set containing $f(C_1)$ and preparing the family $f(B)$, where $B$ runs over the balls $1$-next to $C_1$. (This family is parametrized by $\\mathrm{RV}$-variables, by Lemma~\\ref{lem:InextFam}, so Corollary~\\ref{cor:prep} applies.) Now set $C := f^{-1}(D)$. We claim that these $C$ and $D$ are as desired.\n\nNote that we have $C \\supset C_1$, so the only thing that is not clear from the construction is that balls $\\lambda$-next to $C$ are sent to elements of $D$ or to balls\n$\\lambda$-next to $D$. We first deal with the case $\\lambda = 1$.\n\nLet $B$ be a ball $1$-next to $C$ and let $B_1$ be the ball $1$-next to $C_1$ containing $B$. If $f(B_1)$ is a singleton, then it is an element of $D$ and we are done. If $f(B_1) \\cap D = \\emptyset$, then also $B_1 \\cap C = \\emptyset$ (since $C = f^{-1}(D)$), so $B = B_1$ and $f(B_1)$ is a ball $1$-next to $D$.\nFinally, suppose that $f(B_1) \\cap D \\ne \\emptyset$. Then $B_1 \\cap C \\ne \\emptyset$, so $B$ is $1$-next to some $c \\in C \\cap B_1$. Then using that (\\ref{eq:quoco}) is constant on $B_1$, we obtain that $f(B)$ is a ball $1$-next to $f(c)$.\n\nNow suppose that $\\lambda < 1$. Any ball $B$ $\\lambda$-next to $C$ is contained in a ball $B'$ $1$-next to $C$. If $f$ is constant on $B'$, we are done; otherwise, $f(B')$ is a ball $1$-next to $D$, and we deduce that $f(B)$ is $\\lambda$-next to $D$, using once more that (\\ref{eq:quoco}) is constant on $B'$.\n\\end{proof}\n\n\\subsection{An equivalent condition to 1-h-minimality}\\label{sec:ta}\n\nThe conclusions of Lemmas~\\ref{lem:fin-inf} and \\ref{lem:gammaLin} together are actually equivalent to $1$-h-minimality.\nMore precisely, we have the following equivalence, where Condition~(T1) is slightly weaker than Lemma~\\ref{lem:gammaLin}:\n\n\\begin{thm}[Criterion for $1$-h-minimality]\\label{thm:tame2vf}\nLet ${\\mathcal L}$ be a language expanding the language ${\\mathcal L}_\\val$ of valued fields, and let ${\\mathcal T}$ be a complete ${\\mathcal L}$-theory whose models are valued fields of equi-characteristic $0$.\nThen ${\\mathcal T}$ is $1$-h-minimal if and only if, for every model $K$ of ${\\mathcal T}$, for every set $A \\subset K \\cup \\mathrm{RV}$ and for every\n${\\mathcal L}(A)$-definable $f\\colon K \\to K$, we have the following:\n\\begin{enumerate}[(T1)]\n\\item There exists a finite ${\\mathcal L}(A)$-definable $C \\subset K$ such that for every ball $B \\subset K$ $1$-next to $C$,\nthere exists a $\\mu_B \\in \\Gamma_K$ such that for every $x_1, x_2 \\in B$, we have $|f(x_1) - f(x_2)| = \\mu_B \\cdot |x_1 - x_2|$.\n\\item The set $\\{d \\in K \\mid f^{-1}(d)$ is infinite$\\}$ is finite.\n\\end{enumerate}\n\\end{thm}\n\nNote that in the above, we intentionally allow $C$ to use the parameters from $\\mathrm{RV}$ in its definition, in contrast to the condition in the definition of $1$-h-minimality.\n\n\\begin{remark}\\label{rem::tame2vf}\nThe conditions given in Theorem \\ref{thm:tame2vf} are very closely related to the tameness notion of Definition~2.1.6 of \\cite{CCL-PW} and its variant from Section~2.1 of \\cite{CFL}, one difference being that we do not assume the existence of an angular component map.\n\\end{remark}\n\nBy Lemmas~\\ref{lem:gammaLin} and \\ref{lem:fin-inf}, (T1) and (T2) follow from $1$-h-minimality, so in the remainder of Subsection~\\ref{sec:ta}, we assume that ${\\mathcal T}$ satisfies (T1) and (T2), our goal being to prove $1$-h-minimality.\nWe also assume throughout the subsection that $K$ is a model of ${\\mathcal T}$.\n\n\\begin{remark}\\label{rem:nocontrol}\nBy applying (T1) to the characteristic function of an $A$-definable set $X \\subset K$ (for $A \\subset K \\cup \\mathrm{RV}$), we find a finite $A$-definable set $C \\subset K$ $1$-preparing $X$. In particular, Assumption~\\ref{ass:no-ctrl} is satisfied, so we may apply Lemmas~\\ref{lem:finite} ($\\exists^\\infty$-elimination), \\ref{lem:average} (existence of injective functions from finite sets to $\\mathrm{RV}$) and \\ref{lem:InextFam} (the balls $1$-next to $C$ are $\\mathrm{RV}$-parametrized).\n\\end{remark}\n\n\n\n\\begin{lem}\\label{lem:tame2finite}\nAssume (as convened for the remainder of Subsection~\\ref{sec:ta}) that ${\\mathcal T}$ satisfies (T1) and (T2) from Theorem \\ref{thm:tame2vf} and that $K$ is a model.\nWe have the following:\n\\begin{enumerate}\n\\item Any definable (with parameters) function $f\\colon\\mathrm{RV}^k \\to K$ has finite image.\n\\item If $C_\\xi \\subset K$ is a definable (with parameters) family of finite sets, parametrized by $\\xi \\in \\mathrm{RV}^k$, then\nthe union $\\bigcup_{\\xi \\in \\mathrm{RV}^k} C_\\xi$ is still finite.\n\\end{enumerate}\n\\end{lem}\n\n\n\\begin{proof}\nWe first prove both claims for $k = 1$.\n\n(1) The composition $f \\circ \\operatorname{rv}\\colon K \\to K$ is locally constant everywhere except possibly at $0$, so the claim follows from\n(T2).\n\n(2)\nBy Lemma~\\ref{lem:finite}, the cardinality of $C_\\xi$ is bounded independently of $\\xi$,\nso we may assume that the cardinality is constant, say, equal to $m$.\nLet $\\sigma_1, \\dots, \\sigma_m \\in {\\mathbb Z}[x_1, \\dots, x_m]$ be the elementary symmetric functions in $m$ variables, which\nwe consider as functions on the set of $m$-element subsets of $K$.\nBy (1), for each $i$, the function $f_i(\\xi) := \\sigma_i(C_\\xi)$ has finite image. Since $\\sigma_1(C), \\dots, \\sigma_m(C)$ together determine $C$,\nthere are only finitely many different sets $C_\\xi$, which implies the claim.\n\nNow we deduce (2) for arbitrary $k$ by induction: Given a definable family $C_{\\xi, \\xi'} \\subset K$, for $\\xi \\in \\mathrm{RV}$ and $\\xi' \\in \\mathrm{RV}^k$, we first obtain that $D_{\\xi'} := \\bigcup_{\\xi \\in \\mathrm{RV}} C_{\\xi, \\xi'}$ is finite for every $\\xi'$ and then that the entire union is finite.\n\nFinally, we obtain (1) for arbitrary $k$ by applying (2) to the family of singletons $C_\\xi := \\{f(\\xi)\\}$.\n\\end{proof}\n\n\\private{All the stuff above the following lemma does not use any control of parameters, i.e., in (1), we wouldn't need to require $C$ to be $A$-definable.}\n\n\\begin{lem}\\label{lem:tame-0h}\nGiven an $A$-definable function $f\\colon K \\to K$, with $A \\subset K \\cup \\mathrm{RV}$, we can find $C$ as in Condition (T1) of Theorem~\\ref{thm:tame2vf} which is moreover\n$(A \\cap K)$-definable. In particular, $\\operatorname{Th}(K)$ is $0$-h-minimal.\n\\end{lem}\n\n\\begin{proof}\nUsing (T1), we first find an $A$-definable set $C$. We consider it as a member of an $(A\\cap K)$-definable family of sets $C_\\xi$, parametrized by $\\xi \\in \\mathrm{RV}^k$.\nBy Lemma~\\ref{lem:tame2finite}, the union $C' := \\bigcup_{\\xi \\in \\mathrm{RV}^k} C$ is still finite. Moreover, it is $(A\\cap K)$-definable, and since it contains $C$, it satisfies the requirements of (T1).\n\nFor the in-particular part, apply this improved (T1) to the characteristic function of an $A$-definable set $X \\subset K$ (where $A \\subset K \\cup \\mathrm{RV}$), to find a finite $(A \\cap K)$-definable $C$ $1$-preparing $X$.\n\\end{proof}\n\nNote that we can now use the results from Section~\\ref{sec:Prepfam} with $\\ell = 0$; we will in particular use Corollary~\\ref{cor:prep} several times to prepare $\\mathrm{RV}$-parametrized families of subsets of $K$.\n\n\nThe next lemma is a variant of Lemma~\\ref{lem:gammaLin} with different assumptions.\n\n\\begin{lem}\\label{lem:gammaLinTame}\nGiven an $A$-definable function $f\\colon K \\to K$, for $A \\subset K \\cup \\mathrm{RV}$, we can find \\(C\\) which is  \\((A\\cap K)\\)-definable, such that Condition (T1) of Theorem~\\ref{thm:tame2vf} holds, and such that moreover, for $B \\subset K$ $1$-next to $C$, for $\\mu_B$ as in (T1) and for $B' \\subset B$ an open ball, the image $f(B')$ is either a singleton (when $\\mu_B = 0$) or an open ball of radius $\\rad_{\\mathrm{op}}(B') \\cdot \\mu_B$.\n\\end{lem}\n\n\\begin{proof}\nLet $C$ be an $(A\\cap K)$-definable set satisfying (T1).\n By Lemma~\\ref{lem:InextFam}, the family of balls $B$ $1$-next to $C$ can be parametrized using $\\mathrm{RV}$-parameters, so using\nCorollary~\\ref{cor:prep}, we can prepare the family of $f(B)$ using a finite $(A\\cap K)$-definable set $D$.\nSimilarly, we find a finite $(A\\cap K)$-definable set $C_0$ preparing the family $f^{-1}(B_1)$, where $B_1$ runs over the balls $1$-next to $D$.\nWe claim that the set $C' := C \\cup C_0$ does the job.\n\nSo let $B' \\subset K$ be $1$-next to $C'$, let $B$ be the ball $1$-next to $C$ containing $B'$,\nand let $\\mu_B \\in \\Gamma_K$ be as in (T1), i.e., such that we have\n\\begin{equation}\\label{eq:valeq}\n|f(x_1) - f(x_2)| = \\mu_B \\cdot |x_1 - x_2|\n\\end{equation}\nfor all $x_1, x_2 \\in B$.\nWe suppose that $\\mu_B \\ne 0$ (otherwise $f(B')$ is a singleton) and we have to show that for every open ball $B'' \\subset B'$,\n$f(B'')$ is an open ball of radius $\\rad_{\\mathrm{op}}(B'')\\cdot \\mu_B$.\n\nLet $B''_1$ be the smallest ball containing $f(B'')$.\nUsing (\\ref{eq:valeq}), one obtains that $B''_1$ is an open ball with radius $\\rad_{\\mathrm{op}} B''_1 = \\rad_{\\mathrm{op}}(B'')\\cdot \\mu_B$, so it remains to show that $f(B'')$ is equal to the entire ball $B''_1$.\n\nBy our definition of $C_0$, $f(B')$ (and hence also $B''_1$) is contained in a ball $B_1$ that is $1$-next to $D$, and by definition of $D$\n(and since $B_1$ is certainly not disjoint from $f(B)$), $B_1$ is contained in $f(B)$. In particular, the entire ball $B''_1$ is contained in $f(B)$. However, using (\\ref{eq:valeq}) again, we obtain that no element of $B \\setminus B''$ can be sent\ninto $B''_1$, so we deduce $f(B'') = B''_1$, as desired.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:tame2vf}, ``$\\Leftarrow$'']\nWe are given $\\lambda \\le 1$ in $\\Gamma^\\times_K$ and an $(A \\cup \\{\\xi\\})$-definable set\n$X \\subset K$, with $A \\subset K\\cup\\mathrm{RV}$ and $\\xi \\in \\mathrm{RV}_\\lambda$. We need to find a finite $(A \\cap K)$-definable set $C \\subset K$ which $\\lambda$-prepares $X$. We may assume that $\\lambda$ is $A$-definable.\n\nBy Lemma~\\ref{lem:tame2finite} (2), it is enough to find a $C$ which is $A$-definable. Indeed, then we can get rid of the $\\mathrm{RV}$-parameters in the same way as in Lemma~\\ref{lem:tame-0h}.\n\nWe consider $X$ as a member of an $A$-definable family $X_y$, where $y$ runs over $K$, with\n$X = X_y$ for every $y \\in Y := \\operatorname{rv}_\\lambda^{-1}(\\xi)$.\n(This uses $\\lambda \\in \\dcl_{\\Gamma_K}(A)$.)\n\nBy Remark~\\ref{rem:nocontrol}, we find, for each $y \\in K$, a finite $(A \\cup \\{y\\})$-definable set $\\hat C_y$\nthat $1$-prepares $X_y$. Using Lemma \\ref{lem:average}, we find an $(A \\cup \\{y\\})$-definable\ninjective map $h_y\\colon \\hat C_y \\to \\mathrm{RV}^k$.\nBy compactness, we may assume that those $\\hat C_y$ and $h_y$ form $A$-definable\nfamilies parametrized by $y \\in K$. This allows us to write the set $\\bigcup_{y \\in K} \\{y\\} \\times \\hat C_y \\subset K^2$\nas a disjoint union of graphs of functions $g_\\eta\\colon Y_\\eta \\subset K \\to K$, which\nform an $A$-definable family parameterized by $\\eta \\in \\mathrm{RV}^k$, namely:\n$g_\\eta(y) := h_y^{-1}(\\eta)$, where the domain $Y_\\eta$ of $g_\\eta$ is the set of those\n$y \\in K$ for which $\\eta$ is in the image of $h_y$. (For some $\\eta$, $Y_\\eta$ may be empty.)\n\nFor each $\\eta$, we find a finite $(A\\cup \\{\\eta\\})$-definable set $D_\\eta \\subset K$ $1$-preparing $Y_\\eta$ and $Z_\\eta := \\{y \\in Y_\\eta \\mid g_\\eta(y) \\in X_y\\}$\nand also preparing $g_\\eta$ in the sense of Lemma~\\ref{lem:gammaLinTame} (by applying the lemma to $g_\\eta$ extended by $0$ outside of $Y_\\eta$).\nUsing compactness again, we suppose that $D_\\eta$ is an $A$-definable family parametrized by $\\eta$, so that the union $D := \\{0\\} \\cup \\bigcup_\\eta D_\\eta$ is $A$-definable. Note that by Lemma~\\ref{lem:tame2finite} (2), $D$ is finite.\n\nIf $D \\cap Y \\ne \\emptyset$, we obtain a finite $A$-definable set\n$\\bigcup_{y \\in D } \\hat C_y$\n which $\\lambda$-prepares (even $1$-prepares) $X$ and hence we are done. So now assume that $D \\cap Y = \\emptyset$.\nIn that case, we claim that the following set $C$ $\\lambda$-prepares $X$:\nBy Lemma~\\ref{lem:InextFam}, the balls $1$-next to $D$ form an $A$-definable family $(B_\\xi)_\\xi$, where $\\xi$ runs over $\\mathrm{RV}^k$ for some $k \\ge 1$.\nWe let (using Corollary~\\ref{cor:prep}) $C \\subset K$ be a finite $A$-definable set which $1$-prepares $g_\\eta(B_\\xi)$ for each $(\\eta, \\xi)$ satisfying $B_\\xi \\subset Y_\\eta$.\n\nTo prove that this $C$ indeed $\\lambda$-prepares $X$, we need to verify: For every $x \\in X$ and every $x' \\in K \\setminus X$, there exists a $c \\in C$ such that $|x - c|\\cdot \\lambda \\le |x - x'|$.\nLet $B_1 \\subset K$ be the smallest (closed) ball containing $x$ and $x'$.\n\nFix a $y_0 \\in Y$. Since $\\hat C_{y_0}$ 1-prepares $X$, $\\hat C_{y_0} \\cap B_1$ is non-empty, so there exists an $\\eta$ such that $y_0 \\in Y_\\eta$ and $g_\\eta(y_0) \\in B_1$. We fix such an $\\eta$ for the remainder of the proof.\n\nRecall that $Y$ is a ball satisfying $Y \\cap D = \\emptyset$,\nand let $B_0$ the ball $1$-next to $D$ containing $Y$.\nOur choice of $D$ in particular implies that $g_\\eta$ is defined on all of $B_0$ and that for $\\mu_{B_0}$ as in Lemma~\\ref{lem:gammaLinTame}, $g_\\eta(Y)$ and $g_\\eta(B_0)$ are open balls satisfying\n\\begin{equation}\\label{eq:th1}\n\\rad_{\\mathrm{op}}(g_\\eta(Y)) = \\rad_{\\mathrm{op}}(Y) \\cdot \\mu_{B_0}\n\\quad \\text{and} \\quad\n\\rad_{\\mathrm{op}}(g_\\eta(B_0)) = \\rad_{\\mathrm{op}}(B_0) \\cdot \\mu_{B_0}.\n\\end{equation}\nMoreover, since $D$ also $1$-prepares $\\{y \\in Y_\\eta \\mid g_\\eta(y) \\in X_y\\}$, and since $X = X_y$ for all $y \\in Y$, we obtain that $g_\\eta(Y)$ is either contained in $X$ or disjoint from $X$. By our choice of $\\eta$, we have $g_\\eta(Y) \\cap B_1 \\ne \\emptyset$. However, $B_1$ is neither contained in $X$ nor disjoint from $X$, so we deduce $B_1 \\not\\subset g_\\eta(Y)$. This implies\n\\begin{equation}\\label{eq:th2}\n\\rad_{\\mathrm{op}}(g_\\eta(Y)) \\le \\rad_{\\mathrm{cl}}(B_1) = |x - x'|.\n\\end{equation}\nSince $C$ prepares $g_\\eta(B_0)$, there exists a $c \\in C$ such that\n\\begin{equation}\\label{eq:th3}\n|g_\\eta(y_0) - c| \\le \\rad_{\\mathrm{op}}(g_\\eta(B_0)).\n\\end{equation}\nFinally, recall that $Y$ is a fiber of $\\operatorname{rv}_\\lambda$.\nSince $0 \\notin B_0$ (which we ensured by putting $0$ into $D$), we deduce $\\lambda\\cdot \\rad_{\\mathrm{op}}(B_0) \\le \\rad_{\\mathrm{op}}(Y)$.\nPutting this together with (\\ref{eq:th1}), (\\ref{eq:th2}) and (\\ref{eq:th3}), we obtain:\n\\[\n|g_\\eta(y_0) - c| \\cdot \\lambda \\le \\mu_{B_0}\\cdot \\rad_{\\mathrm{op}}(B_0)\\cdot \\lambda \\le  \\mu_{B_0}\\cdot \\rad_{\\mathrm{op}}(Y) \\le  |x - x'|.\n\\]\nNow recall that \\(g_\\eta(y_0) \\in B_1\\), so we obtain the desired result:\n\\[\n|x -c| \\cdot \\lambda \\leq \\max \\{\\underbrace{|x-g_\\eta(y_0)|}_{\\le |x - x'|},|g_\\eta(y_0) - c|\\}\\cdot\\lambda \\leq |x - x'|.\n\\qedhere\\]\n\\end{proof}\n\n\n\\subsection{A criterion for preparability}\\label{sec:trans}\n\nWe already proved that various kinds of objects ``living over $K$'' can be prepared by finite subsets of $K$ in various different senses.\nWe finish Section~\\ref{sec:first-properties} with a criterion simplifying such proofs (Lemma~\\ref{lem:fullyT}). Since we want to apply this to quite different\nnotions of ``being prepared'', we do this at an abstract level, using the following definition.\n\n\\begin{defn}[Preparing bad balls]\nLet ${\\mathcal B}$ be a set of closed balls in $K$ (the ``bad balls''). We say that a finite set $C \\subset K$ \\emph{prepares} ${\\mathcal B}$\nif for every $B \\in {\\mathcal B}$, the intersection $C \\cap B$ is non-empty.\n\\end{defn}\n\n\\begin{example}\nGiven a definable subset $X \\subset K$, we can let ${\\mathcal B}$ be the set of all closed balls $B$ that are neither disjoint from $X$ nor contained in $X$.\nThen a finite set $C \\subset K$ $1$-prepares $X$ if and only if it prepares ${\\mathcal B}$.\nIndeed, the implication ``$\\Rightarrow$'' is clear. For the other implication, suppose that $B'$ is a ball\n$1$-next to $C$ containing both, a point $x \\in X$ and a point $x' \\in K \\setminus X$; then the smallest (closed) ball containing $x$ and $x'$ is disjoint from $C$ but lies in ${\\mathcal B}$.\n\\end{example}\n\n\nNote that we can (and will) assume ${\\mathcal B}$ to be an imaginary definable set.\nAs such, in the above example, ${\\mathcal B}$ is definable over the same parameters as $X$.\n\nBy a compactness argument (as in the proof of Lemma~\\ref{lem:type-0-h-min}), a $\\emptyset$-definable ${\\mathcal B}$ can be prepared by a finite $\\emptyset$-definable $C \\subset K$ if and only if, in a sufficiently saturated model $K$, no ball $B \\in {\\mathcal B}$ is disjoint from $\\acl_K(\\emptyset)$. The following lemma provides an even weaker condition that needs to be checked if one wants to prove that ${\\mathcal B}$ can be prepared:\n\n\n\\begin{lem}[Criterion for preparability]\\label{lem:fullyT}\nSuppose that $\\operatorname{Th}(K)$ is $0$-h-minimal and that $K$ is $|{\\mathcal L}|^+$-saturated.\nLet ${\\mathcal B}$ be a $\\emptyset$-definable set of closed balls in $K$. We call an arbitrary ball $B' \\subset K$ a ``bad ball'' if it contains a ball $B \\in {\\mathcal B}$ as a subset.\n\nSuppose that for every $a \\in K$, there is no open ball $B' \\subset K \\setminus \\acl_K(a)$ which is $1$-next to $a$ and bad.\nThen there exists a finite $\\emptyset$-definable set $C \\subset K$ preparing ${\\mathcal B}$.\n\\end{lem}\n\n\\begin{remark}\nIt might sound more natural to let the balls in ${\\mathcal B}$ be open instead of closed, but that would make the lemma false: We use, in the proof, that any open bad ball already contains a closed bad subball.\n\\end{remark}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:fullyT}]\nSuppose that no finite $\\emptyset$-definable set $C$ prepares ${\\mathcal B}$.\nThen, by saturation, we find a single ball $B \\in {\\mathcal B}$ which is disjoint from all finite $\\emptyset$-definable $C$; in other words, this ball satisfies $B \\cap \\acl_K(\\emptyset) = \\emptyset$. We fix $B$ for the remainder of the proof.\n\nIf $B$ is $1$-next to some $a \\in \\acl_K(\\emptyset)$, then we also have $B \\cap \\acl_K(a) = \\emptyset$, and we get a contradiction to the assumption that such a $B$ should not be bad. Therefore, if for $a \\in \\acl_K(\\emptyset)$ we denote by $B_a$ the ball $1$-next to $a$ containing $B$, none of the sets $B_a \\setminus B$ is empty. By saturation, also the intersection\n$\\bigcap_{a \\in \\acl_K(\\emptyset)} B_a \\setminus B$ is non-empty.\nLet $B'$ be the smallest ball containing $B$ and any chosen element of that intersection. This ball has the following properties: It is closed, it is disjoint from $\\acl_K(\\emptyset)$, and it is strictly bigger than $B$.\nSet $\\gamma := \\rad_{\\mathrm{cl}}(B)$ and $\\mu := \\rad_{\\mathrm{cl}}(B')$.\n\nAll elements of $B'$ have the same type over $\\mathrm{RV}$; indeed, if two elements of $B'$ could be distinguished by a formula with parameters from $\\mathrm{RV}$, then any finite set $C$ preparing the $\\mathrm{RV}$-parametrized family of subsets of $K$ defined by that formula would have to contain points of $B'$; but then $C$ cannot be $\\emptyset$-definable, since $B' \\cap \\acl_K(\\emptyset) = \\emptyset$.\n\nWe deduce that all the open balls $B_{<\\mu}(b) \\subset B'$ (for $b \\in B'$) are bad. Indeed, for $b \\in B$, we have\n$B_{<\\mu}(b) \\supset B_{\\le \\gamma}(b) \\in {\\mathcal B}$, and since any other $b' \\in B'$ has the same type as $b$ over $\\mathrm{RV}$ (and ${\\mathcal B}$ is $\\emptyset$-definable), we also have $B_{\\le \\gamma}(b') \\in {\\mathcal B}$,\nwitnessing that $B_{<\\mu}(b')$ is bad.\n\nNow fix an arbitrary $a \\in B'$. By saturation, there exists a $b' \\in B'$ such that $B'' := B_{<\\mu}(b')$ is disjoint from $\\acl_K(a)$. This however contradicts the assumption of the lemma, since $B''$ is bad and $1$-next to $a$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Derivation, the Jacobian Property and Taylor approximation}\\label{sec:derJac\n\nWe continue assuming that $K$ is a valued field of equi-characteristic $0$,\nconsidered as a structure in a fixed language ${\\mathcal L} \\supset {\\mathcal L}_\\val$, and we\nassume that $\\operatorname{Th}(K)$ is $1$-h-minimal.\nThe goal of this section is to prove various preparation results for definable functions $f\\colon K \\to K$,\nstarting with Theorem~\\ref{thm:der} which states that $f$ is almost everywhere differentiable, and\nculminating in Theorem~\\ref{thm:high-ord}, which states that away from a finite set, $f$\nhas good approximations by its Taylor polynomials.\n\n\n\\subsection{Derivation, strict and classical}\n\\label{sec:derivative}\n\n\nWe define the derivative of functions $f\\colon K \\to K$ as usual:\n\n\\begin{defn}[Classical derivative]\nWe say that the \\emph{(classical) derivative} of a function $f\\colon K \\to K$ exists at $x\\in K$ if there exists $a\\in K$ such that for each $\\varepsilon$ in $\\Gamma^\\times_K$ there exists $\\delta$ in $\\Gamma^\\times_K$ such that for all $y\\in K$ with $|x-y|<\\delta$ and $x\\ne y$ one has\n$$\n|\\frac{f(x)-f(y)}{x-y} - a | < \\varepsilon;\n$$\nwe then write $f'(x)$ for $a$.\n\\end{defn}\n\nIn the context of totally disconnected fields, one sometimes needs to put a stronger condition on the existence of derivatives, namely:\n\n\\begin{defn}[Strict derivative]\\label{defn:strict-der}\nWe say that the \\emph{strict derivative} of a function $f\\colon K \\to K$ exists at $x\\in K$ if there exists $a\\in K$ such that for each\n$\\varepsilon$ in $\\Gamma^\\times_K$ there exists $\\delta$ in $\\Gamma^\\times_K$ such that for all $y_i\\in K$\nwith $|x-y_i|<\\delta$ for $i=1,2$ and $y_1 \\not=y_2$ one has\n$$\n|\\frac{f(y_1)-f(y_2)}{y_1-y_2} - a | < \\varepsilon.\n$$\n\\end{defn}\n\n\\begin{remark}\nAs usual,\none easily verifies that the set of $x$ where the classical derivative of a definable function $f$ exists is definable over the same parameters as $f$, and similarly for the strict derivative. Moreover, the derivative of $f$ is definable over the same parameters as $f$.\n\\end{remark}\n\n\n\\begin{thm}[Existence of derivatives]\\label{thm:der}\nLet $K$ be a valued field of equi-characteristic $0$ such that $\\operatorname{Th}(K)$ is 1-h-minimal, and let $f\\colon K \\to K$ be a definable (with parameters) function.\nThen the strict derivative of $f$ exists almost everywhere, i.e., the set of $x \\in K$ such that the strict derivative of $f$ does not exist at $x$ is finite.\n\\end{thm}\n\nMost of the remainder of this subsection is devoted to the proof of this theorem, so from now on we fix a definable function $f\\colon K \\to K$.\nBy Lemma~\\ref{lem:addconst}, we may as well impose $f$ to be $\\emptyset$-definable.\n\nWe also fix a handy notation for difference quotients: Given unequal $x_1,x_2$ in $K$, we set\n\\[\nq_f(x_1, x_2) := \\frac{f(x_1) -f(x_2)}{x_1 - x_2}.\n\\]\nFor convenience we put $q_f(x, x):=0$ for any $x\\in K$.\n\nWe start by proving an auxiliary lemma, which is a weak form of Theorem~\\ref{thm:der}, stating that in some sense, one has finitely-valued derivatives.\n\n\\begin{lem}\\label{lem:f'fin}\nFor $x \\in K$, let $A_x$ be the set of accumulation points of\n\\begin{equation}\\label{eq:def_A_x}\n\\lim_{\\substack{y_1,y_2\\to x\\\\y_1\\ne y_2}}q_f(y_1, y_2).\n\\end{equation}\nThere is a finite $\\emptyset$-definable set $C\\subset K$ such that, for every $x\\in K\\setminus C$, $A_x$ is finite and we moreover have\n\\begin{equation}\\label{eq:limmin}\n\\lim_{\\substack{y_1,y_2\\to x\\\\y_1\\ne y_2}}\n\\min_{a\\in A_x} \\left|  q_f(y_1, y_2) - a  \\right| = 0.\n\\end{equation}\n\\end{lem}\n\n\n\\begin{proof}\nClearly, the set $C$ of $x$ where $A_x$ is not as desired is $\\emptyset$-definable (using Lemma~\\ref{lem:finite} to express finiteness of $A_x$). We need to show that $C$ is finite.\n\nWe may assume that $K$ is sufficiently saturated and we suppose that $C$ is infinite. Then $C$ contains a transcendental element $x_0$ (i.e., $x_0 \\notin \\acl_K(\\emptyset)$).\nWe will prove that for transcendental $x_0 \\in K$,\nthere exists a finite set $A$ satisfying (\\ref{eq:limmin}).\nOne easily checks that this implies that $A_{x_0} \\subset A$ and that then, $A_{x_0}$ also satisfies (\\ref{eq:limmin}), contradicting $x_0 \\in C$. Constructing $A$ is done in several steps.\n\n\\medskip\n\nStep 1:\nUsing (once more) that $K$ is sufficiently saturated, we find an entire open ball $B_1 := B_{<\\mu}(x_0)$ that is disjoint from $\\acl_K(\\emptyset)$.\nWe fix these $\\mu$ and $B_1$ once and for all and more generally define $B_\\lambda :=  B_{<\\lambda\\cdot\\mu}(x_0)$, for $\\lambda \\le 1$.\n\n\\medskip\n\nFor Steps 2--4, we fix $\\lambda \\le 1$ in $\\Gamma^\\times_K$, and\n$\\zeta$ will always be an element of $\\mathrm{RV}$ satisfying $|\\zeta| < \\lambda \\cdot \\mu$ (so that $\\zeta = \\operatorname{rv}(x - x')$ for some $x, x' \\in B_\\lambda$).\n\n\\medskip\n\nStep 2: For $\\zeta$ as above, the subset of $\\mathrm{RV}_\\lambda$ defined by $Q_{x,\\zeta} := \\{\\operatorname{rv}_\\lambda(q_f(x, x')) \\mid x' \\in x+ \\operatorname{rv}^{-1}(\\zeta)\\}$ is independent of $x$ when $x$ runs over $B_\\lambda$.\n\n\\medskip\n\nProof: We $\\lambda$-prepare the family $(Q_{x,\\zeta})_{x,\\zeta}$ using Corollary~\\ref{cor:prep} (i.e., we consider $Q_{x,\\zeta}$ as a fiber of a definable set $Q \\subset  K \\times \\mathrm{RV} \\times \\mathrm{RV}_\\lambda$ and $\\lambda$-prepare $Q$). The finite $\\emptyset$-definable\nset $C'$ obtained in this way is disjoint from $B_1$ (since $B_1 \\cap \\acl_K(\\emptyset) = \\emptyset$)\n, so $B_\\lambda$ is contained in a ball $\\lambda$-next to $C'$. This implies Step 2.\n\\qed2\n\n\n\\medskip\n\nSet $Q_\\zeta := Q_{x,\\zeta}$.\n\n\\medskip\n\nStep 3: $Q_\\zeta \\subset Q_{2\\zeta}$ (where $2\\zeta$ is $\\operatorname{rv}(2)\\cdot\\zeta$).\n\n\n\n\\medskip\n\nProof: Fix any $\\xi\\in Q_\\zeta$. (We need to show that $\\xi \\in  Q_{2\\zeta}$.)\nChoose $x_1 \\in B_\\lambda$ witnessing $\\xi \\in Q_{x_0,\\zeta}$, i.e., such that $\\operatorname{rv}(x_0 - x_1) = \\zeta$ and\n$\\operatorname{rv}_\\lambda(q_f(x_0, x_1)) = \\xi$; similarly choose $x_2 \\in B_\\lambda$ such that $\\operatorname{rv}(x_1 - x_2) = \\zeta$ and $\\operatorname{rv}_\\lambda(q_f(x_1, x_2)) = \\xi$. Then the following computation shows that $\\operatorname{rv}_\\lambda(q_f(x_0, x_2)) = \\xi$ (which implies $\\xi \\in Q_{2\\zeta}$):\nSet $r_i := x_{i-1} - x_i$ and $s_i := f(x_{i-1}) - f(x_i)$ for $i = 1,2$. We have\n\\begin{align*}\n|q_f(x_0,x_2) - q_f(x_0, x_1)| &=\n\\left|\\frac{s_1 + s_2}{r_1 + r_2} - \\frac{s_1}{r_1}\\right| =\n\\left|\\left(\\frac{s_2}{r_2} - \\frac{s_1}{r_1}\\right)\\cdot\\frac{r_2}{r_1+r_2}\\right|\\\\\n&= \\underbrace{\\left|q_f(x_1,x_2) - q_f(x_0,x_1)\\right|}_{< \\lambda\\cdot|q_f(x_0,x_1)|}\\cdot\\underbrace{\\left|\\frac{r_2}{r_1+r_2}\\right|}_{=1}\n\\end{align*}\nand hence $\\operatorname{rv}_\\lambda(q_f(x_0,x_2)) = \\operatorname{rv}_\\lambda(q_f(x_0,x_1))$.\n\\qed3\n\n\\medskip\n\nApplying Step 3 repeatedly shows: $Q_\\zeta \\subset Q_{2^n\\zeta}$ for every integer $n  \\ge 0$.\n\n\\medskip\n\nStep 4: $Q_\\zeta$ is a singleton for every $\\zeta$.\n\n\\medskip\n\nProof: Suppose otherwise, i.e., $\\xi_1, \\xi_2 \\in Q_\\zeta$ with $\\xi_1 \\ne \\xi_2$. The set\n$$\nX := \\{x \\in B_\\lambda \\mid \\operatorname{rv}_\\lambda(q_f(x_0, x)) = \\xi_1\\};\n$$\nis definable, and hence it can be\n$1$-prepared by a finite set $D$. (We do not care about the parameters needed to define $X$ and $D$.)\nHowever,\nfor each of the (disjoint) balls $B_n := \\{x\\in B_\\lambda\\mid \\operatorname{rv} (x_0 - x) = 2^n \\zeta\\}$ (where $n$ runs over the non-negative integers), we have neither $X \\cap B_n = \\emptyset$ (since $\\xi_1 \\in Q_{2^n\\zeta}$) nor $B_n \\subset X$ (since $\\xi_2 \\in Q_{2^n\\zeta}$); hence for $D$ to $1$-prepare $X$, we would need $D \\cap B_n \\ne \\emptyset$ for every $n$, contradicting the finiteness of $D$.\n\\qed4\n\n\\medskip\n\nLet us reformulate what we obtained until now in a slightly different way: Given any $\\zeta \\in \\mathrm{RV}$ and any $\\lambda \\in \\Gamma_K$ satisfying $|\\zeta|\/\\mu < \\lambda \\le 1$, Step~4 states that the entire set\n$\\tilde{Q}_{\\zeta,\\lambda} := \\{q_f(x, x') \\mid x, x' \\in B_\\lambda, \\operatorname{rv}(x - x') = \\zeta\\}$\nis contained in a single ball $\\tilde B_{\\zeta,\\lambda}$ $\\lambda$-next to $0$.\n\n\\medskip\n\nStep 5: Using Corollary~\\ref{cor:prep}, choose a finite set $A \\subset K$ $1$-preparing the family $(\\tilde{Q}_{\\zeta,\\lambda})_{\\zeta,\\lambda}$, for $\\zeta \\in \\mathrm{RV}$ and \\(\\lambda \\in \\Gamma_K^\\times\\) satisfying $|\\zeta|\/\\mu < \\lambda \\leq 1$. (Again, we do not care about parameters.)\n\n\\medskip\n\nThe last step consists in showing that the set $A$ satisfies (\\ref{eq:limmin}), as desired. More precisely, we show:\n\n\\medskip\n\nStep 6: There exists a constant $\\kappa \\in \\Gamma_K$ such that for every $\\lambda < 1$ and every $x, x' \\in B_\\lambda$ with $x \\ne x'$, we have $\\min_{a\\in A} |  q_f(x,x') - a  | \\le \\lambda \\cdot \\kappa$.\n\n\\medskip\n\nProof: The constant is $\\kappa := \\max \\{|a| \\mid a \\in A\\}$. Let $x, x' \\in B_\\lambda$ be given. For $\\zeta := \\operatorname{rv}(x - x')$, we have $q_f(x,x') \\in \\tilde{Q}_{\\zeta,\\lambda} \\subset \\tilde B_{\\zeta,\\lambda}$. Note that $\\tilde B_{\\zeta,\\lambda}$ is an open ball of radius $\\lambda\\cdot |q_f(x, x')|$.\nSince $A$ 1-prepares $\\tilde{Q}_{\\zeta,\\lambda}$, there exists an $a \\in A$ such that $|q_f(x,x') - a| \\le \\rad_{\\mathrm{op}}(\\tilde B_{\\zeta,\\lambda}) = \\lambda\\cdot |q_f(x, x')|$. Since $\\lambda < 1$, this in particular implies $|q_f(x,x')| = |a|$, so the right hand side is $\\lambda \\cdot |a| \\le \\lambda \\cdot \\kappa$ and we are done.\n\\qed6\n\n\\medskip\n\nThis finishes the proof of the lemma.\n\\end{proof}\n\nUsing the lemma, we can now prove that the derivative of $f$ exists almost everywhere.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:der}]\nLet $A_x$ (for $x \\in K$) and $C$ be as in Lemma \\ref{lem:f'fin}.\nWe need to show that, for almost all $x\\in K \\setminus C$, the set $A_x$ is a singleton.\nSuppose otherwise, i.e., that $A_x$ is not a singleton for infinitely many $x$. Then as usual, we can find a ball $B$ such that $A_x$ consists of several elements for every $x \\in B$.\n\nWe first shrink $B$ in such a way that the cardinality $\\#A_x$ is constant for $x \\in B$, and then we further shrink it to make\nthe map $x \\mapsto A_x$ ``approximately constant'' on $B$ in the following sense: There is\na $\\mu \\in \\Gamma^\\times_K$ such that for every $x, x' \\in B$, the relation $a \\sim_\\mu a' :\\iff |a - a'| < \\mu$ defines a bijection between $A_x$ and $A_{x'}$.\nThis shrinking of $B$ is possible as follows: By $1$-preparing the (graph of the) function $B \\to \\Gamma^\\times_K, x \\mapsto \\min_{a_1, a_2 \\in A_{x}, a_1 \\ne a_2} |a_1 - a_2|$, we find a subball (which we will now call $B$) on which this minimum is constant equal to some $\\mu \\in \\Gamma^\\times_K$. We then choose any $x_0 \\in B$, and using the definition (\\ref{eq:def_A_x}) of $A_{x_0}$, we replace $B$ by an even smaller ball around $x_0$\nsuch a way that for every $x_1, x_2 \\in B$, we have\n\\[\n\\min_{a \\in A_{x_0}}|\\frac{f(x_1) - f(x_2)}{x_1 - x_2} - a| < \\mu.\n\\]\nThis implies that $\\sim_\\mu$ defines bijections as desired.\n\nFix any $a_0 \\in \\bigcup_{x \\in B}A_x$. We\napply Lemma~\\ref{lem:gammaLin} (2) (the Valuative Jacobian Property) to the ${\\mathcal L}(a_0)$-definable function $\\tilde f(x) := f(x) - a_0x$. This allows\nus to further shrink $B$ in such a way that the quotient $|\\tilde f(x_1) - \\tilde f(x_2)|\/|x_1 - x_2|$ is constant for $x_1, x_2 \\in B, x_1 \\ne x_2$.\n\nThis now leads to a contradiction, as follows.\nFix any $x \\in B$. For every $a \\in A_x$, there exist $x_1, x_2 \\in B$ with $|\\frac{f(x_1) - f(x_2)}{x_1 - x_2} - a| < \\mu$ (by definition of $A_x$). Since\n\\[\n\\frac{\\tilde f(x_1) - \\tilde f(x_2)}{x_1 - x_2} =\n\\frac{f(x_1) - f(x_2)}{x_1 - x_2} - a_0,\n\\]\nthis implies $|\\frac{\\tilde f(x_1) - \\tilde f(x_2)}{x_1 - x_2}| < \\mu$ if $a \\sim_\\mu a_0$ and\n$|\\frac{\\tilde f(x_1) - \\tilde f(x_2)}{x_1 - x_2}| \\ge \\mu$ if $a \\not\\sim_\\mu a_0$.\nSince $A_x$ contains elements $a$ of both kinds (with and without $a \\sim_\\mu a_0$), this contradicts $|\\frac{\\tilde f(x_1) - \\tilde f(x_2)}{x_1 - x_2}|$ being constant.\n\\end{proof}\n\nUsing the existence of derivatives, we can now reformulate the Valuative Jacobian Property (Lemma~\\ref{lem:gammaLin}) in a nicer way:\n\n\\begin{cor}[Valuative Jacobian Property]\\label{cor:gammaLin}\nFor every $\\emptyset$-definable function $f\\colon K \\to K$, there exists a finite $\\emptyset$-definable set $C \\subset K$\nsuch that for every ball $B$ $1$-next to $C$, $f'$ exists on $B$, $|f'|$ is constant on $B$, and we have the following:\n\\begin{enumerate}\n \\item For every $x_1, x_2 \\in B$, we have $|f(x_1) - f(x_2)| = |f'(x_1)|\\cdot |x_1 - x_2|$.\n   (In particular, $f$ is constant on $B$ if $f' = 0$ on $B$.)\n \\item\n   If $f' \\ne 0$ on $B$, then for every open ball $B' \\subset B$, $f(B')$ is an open ball of radius $|f'(x)| \\cdot\\rad_{\\mathrm{op}}(B')$\n   for any $x \\in B$.\n\\end{enumerate}\n\\end{cor}\n\n\\begin{proof}\nThe only difference between this and Lemma~\\ref{lem:gammaLin} is that the factor called $\\mu_B$ in Lemma~\\ref{lem:gammaLin} is now claimed\nto be equal to $|f'(x)|$ for any $x \\in B$. But indeed, by definition of the derivative, for $x \\in B$ and $x'$ sufficiently close to $x$, we have\n$|f'(x)| = \\frac{|f(x') - f(x)|}{|x' - x|} = \\mu_B$.\n\\end{proof}\n\n\n\n\n\\subsection{The Jacobian Property and Taylor approximations}\n\\label{sec:Jac}\n\n\nWe now come to one of the central results of this paper: Every definable function $K \\to K$ is, away from a finite set $C$,\nwell approximated by its Taylor polynomials. Here follows the precise statement and various variants and corollaries. The proof will be given in the next subsection, and higher dimensional variants will be deduced in Subsections~\\ref{sec:sjp} and \\ref{sec:taylor-box}.\n\n\\begin{defn}[Taylor Polynomial]\\label{defn:taylor}\nLet $f\\colon X \\subset K \\to K$ be a function which is $r$-fold differentiable at $x_0$, for some $r \\in {\\mathbb N}$ and some $x_0 \\in K$.\nThen we write\n\\begin{equation}\nT^{<r+1}_{f,x_0}(x) := T^{\\le r}_{f,x_0}(x) := \\sum_{i = 0}^r \\frac{f^{(i)}(x_0)}{i!}(x - x_0)^i\n\\end{equation}\nfor the Taylor polynomial of degree $r$ of $f$ around $x_0$. (Here, $f^{(i)}$ denotes the $i$-th derivative of $f$.) Similarly, when $f:X\\subset K^m\\to K$ is $r$-fold differentiable at $x_0\\in X$, then we write  \\begin{equation}\\label{eq:taylor-m}\nT^{<r+1}_{f,x_0}(x):=T^{\\le r}_{f,x_0}(x) := \\sum_{i \\in{\\mathbb N}^m,\\ |i|\\le r} \\frac{f^{(i)}(x_0)}{i!}(x - x_0)^i\n\\end{equation}\nfor the Taylor polynomial of degree $r$ of $f$ around $x_0$ (where (\\ref{eq:taylor-m}) uses multi-index notation, with $|i|=\\sum_{j=1}^m i_j$).\n\\end{defn}\n\n\\begin{thm}[Taylor approximations]\\label{thm:high-ord}\nLet $f\\colon K \\to K$ be a $\\emptyset$-definable function and let $r \\in {\\mathbb N}$ be given. Then there exists a finite $\\emptyset$-definable set $C$ such that for every open ball $B$ $1$-next to $C$, $f$ is $(r+1)$-fold differentiable on $B$, $|f^{(r+1)}|$ is constant on $B$, and we have:\n\\begin{equation}\\label{eq:t-higher}\n |f(x) -  T^{\\le r}_{f,x_0}(x) | \\leq |f^{(r+1)}(x_0)\\cdot (x-x_0)^{r+1}|\n\\end{equation}\nfor every $x_0, x \\in B$.\n\\end{thm}\n\n\\begin{remark}\nAs explained in Remark~\\ref{rem:acl}, here (in Theorem~\\ref{thm:high-ord}) and in the following (Corollaries~\\ref{cor:high-ord}, \\ref{cor:high-ord-S}, \\ref{cor:JP}), we can as well allow $f$ to be $(A \\cup \\mathrm{RV})$-definable for $A \\subset K$ and obtain an $A$-definable set $C$.\n\\end{remark}\n\n\nAs in the reals, we can also express the error in terms of the $r$-th derivative. We mention two different\nsuch variants of the theorem. (Both will also play a role in the proof of the theorem.)\n\n\n\\begin{cor}[Taylor approximations]\\label{cor:high-ord}\nLet $f\\colon K \\to K$ be a $\\emptyset$-definable function and let $r \\in {\\mathbb N}$ be given. Then there exists a finite $\\emptyset$-definable set $C$ such that for every open ball $B$ $1$-next to $C$, $f$ is $r$-fold differentiable on $B$, $|f^{(r)}|$ is constant on $B$, and we have:\n\\begin{equation}\\label{eq:c-higher}\n |f(x) -  T^{\\le r}_{f,x_0}(x) | \\leq |f^{(r)}(x_0)\\cdot (x-x_0)^{r+1}|\/\\rad_{\\mathrm{op}}(B)\n\\end{equation}\nfor every $x_0, x \\in B$.\n\\end{cor}\n\n\\begin{proof}\nApply (a) Theorem~\\ref{thm:high-ord} to $f$, (b) Corollary~\\ref{cor:gammaLin} to $f^{(r)}$ and (c) Corollary~\\ref{cor:prep} to $\\operatorname{rv}\\circ f^{(r)}$,\nand let $C$ be the union of the sets obtained in these ways.\nThen we have, for $B$ $1$-next to $C$ and for $x_0 \\in B$,\n\\[\n|f^{(r+1)}(x_0)| \\cdot \\rad_{\\mathrm{op}}(B) \\overset{(b)}{=} \\rad_{\\mathrm{op}}(f^{(r)}(B)) \\overset{(c)}{\\le} f^{(r)}(x_0).\n\\]\nUsing this, (\\ref{eq:t-higher}) implies (\\ref{eq:c-higher}).\n\\end{proof}\n\n\\begin{cor}[Taylor approximations]\\label{cor:high-ord-S}\nLet $f\\colon K \\to K$ be a $\\emptyset$-definable function and let $r \\in {\\mathbb N}$ be given. Then there exists a finite $\\emptyset$-definable set $C$ such that for every open ball $B$ $1$-next to $C$, $f$ is $r$-fold differentiable on $B$, $|f^{(r)}|$ is constant on $B$, and we either have\n\\begin{equation}\\label{eq:c-higher-S}\n |f(x) -  T^{\\le r}_{f,x_0}(x) | < |f^{(r)}(x_0)\\cdot (x-x_0)^{r}|\n\\end{equation}\nor $f(x) =  T^{\\le r}_{f,x_0}(x)$ (if $|f^{(r)}(x_0)| = 0$)\nfor every $x_0, x \\in B$.\n\\end{cor}\n\n\\begin{proof}\nThis follows directly from Corollary~\\ref{cor:high-ord}, using that $|x - x_0| < \\rad_{\\mathrm{op}}(B)$.\n\\end{proof}\n\nNote that in the case $r = 1$, we in particular get back the $\\mathrm{RV}_\\lambda$-Jacobian Property, studied in an analytic setting in \\cite[Theorem 6.3.7, Remark 6.3.16]{CLip}.\nMore precisely, the following version of the Jacobian Property is even uniform for all $\\lambda \\le 1$. (One also obtains the corresponding result in mixed characteristic,\nas a corollary of the mixed characteristic Taylor approximation formulated in Corollary~\\ref{cor:high-ord:mix}.)\n\n\\begin{cor}[Jacobian Property]\\label{cor:JP}\nLet $f\\colon K \\to K$ be a $\\emptyset$-definable function. Then there exists a finite $\\emptyset$-definable set $C$ such that for every $\\lambda \\le 1$ in $\\Gamma^\\times_K$, for every ball $B$ $\\lambda$-next to $C$ and for every $x_0$ and $x$ in $B$, we have:\n\\begin{enumerate}\n \\item The derivative $f'$ exists on $B$ and $\\operatorname{rv}_\\lambda\\circ f'$ is constant on $B$.\n \\item $\\operatorname{rv}_\\lambda(\\frac{f(x) - f(x_0)}{x - x_0}) = \\operatorname{rv}_\\lambda(f')$\n \\item For every open ball $B' \\subset B$, $f(B')$ is either a point or an open ball.\n\\end{enumerate}\n\\end{cor}\n\n\\begin{defn}[Jacobian Property]\\label {defn:JP}\nIf Conditions (1)--(3) of Corollary~\\ref{cor:JP} hold, we say that $f|_B$ has the \\emph{$\\mathrm{RV}_\\lambda$-Jacobian Property} (or just \\emph{Jacobian Property}, in the case $\\lambda = 1$).\n\\end{defn}\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:JP}]\n\nChoose $C$ using Corollary~\\ref{cor:high-ord} (applied to $f$, with $r = 1$); this will ensure (2), as we shall see below.\n\nTo obtain that $\\operatorname{rv}_\\lambda\\circ f'$ is constant on balls $\\lambda$-next to $C$, enlarge $C$ using Proposition-Definition~\\ref{prop:uniform}: Consider the graph of $\\operatorname{rv}_\\lambda \\circ f'$ as a subset of $K \\times \\mathrm{RV}_\\lambda \\subset K \\times \\mathrm{RV}_\\bullet$ and take the union of all of those as the set $W$ in the Proposition-Definition.\n\nFinally, to obtain that $f(B')$ is either a point or a ball, enlarge $C$ once more using Lemma~\\ref{lem:open-to-open}.\n\nIt remains to verify (2), so fix $\\lambda \\le 1$, fix a ball $B$ $\\lambda$-next to $C$, fix $x_0, x \\in B$, and let $B'$ denote the ball $1$-next to $C$ containing $B$. Then we have\n\\begin{equation}\\label{eq:to-jac}\n|f(x) -  f(x_0) - f'(x_0)(x - x_0) | \\leq |f'(x_0)\\cdot (x-x_0)^2|\/\\rad_{\\mathrm{op}}(B').\n\\end{equation}\nSince $\\rad_{\\mathrm{op}}(B) = \\lambda\\cdot \\rad_{\\mathrm{op}}(B')$, we have\n$|x - x_0| < \\lambda\\cdot \\rad_{\\mathrm{op}}(B')$,\nso after dividing both sides of (\\ref{eq:to-jac}) by $|x - x_0|$, we obtain\n\\[\n|\\frac{f(x) -  f(x_0)}{x - x_0} - f'(x_0)|< |f'(x_0)|\\cdot \\lambda,\n\\]\nwhich is equivalent to (2).\n\\end{proof}\n\n\\subsection{Proof of Taylor approximation}\n\nBefore we dive into the proof of Theorem~\\ref{thm:high-ord}, here is a somewhat technical lemma that will be needed for one special case.\n\n\\begin{lem}[Preparing equivalence relations]\\label{lem:equivalence}\nSuppose that $\\Gamma_K$ is discrete. Fix an open ball $B \\subset K$ which is\ndisjoint from $\\acl(\\emptyset)$. Suppose that $\\sim$ is an $\\emptyset$-definable binary relation on $K$ with the following properties:\n(i) the restriction of $\\sim$ to $B \\times B$ is an equivalence relation with finitely many equivalence classes; (ii) every ball $B'$ strictly contained in $B$ (i.e., $B' \\subsetneq B$) is contained in a single equivalence class.\nThen the entire ball $B$ is a single equivalence class.\n\\end{lem}\n\n\\begin{proof}\nWe write $k := {\\mathcal O}_K\/{\\mathcal M}_K$ for the residue field, $\\operatorname{res}\\colon {\\mathcal O}_K \\to k$ for the residue map, and we\nchoose a linear bijection $f\\colon B \\to {\\mathcal O}_K$. (Note that since $\\Gamma_K$ is discrete, the open ball $B$ is also closed.) Then $\\sim$ induces an equivalence relation $\\sim_k$ on $k$\n(satisfying $x \\sim x' \\iff \\operatorname{res}(f(x)) \\sim_k \\operatorname{res}(f(x'))$ for $x, x' \\in B$).\n\nFor $x \\in K$, set $W_x := \\{\\operatorname{rv}(x' - x) \\mid x' \\in K, x' \\sim x\\}$.\nSince $B$ is disjoint from $\\acl_K(\\emptyset)$, there exists a single set $W \\subset \\mathrm{RV}$ such that $W_x = W$ for all $x \\in B$\n(by Corollary~\\ref{cor:prep}). This set $W$ yields a set $T \\subset k$ such that $z \\sim_k z'$ if and only if $z - z' \\in T$ (for $z, z' \\in k$).\n\nA straight-forward computation shows: The fact that ``$z - z' \\in T$'' defines an equivalence relation on $k$ implies that $T$ is a ${\\mathbb Z}$-submodule of $k$. The number of equivalence classes is equal to the cardinality of the ${\\mathbb Z}$-module quotient $k\/T$, so to finish the proof of the lemma\\footnote{The second author would like to thank W.~Singhof for providing that argument during a coffee break.}, suppose that this quotient has finite cardinality $N > 1$.\nThen for any $z \\in k$, we have $Nz \\in T$. However, this fails if we choose any $z' \\in k \\setminus T$ and set $z := \\frac1Nz'$.\n\\end{proof}\n\n\n\nWe now come to the proof of Theorem~\\ref{thm:high-ord}. It goes by induction on $r$, where Corollaries~\\ref{cor:high-ord} and \\ref{cor:high-ord-S} are used as intermediate steps. More precisely, Theorem~\\ref{thm:high-ord} follows from the following four lemmas:\n\n\\begin{lem}\\label{lem:h:t1}\nTheorem~\\ref{thm:high-ord} holds for $r = 0$.\n\\end{lem}\n\n\\begin{lem}\\label{lem:h:ts}\nFor any $r \\ge 1$, if Theorem~\\ref{thm:high-ord} holds for $r-1$ (for every $1$-h-minimal theory)\nand Corollary~\\ref{cor:high-ord-S} holds for all $r' < r$,\nthen Corollary~\\ref{cor:high-ord-S} holds for $r$.\n\\end{lem}\n\n\n\\begin{lem}\\label{lem:h:sc}\nFor any $r \\ge 1$, if Corollary~\\ref{cor:high-ord-S} holds for $r$ (for every $1$-h-minimal theory), then Corollary~\\ref{cor:high-ord} holds for $r$.\n\\end{lem}\n\n\\begin{lem}\\label{lem:h:ct}\nFor any $r \\ge 1$, if Corollary~\\ref{cor:high-ord} holds for $r$\n(for every $1$-h-minimal theory), then Theorem~\\ref{thm:high-ord} holds for $r$.\n\\end{lem}\n\nFor all of the proofs, first note that the condition about sufficient differentiability of $f$ on $K\\setminus C$ appearing in the theorem and in the corollaries is easily obtained using Theorem~\\ref{thm:der}; this will not be further mentioned. Similarly, $|f^{(r)}|$ (or $|f^{(r+1)}|$) can easily be made constant on balls $1$-next to $C$ using Corollary~\\ref{cor:prep}.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:h:t1}]\nIn the case $r = 0$, (\\ref{eq:t-higher}) becomes $|f(x) - f(x_0)| \\le |f'(x_0)\\cdot (x - x_0)|$,\nso the theorem follows from Corollary~\\ref{cor:gammaLin}.\n\\end{proof}\n\nThe proofs of the other three lemmas are long and technical, but the main idea is simple (and the same in all three lemmas): Given a function $f$ which we want to control on a ball $B$, we define\n$g(x) := f(x) - ax^r$ in such a way that the $r$-th derivative of $g$ is small on $B$. In this way, applying the inductive assumption to $g$ yields a particularly strong bound, which will be good enough to obtain the desired bound on $f$.\n\nThe difficulty with this approach is that we need to control the parameters over which $g$ is definable. A powerful ingredient for this is\nLemma~\\ref{lem:fullyT}, which allows us to use points near $B$ as parameters. Nevertheless, various additional technical tricks are needed (different for each of the lemmas) to make the proofs work.\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:h:ts}]Namely:\\\\\nFrom $\\underbrace{|f -  T^{\\le r-1}_{f,x_0} | \\leq |f^{(r)}\\cdot (x-x_0)^{r}|}_{(\\ref{eq:t-higher})\\text{ for }r-1}$  to\n$\\underbrace{|f -  T^{\\le r}_{f,x_0} | < |f^{(r)}\\cdot (x-x_0)^{r}|}_{(\\ref{eq:c-higher-S})}$.\n\\medskip\n\nWe assume that $K$ is sufficiently saturated.\nThen it suffices to prove that on every open ball $B \\subset K$ which is disjoint from $\\acl(\\emptyset)$, either (\\ref{eq:c-higher-S}) holds, or we have $f(x) = T^{\\leq r}_{f,x_0}(x)$ (by the compactness argument\nfrom Lemma~\\ref{lem:type-0-h-min}), so fix such a $B$ and suppose that $x_0, x \\in B$ violate (\\ref{eq:c-higher-S}).\n\nWe may assume that $f^{(r)} \\ne 0$ on $B$, since\notherwise, Theorem~\\ref{thm:high-ord} for $r-1$ implies $f(x) = T^{\\leq r-1}_{f,x_0}(x) = T^{\\leq r}_{f,x_0}(x)$ for any $x_0, x \\in B$ and we are done.\n\n\\medskip\n\nCase 1: There exists an open ball $B' := B_{<\\delta}(x_0)$ containing $x$ which is strictly smaller than $B$.\n\n\\medskip\n\nWe fix the above $\\delta$ for the remainder of the proof of Case 1.\nAlso, fix any $a \\in B$ and choose any open ball $B'' = B_{< \\delta}(x_0') \\subset B$ of radius $\\delta$ disjoint from $\\acl(a)$. (Such a $B''$ exists\nby saturation.) Since $x_0$ and $x_0'$ have the same type over $\\Gamma_K$ (by Lemma~\\ref{lem:type-0-h-min}), there exists\nan $x' \\in B''$ such that $x'_0, x'$ violate (\\ref{eq:c-higher-S}).\n\nWe apply Theorem~\\ref{thm:high-ord} for $r-1$ to the function\n\\[g(x) := f(x) - \\frac{f^{(r)}(a)}{r!}\\cdot x^r.\n\\]\nSince $g$ is $a$-definable and $B''$ is disjoint from $\\acl(a)$, we obtain\n\\begin{equation}\\label{eq:gpt}\n|g(x') - T^{\\le r-1}_{g,x'_0}(x')| \\le  | g^{(r)}(x'_0)(x'-x'_0)^r|.\n\\end{equation}\nThe definition of $g$ has been chosen such\nthat $g^{(r)}(x) = f^{(r)}(x) - f^{(r)}(a)$ for all $x \\in B$, so using that\n$\\operatorname{rv}(f^{(r)})$ is constant on $B$ (this uses Corollary~\\ref{cor:prep} and that $B \\cap \\acl_K(\\emptyset) = 0$), we deduce\n\\begin{equation}\\label{eq:g<f}\n|g^{(r)}(x)| < |f^{(r)}(x)|\n\\end{equation}\nfor all $x \\in B$.\nFrom this, we obtain the following (explanation of $(\\star)$ below):\n\\begin{equation}\\label{eq:get-chS}\n\\begin{aligned}\n |f(x') -  T^{\\le r}_{f,x'_0}(x') | &\\overset{(\\star)}{=} |g(x') -  T^{\\le r}_{g,x'_0}(x') |\\\\\n &= |g(x') -  T^{\\le r-1}_{g,x'_0}(x') + \\frac{1}{r!}g^{(r)}(x_0')(x'-x'_0)^{r}|\\\\\n &\\leq \\max\\{|g(x') -  T^{\\le r-1}_{g,x'_0}(x')|,|\\frac{1}{r!}g^{(r)}(x_0')(x'-x'_0)^{r}|\\}\\\\ \n & \\overset{(\\ref{eq:gpt})}{\\le}  | g^{(r)}(x'_0)(x'-x'_0)^r| \\overset{(\\ref{eq:g<f})}{<} |f^{(r)}(x'_0)\\cdot (x'-x'_0)^{r}|\n\\end{aligned}\n\\end{equation}\n\n$(\\star)$: $f$ and $g$ differ by a polynomial of degree $r$, so their Taylor approximations differ by the same polynomial.\n\nThus (\\ref{eq:c-higher-S}) holds for $x_0', x'$, which\nis a contradiction and hence finishes the proof of Case~1.\n\n\\medskip\n\nCase 2: The only open ball $B' \\subset B$ containing both $x_0$ and $x$ is already $B$ itself.\n\n\\medskip\n\nNote that this can only happen if $\\Gamma_K$ is discrete (otherwise the radius of $B'$ can be taken strictly between\n$|x_0 - x|$ and $\\rad_{\\mathrm{op}}(B)$), and it also means that $|x_0 - x| = \\rad_{\\mathrm{cl}}(B) =: \\delta$.\n\nBy Case~1, we may assume that (\\ref{eq:c-higher-S}) holds on every proper subball of $B$. Moreover,\nwe apply Corollary~\\ref{cor:high-ord-S} inductively to derivatives of $f$ to get, for $1 \\le i \\le r$:\n\\begin{equation}\\label{eq:der}\n  |f^{(i)}(x) -  T^{\\le r-i}_{f^{(i)},x_0}(x) | < |f^{(r)}|\\cdot |(x-x_0)^{r-i}| \\le  |f^{(r)}|\\cdot \\delta^{r-i}\n\\end{equation}\nfor every $x_0, x \\in B$. (Note that $|f^{(r)}|$ is constant on $B$; here and in the sequel, we just write $|f^{(r)}|$\ninstead of $|f^{(r)}(x)|$ for $x \\in B$.)\n\n\\medskip\n\nStep 2.1: Set\n\\[\nd(x_0, x) :=T^{\\le r}_{f,x_0}(x) - f(x_0) = \\sum_{i = 1}^r \\frac{f^{(i)}(x_0)}{i!}(x - x_0)^i\n\\]\nfor $x_0, x \\in B$. Then, for any $x_1, x_2, x_3 \\in B$, we have\n\\begin{equation}\\label{eq:additive}\n|d(x_1, x_2) + d(x_2, x_3) - d(x_1, x_3)| < |f^{(r)}| \\cdot |x_2-x_3|\\cdot \\delta^{r-1}.\n\\end{equation}\n\n\\medskip\n\nProof:\nThis is a straight-forward computation, consisting in using (\\ref{eq:der}) to approximate the derivatives appearing in $d(x_2,x_3)$\nby their Taylor series around $x_1$. The details are as follows. In the computation,\n``$\\approx$'' means that the norm of the difference is less than $ |f^{(r)}| \\cdot |x_2-x_3|\\cdot \\delta^{r-1}$.\n\\begin{align*}\nd(x_2, x_3) &= \\sum_{i = 1}^r \\frac{f^{(i)}(x_2)}{i!}(x_3 - x_2)^i \\approx\n\\sum_{i = 1}^r\\sum_{j = 0}^{r-i} \\frac{f^{(i+j)}(x_1)}{i!\\cdot j!}(x_2 - x_1)^j(x_3 - x_2)^i\n\\\\\n&=\\sum_{k = 1}^r \\underbrace{\\sum_{\\substack{i+j=k,\\\\i>0}}\\frac{k!}{i!\\cdot j!}(x_2 - x_1)^j(x_3 - x_2)^i}_{= ((x_2 - x_1) + (x_3 - x_2))^k - (x_2 - x_1)^k(x_3 - x_2)^0}\\frac{f^{(i+j)}(x_1)}{k!}\n\\\\\n&=\\sum_{k = 1}^r \\big((x_3-x_1)^k - (x_2 - x_1)^k\\big)\\frac{f^{(k)}(x_1)}{k!} = d(x_1, x_3)  - d(x_1, x_2)\n\\end{align*}\n\\qed2.1\n\n\\medskip\n\nFor $x_1, x_2 \\in K$, define\n\\[\nx_1 \\sim x_2 :\\iff |f(x_2) - f(x_1) - d(x_1, x_2)| < |f^{(r)}| \\cdot \\delta^r\n\\]\n(with some arbitrary convention if derivatives at $x_1 \\notin B$ do not exist).\nThis relation is definable using only the parameter $|f^{(r)}| \\cdot \\delta^r \\in \\Gamma$.\nOur aim is to verify the prerequisites of Lemma~\\ref{lem:equivalence} (in the language where $|f^{(r)}| \\cdot \\delta^r$ has been added as a constant), but before that, let us verify that this then finishes the proof:\nThe lemma then implies that all elements of $B$ are $\\sim$-equivalent. In particular,\nfor our original $x_0, x \\in B$ satisfying $|x_0 - x| = \\delta$, we obtain\n\\begin{equation}\\label{eq:lem-finishes}\n|f(x) - T^r_{f,x_0}(x)| =\n|f(x) - f(x_0) - d(x_0, x)| < |f^{(r)}| \\cdot \\delta^r,\n\\end{equation}\ni.e., (\\ref{eq:c-higher-S}) holds, as desired.\n\n\\medskip\n\n\nStep 2.2: The restriction of $\\sim$ to $B \\times B$ is an equivalence relation.\n\n\n\\medskip\n\nProof: Fix $x_0 \\in B$ and define, for $x \\in B$:\n\\begin{equation}\\label{eq:ftilde}\n\\tilde f(x) := f(x) - d(x_0, x).\n\\end{equation}\nThen, using ``$\\approx$'' to mean that the difference has norm less than $|f^{(r)}|\\cdot \\delta^r$, we have:\n\\begin{equation}\\label{eq:with_x0}\n\\begin{aligned}\n \\tilde f(x_2) - \\tilde f(x_1)\n&= f(x_2) - f(x_1) - d(x_0, x_2) + d(x_0, x_1)\\\\\n&\\overset{(\\ref{eq:additive})}{\\approx} f(x_2) - f(x_1) - d(x_1, x_2)\n\\end{aligned}\n\\end{equation}\nSo $x_1 \\sim x_2$ if and only if $\\tilde f(x_2) \\approx \\tilde f(x_1)$, which is clearly an equivalence relation.\n\\qed2.2\n\n\\medskip\n\n\nStep 2.3: Each proper subball $B' \\subsetneq B$ is contained in a single equivalence class of $\\sim$.\n\n\\medskip\n\nProof: This follows from our assumption that $f$ satisfies (\\ref{eq:c-higher-S}) on $B'$ (using a similar computation as in (\\ref{eq:lem-finishes})).\n\\qed2.3\n\n\\medskip\n\nStep 2.4: $\\sim$ has only finitely many equivalence classes on $B$.\n\n\\medskip\n\nProof: Consider $\\tilde f$ as in (\\ref{eq:ftilde}).\nIf $x_1, x_2 \\in B$ satisfy $|x_1 - x_2|\\le |\\varpi|\\cdot\\delta$ (where $\\varpi\\in{\\mathcal O}_K$ is a uniformizer), then\nin the ``$\\approx$'' of (\\ref{eq:with_x0}), the difference is less than $|f^{(r)}|\\cdot |x_1 - x_2|\\cdot \\delta^{r-1} \\le |f^{(r)}|\\cdot|\\varpi|\\cdot\\delta^r$, and the right hand side of (\\ref{eq:with_x0}) satisfies\n\\[\n|f(x_2) - f(x_1) - d(x_1, x_2)| = |f(x_2) - T^r_{f,x_1}(x_2)| \\overset{\\text{Case~1}}{<} |f^{(r)}|\\cdot |x_1-x_2|^r.\n\\]\nThus $|\\tilde f(x_2) - \\tilde f(x_1)| < |f^{(r)}|\\cdot|\\varpi|\\cdot\\delta ^r$, i.e., for any open ball $B' \\subset B$ of (open) radius $\\delta$,\n$\\tilde f(B')$ is contained in an open ball of (open) radius $|f^{(r)}|\\cdot|\\varpi|\\cdot\\delta^{r}$. By Lemma~\\ref{lem:balltosmall}, $\\tilde f(B)$ is therefore contained in the union of finitely many closed balls of (closed) radius $|f^{(r)}|\\cdot|\\varpi|\\cdot\\delta^{r}$. By the characterization of $\\sim$ from Step~2.2, each such closed ball corresponds exactly to one equivalence class of $\\sim$, so we are done.\n\\qed2.4\n\n\\medskip\n\nThese were all the prerequisites needed for Lemma~\\ref{lem:equivalence}, so this finishes the proof of Case~2 and hence of the entire Lemma.\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:h:sc}]Namely:\\\\\nFrom $\\underbrace{|f -  T^{\\le r}_{f,x_0} | < |f^{(r)}\\cdot (x-x_0)^{r}|}_{(\\ref{eq:c-higher-S})}$  to\n$\\underbrace{|f -  T^{\\le r}_{f,x_0} | \\leq |f^{(r)}\\cdot (x-x_0)^{r+1}|\/\\rad_{\\mathrm{op}}(B)}_{(\\ref{eq:c-higher})}$.\n\\medskip\n\nWe assume that $K$ is sufficiently saturated.\n\n\n\\medskip\n\nStep 1: The corollary follows if we can verify that for every $a \\in K$, Inequality (\\ref{eq:c-higher}) holds for every ball $B \\subset K \\setminus \\acl_K(a)$ that is $1$-next to $a$.\n\n\\medskip\n\nProof:\nAn easy computation shows that given any open ball $B$, Inequality (\\ref{eq:c-higher}) holds on $B$ if and only if for every closed subball $B' \\subset B$, and every \\(x_0, x \\in B'\\), we have the following corresponding strict inequality:\n\\begin{equation}\\label{eq:c-higher:c}\n| f(x) -  T^{\\le r}_{f,x_0}(x) | < |f^{(r)}(x_0)\\cdot(x - x_0)^{r+1}| \/ \\rad_{\\mathrm{cl}}(B').\n\\end{equation}\nIn particular, for $a$ and $B$ as in the assumption of Step~1, every closed subball of $B$ satisfies (\\ref{eq:c-higher:c}).\nThis allows us to apply Lemma~\\ref{lem:fullyT}, where we take ${\\mathcal B}$ to be the set of closed balls on which (\\ref{eq:c-higher:c}) fails.\nThe lemma yields a finite $\\emptyset$-definable set $C \\subset K$\nintersecting each ball in ${\\mathcal B}$ and hence also intersecting (as desired) each open ball where (\\ref{eq:c-higher}) does not hold.\\qed1\n\n\\medskip\n\nFor the remainder of the proof, let $a \\in K$ be given and let $B$ be a ball which is disjoint from $\\acl_K(a)$ and $1$-next to $a$. We may assume that $f^{(r)}$ is nowhere $0$ on $B$, since otherwise, it would be $0$ on all of $B$ (since $B \\cap \\acl_K(\\emptyset) = \\emptyset$) and Corollary~\\ref{cor:high-ord-S} would yield $f(x)  = T^{\\le r}_{f,x_0}(x)$ for $x \\in B$.\n\nSuppose that (\\ref{eq:c-higher}) fails on $B$.\nChoose $x, x_0 \\in B$ witnessing this failure and fix, for the remainder of the proof,\n\\begin{equation}\\label{eq:counter-d}\n\\delta := |x - x_0|\n\\end{equation}\nand\n\\begin{equation}\\label{eq:counter-a}\n\\alpha := \\frac{ |f(x) -  T^{\\le r}_{f,x_0}(x) | }{ |f^{(r)}(x_0)\\cdot (x-x_0)^{r+1}|\/\\rad_{\\mathrm{op}}(B)};\n\\end{equation}\nin other words, $\\alpha > 1$ is the factor by which (\\ref{eq:c-higher}) fails for $x_0$, $x$.\nMoreover, set\n\\[\n\\gamma := \\min\\{\\delta \\cdot \\alpha, \\rad_{\\mathrm{op}}(B)\\}\n\\]\n(so that the ball $B_{< \\gamma}(x_0)$ is contained in $B$ and is not much bigger than the smallest ball containing $x_0$ and $x$).\n\n\\medskip\n\nStep 2: There exists an open ball $B' \\subset B$ of radius $\\gamma$ containing a ``$(\\delta,\\alpha)$-counter-example to (\\ref{eq:c-higher})''\nwith the additional properties that $B'$ is 1-next to some $a' \\in K$ and $B' \\cap \\acl_K(a, a') = \\emptyset$. By a $(\\delta,\\alpha)$-counter-example to (\\ref{eq:c-higher}),\nwe mean a pair $x_0, x \\in B'$ with\n$|x - x_0| = \\delta$ satisfying\n\\begin{equation}\\label{eq:da}\n|f(x) -  T^{\\le r}_{f,x_0}(x) | \\ge \\alpha\\cdot |f^{(r)}(x_0)\\cdot (x-x_0)^{r+1}|\/\\rad_{\\mathrm{op}}(B).\n\\end{equation}\n\n\\medskip\n\nProof: If $\\gamma = \\rad_{\\mathrm{op}}(B)$, we can simply take $B' = B$ and $a' = a$, so now suppose that $\\gamma < \\rad_{\\mathrm{op}}(B)$.\n\nWe claim that the fact that $B$ contains a $(\\delta,\\alpha)$-counter-example already implies that\neach open ball $B_{< \\gamma}(z) \\subset B$ of radius $\\gamma$ contains a $(\\delta,\\alpha)$-counter-example.\nIndeed, the set\n\\[\nZ := \\{z \\in K \\mid \\exists x_0, x \\in B_{< \\gamma}(z)\\colon (x_0, x)\\text{ is a $(\\delta,\\alpha)$-counter-example}\\}\n\\]\nis definable using only some value group parameters (namely $\\alpha$, $\\delta$ and $\\rad_{\\mathrm{op}}(B)$), so since $B \\cap \\acl_K(\\emptyset) = \\emptyset$, we have either $B \\subset Z$ or $B \\cap Z= \\emptyset$. In particular, the existence of a single $(\\delta,\\alpha)$-counter-example in $B$ already implies $B \\subset Z$,\nwhich proves our claim.\n\nNow fix any $a' \\in B$ and fix an open ball $B'$ with $\\rad_{\\mathrm{op}}(B') = \\gamma$, which is $1$-next to $a'$ and such that $B' \\cap \\acl_K(a, a') = \\emptyset$; such a $B'$ exists by saturation of $K$.\nSince $\\gamma < \\rad_{\\mathrm{op}}(B)$, we have $B' \\subset B$ and by the previous paragraph, $B'$ contains a $(\\delta,\\alpha)$-counter-example. Hence $a'$ and $B'$ are as desired.\\qed2\n\n\\medskip\n\nThe remainder of the proof now consists in proving that there is no $(\\delta,\\alpha)$-counter-example on $B'$ (so that we have a contradiction). More precisely, fix $x_0, x \\in B'$ with $|x_0 - x| = \\delta$; our goal is to prove that (\\ref{eq:da}) does not hold.\n\n\\medskip\n\nStep 3: There exists a $d \\in \\acl_K(a')$ such that\nthe image $f^{(r)}(B')$ is either a ball $1$-next to $d$, or equal to the singleton $\\{d\\}$.\n\n\\medskip\n\nProof: Apply Proposition~\\ref{prop:range} to compatibly prepare the domain and the image of $f^{(r)}$ using finite $a'$-definable sets $C$ and $D$, where we additionally impose $a' \\in C$. Since $B' \\cap \\acl_K(a') = \\emptyset$, $B'$ is a ball $1$-next to $C$ and the claim follows.\n\\qed3\n\n\\medskip\n\nStep~4: We apply Corollary~\\ref{cor:high-ord-S} to the function\n\\[g(x) := f(x) - \\frac{d}{r!}\\cdot x^r.\n\\]\nSince $g$ is $\\{d\\}$-definable and $B' \\cap \\acl_K(d) = \\emptyset$, we obtain\n\\begin{equation}\\label{eq:gprep}\n|f(x) -  T^{\\le r}_{f,x_0}(x) | = |g(x) - T^{\\le r}_{g,x_0}(x)| <  | g^{(r)}(x_0)\\cdot(x-x_0)^r|\n\\end{equation}\n(where $x_0$ and $x$ are as fixed above Step~3). The first equality follows from the fact that $f$ and $g$ differ by a polynomial of degree $r$, so their $r$-th Taylor approximations differ by the same polynomial.\n\n\\medskip\n\nStep~5a: If $f^{(r)}$ is non-constant on $B'$, we obtain\n\\begin{equation}\\label{eq:s6}\n|f(x) - T^{\\le r}_{f,x_0}(x)| < \\rad_{\\mathrm{op}}(f^{(r)}(B')) \\cdot |x-x_0|^r.\n\\end{equation}\n\n\\medskip\n\nProof: By definition of $g$ and by our choice of $d$ (in Step~3), we have\n\\begin{equation}\\label{eq:use-d}\n|g^{(r)}(x_0)| = |f^{(r)}(x_0) - d| = \\rad_{\\mathrm{op}}(f^{(r)}(B')).\n\\end{equation}\nCombining this with (\\ref{eq:gprep}) yields (\\ref{eq:s6}).\n\\qed5a\n\n\\medskip\n\nStep~5b: If $f^{(r)}$ is constant on $B'$, we obtain\n\\[\n|f(x) - T^{\\le r}_{f,x_0}(x)| = 0.\n\\]\n\n\\medskip\n\nProof: Instead of (\\ref{eq:use-d}), we have $g^{(r)}(x_0) = f^{(r)}(x_0) - d = 0$; apply this in the same way as in Step 5a.\\qed5b\n\n\\medskip\n\nIn the constant case (as in Step 5b), we are already done for the lemma, since the right hand side of (\\ref{eq:da}) is non-zero, a contradiction. (Recall that we assume $f^{(r)} \\ne 0$.)\n\nThe last ingredient for the non-constant case is the following:\n\n\\medskip\n\nStep 6: We have $\\rad_{\\mathrm{op}}(f^{(r)}(B')) \\le \\alpha\\cdot |f^{(r)}(x_0)|\\cdot|x-x_0|\/\\rad_{\\mathrm{op}}(B)$.\n\n\\medskip\n\nProof: Using that $\\operatorname{rv}(f^{(r)})$ is constant on $B$, we obtain $\\rad_{\\mathrm{op}}(f^{(r)}(B)) \\le |f^{(r)}(x_0)|$.\nFrom this and by applying Lemma~\\ref{lem:gammaLin} to $f^{(r)}$, we deduce that $\\rad_{\\mathrm{op}}(f^{(r)}(B')) \\le |f^{(r)}(x_0)|\\cdot \\rad_{\\mathrm{op}}(B')\/\\rad_{\\mathrm{op}}(B)$.\nCombining this with $\\rad_{\\mathrm{op}}(B') = \\gamma \\le \\alpha\\cdot |x - x_0|$ yields the claim.\\qed6\n\n\\medskip\n\nNow Steps 5a and 6 together imply that (\\ref{eq:da}) fails, as desired, so we are done.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:h:ct}]Namely:\\\\\nFrom  $\\underbrace{|f -  T^{\\le r}_{f,x_0} | \\leq |f^{(r)}\\cdot (x-x_0)^{r+1}|\/\\rad_{\\mathrm{op}}(B)}_{(\\ref{eq:c-higher})}$  to\n$\\underbrace{|f -  T^{\\le r}_{f,x_0} | \\leq |f^{(r+1)}\\cdot (x-x_0)^{r+1}|}_{(\\ref{eq:t-higher})}$.\n\\medskip\n\nLet ${\\mathcal B}$ be the set of all closed balls on which (\\ref{eq:t-higher}) does not hold.\nThe strategy is to use Lemma~\\ref{lem:fullyT} to find a finite $\\emptyset$-definable $C$\nmeeting every $B \\in {\\mathcal B}$. Note that then we are done,\nsince if (\\ref{eq:t-higher}) fails for some $x, x_0 \\in B'$, where $B'$ is a ball $1$-next\nto $C$, then (\\ref{eq:t-higher}) also fails on a ball from ${\\mathcal B}$, namely the smallest (closed) ball containing $x$ and $x_0$.\n\nSo as needed for Lemma~\\ref{lem:fullyT}, let $a \\in K$ be given and let $B$ be an open ball $1$-next to $a$ satisfying $B \\cap \\acl_K(a) = \\emptyset$. We need to verify that (\\ref{eq:t-higher}) holds on $B$.\n\nBy applying Proposition~\\ref{prop:range} to $f^{(r)}$ and to a set $C_0$ containing $a$, we find a $d \\in \\acl_K(a)$ such that either $f^{(r)}(B) = \\{d\\}$ or\n$f^{(r)}(B)$ is a ball $1$-next to $d$.\n\nNow apply Corollary~\\ref{cor:high-ord} to $g(x) := f(x) - \\frac{d}{r!}x^r$. Since $B$ is disjoint from the algebraic closure of the parameters used to define $g$,\nwe obtain\n\\begin{equation}\\label{eq:gT}\n|f(x) - T^{\\le r}_{f,x_0}(x)| = |g(x) - T^{\\le r}_{g,x_0}(x)| \\le   |g^{(r)}(x_0)\\cdot (x - x_0)^{r+1}| \/ \\rad_{\\mathrm{op}}(B)\n\\end{equation}\nfor every $x, x_0 \\in B$. As before, the first equality holds because \\(f\\) and \\(g\\) differ by a polynomial of degree \\(r\\). To finish the proof, it now remains to bound the right hand side of (\\ref{eq:gT}) by $|f^{(r+1)}(x_0)\\cdot (x-x_0)^{r+1}|$.\n\nIf $f^{(r)}(B) = \\{d\\}$, then $g^{(r)}(x_0) = 0$ and we are done.\nOtherwise,  $g^{(r)}(B)$ is a ball $1$-next to $0$, so $\\rad_{\\mathrm{op}}(g^{(r)}(B)) = |g^{(r)}(x_0)|$.\nMoreover, by Corollary~\\ref{cor:gammaLin} applied to $g^{(r)}$, we have $\\rad_{\\mathrm{op}}(g^{(r)}(B)) = \\rad_{\\mathrm{op}}(B)\\cdot |g^{(r+1)}(x_0)| = \\rad_{\\mathrm{op}}(B)\\cdot |f^{(r+1)}(x_0)|$.\nCombining these two equations yields the desired bound on the right hand side of (\\ref{eq:gT}).\n\\end{proof}\n\n\\section{Resplendency}\\label{sec:respl}\n\nThe goal of this section is to show that Hensel minimality behaves well with respect to expansions of the language by predicates living only on $\\mathrm{RV}$, one of the main results (Theorem~\\ref{thm:resp:h}) being that if the ${\\mathcal L}$-theory of a valued field $K$ is $0$-, $1$- or $\\omega$-h-minimal, then so is its $\\cLe$-theory, where $\\cLe$ is any $\\mathrm{RV}$-expansion of ${\\mathcal L}$.\n\nThe key ingredient to Theorem~\\ref{thm:resp:h} is Proposition~\\ref{prop:equiv}, which in some sense is even stronger: Any set $X \\subset K$ definable in the expanded language $\\cLe$ can already be prepared by a finite set definable in the smaller language ${\\mathcal L}$. Using this, it often becomes possible, given a completely arbitrary set $Z \\subset \\mathrm{RV}^k$ to ''without loss assume that $Z$ is definable''. This turns out to be pretty useful to get rid of some technicalities related to cell decomposition in valued fields; the preparations for this are done in Subsection~\\ref{sec:alg:skol}.\n\nUnder the assumption of $\\omega$-h-minimality,\npreparation can also be generalized to more general leading term structures, namely:\nOne can define $\\mathrm{RV}_{I} := (K^\\times\/(1+I)) \\cup \\{0\\}$ for arbitrary proper definable ideals $I \\subset {\\mathcal O}_K$, and for most such $I$, any subset of $K$ which is definable using parameters from $\\mathrm{RV}_I$\ncan be ``$I$-prepared'' (see Theorem~\\ref{thm:all:I}). Using this, we deduce that if the theory of a valued field is $\\omega$-h-minimal, then so is the\ntheory of the field with any coarsened (not necessarily definable) valuation (Corollary~\\ref{cor:coarse}).\n\nNote that the proofs in this section need somewhat deeper methods from model theory than the remainder of the paper. In particular, the emphasis shifts from a geometric description of definable sets to questions revolving around the extension of automorphisms.\n\n\n\\subsection{Resplendency for a fixed ideal}\n\\label{sec:fixI}\n\nAs convened earlier, $K$ is a valued field of equi-characteristic zero, considered as a structure in a language ${\\mathcal L}$ containing ${\\mathcal L}_\\val $.\nFor this entire subsection, we fix a proper definable (with parameters) ideal $I \\subset {\\mathcal O}_K$.\nWe start by defining the $I$-version of preparation.\n\n\\begin{defn}[$I$-preparing sets]\\label{defn:Inext}\n\\begin{enumerate}\n \\item We define $\\mathrm{RV}_I$ to be the disjoint union of the\nquotient $K^\\times\/(1+I)$ with $\\{0\\}$, and we write $\\operatorname{rv}_I$ for the map $K\\to \\mathrm{RV}_I$ which extends the projection map $K^\\times\\to K^\\times\/(1+I)$ by sending $0$ to $0$.\n \\item\n We say that a ball $B \\subset K$ is \\emph{$I$-next} to $c$ for some $c\\in K$ if $B = c + \\operatorname{rv}_I^{-1}(\\xi)$\nfor some (nonzero) element $\\xi$ of $\\mathrm{RV}_I$.\nWe say that $B$ is \\emph{$I$-next} to $C$ for some finite (non-empty) set $C\\subset K$\nif $B$ equals $\\bigcap_{c\\in C} B_c$\nwith $B_c$ a ball $I$-next to $c$ for each $c\\in C$.\n \\item Let $C$ be a finite non-empty subset of $K$.\nWe say that a set $X\\subset K$ is \\emph{$I$-prepared} by $C$ if belonging to $X$ depends only on the tuple $(\\operatorname{rv}_{I}(x-c))_{c\\in C}$,\nor, equivalently, if every ball $I$-next to $C$ is either contained in $X$ or disjoint from $X$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{remark}\nIf $I = B_{<\\lambda}(0)$ for some $\\lambda \\le 1$ in $\\Gamma_K$, then of course we have $\\mathrm{RV}_I = \\mathrm{RV}_\\lambda$, $I$-next means $\\lambda$-next, and $I$-prepared means $\\lambda$-prepared.\n\\end{remark}\n\nBy considering $I$ as a member of a $\\emptyset$-definable family of proper ideals of ${\\mathcal O}_K$, we see that\n$\\mathrm{RV}_{I}$ is a definable subset of an imaginary sort. In particular, it makes sense to work using parameters from $\\mathrm{RV}_I$.\n\n\\begin{defn}[Having $I$-preparation]\\label{defn:Iprepared}\nWe say that $K$ has \\emph{$I$-preparation}\nif for every set $A\\subset K$ and every $(A\\cup \\mathrm{RV}_I)$-definable subset $X\\subset K$,\nthere exists a finite $A$-definable set $C \\subset K$ such that\n$X$ is $I$-prepared by $C$.\n\\end{defn}\n\n\\begin{remark}\\label{rem:prep:vs:h}\nNote how this is related to Hensel minimality: $\\operatorname{Th}(K)$ is $0$-h-minimal iff every $K' \\equiv K$ has ${\\mathcal M}_{K'}$-preparation, and $\\operatorname{Th}(K)$ is $\\omega$-h-minimal iff every $K' \\equiv K$ has $B_{<\\lambda}(0)$-preparation for every $\\lambda \\le 1$ in $\\Gamma^\\times_{K'}$.\n\\end{remark}\n\nBy ``resplendent $I$-preparation'', we mean that one can $I$-prepare sets that are definable in arbitrary expansions of ${\\mathcal L}$ by predicates on $\\mathrm{RV}_I$. More precisely:\n\n\\begin{defn}[Resplendent $I$-preparation]\\label{defn:Iresplendent}\n\\begin{enumerate}\n\\item\nAn \\emph{$\\mathrm{RV}_I$-expansion} of ${\\mathcal L}$ (or of any other language we shall consider below) is an expansion obtained from ${\\mathcal L}$ by adding arbitrary\npredicates which live on Cartesian powers of the (imaginary) definable set $\\mathrm{RV}_I$.\n\\item\nWe say that $K$ has \\emph{resplendent $I$-preparation} if for every set $A\\subset K$, for every $\\mathrm{RV}_I$-expansion ${\\mathcal L}'$ of ${\\mathcal L}$, and\nfor every ${\\mathcal L}'(A)$-definable subset $X\\subset K$,\nthere exists a finite ${\\mathcal L}(A)$-definable set $C \\subset K$ such that $X$ is $I$-prepared by $C$.\n\\end{enumerate}\n\\end{defn}\n\nNote that we intentionally require $C$ to be definable in the smaller language ${\\mathcal L}$.\n\nFor the remainder of this subsection, we will assume that $I$ is definable without parameters. In this case,\nit also makes sense to introduce the notions of $I$-preparation for the theory of $K$:\n\n\\begin{defn}[$I$-preparation for theories]\\label{defn:I:T}\nSuppose that $I$ is $\\emptyset$-definable (as convened for the remainder of this subsection).\n\\begin{enumerate}\n\\item\nGiven $K' \\equiv K$, we write $I_{K'}$ for the ideal of ${\\mathcal O}_{K'}$ defined by the formula which defines $I$ in $K$.\n\\item\nWe say that $\\operatorname{Th}(K)$ has \\emph{$I$-preparation} if every model $K' \\equiv K$ has $I_{K'}$-preparation.\n\\item\nWe say that $\\operatorname{Th}(K)$ has \\emph{resplendent $I$-preparation} if every model $K' \\equiv K$ has resplendent $I_{K'}$-preparation.\n\\end{enumerate}\n\\end{defn}\n\n(In particular, for $\\operatorname{Th}(K)$, having ${\\mathcal M}_K$-preparation is the same as $0$-h-minimality.)\n\nSince adding parameters from $\\mathrm{RV}_I$ is a specific kind of $\\mathrm{RV}_I$-expansion, resplendent $I$-preparation clearly implies $I$-preparation. The central result of this subsection is the converse given in the following proposition:\n\n\n\n\\begin{prop}[Resplendency]\\label{prop:equiv}\nSuppose that $I$ is $\\emptyset$-definable. Then the following are equivalent:\n\\begin{enumerate}[(i)]\n\\item $\\operatorname{Th}(K)$ has $I$-preparation.\n\\item $\\operatorname{Th}(K)$ has resplendent $I$-preparation.\n\\end{enumerate}\n\\end{prop}\n\nNote that the proposition in particular implies that $0$-h-minimal theories have resplendent ${\\mathcal M}_K$-preparation;\nin Theorem~\\ref{thm:all:I}, we will see a strong version of this for $\\omega$-h-minimality.\n\nThe proof of (i) $\\Rightarrow$ (ii) requires a number of lemmas which we will prove now. It will be sufficient to consider models $K$ that are sufficiently saturated, so from now on, we fix such a sufficiently saturated $K$: We assume $K$ to be $\\kappa$-saturated for some $\\kappa >|{\\mathcal L}|$. As usual, we call a set \\emph{small} if its cardinality is less than $\\kappa$.\n\n\\begin{conv}\nFor the remainder of Subsection~\\ref{sec:fixI}, we consider ${\\mathcal L}$ as a genuine two-sorted language, with sorts $K$ and $\\mathrm{RV}_{I}$.\n\\end{conv}\n\nLet us first rephrase preparation in terms of definability in a certain sublanguage, namely:\n\\begin{defn}[$\\Lbas$]\nLet $\\Lbas$ be the language $\\{0, +, -\\} \\cup \\{s\\cdot \\mid s\\in {\\mathbb Q}\\}$ of ${\\mathbb Q}$-vector spaces on the valued field $K$\n(where ``$s\\cdot$'' denotes multiplication by $s$), together with\nthe sort $\\mathrm{RV}_{I}$ and the map $\\operatorname{rv}_{I}$.\n\\end{defn}\n\n\\begin{lem}[Preparation in terms of $\\Lbas$]\\label{lem:prep eq qf}\nFor every (not necessarily definable) $X\\subset K$ and every ${\\mathbb Q}$-vector space $A \\subset K$, the following are equivalent:\n\\begin{enumerate}\n\\item There exists a finite set $C\\subset A$ such that $X$ is $I$-prepared by $C$.\n\\item There exists an $\\mathrm{RV}_I$-expansion $\\Lebas$ of $\\Lbas$ such that $X$ is defined by a quantifier free $\\Lebas(A)$-formula.\n\\item There exists an $\\mathrm{RV}_I$-expansion $\\Lebas$ of $\\Lbas$ such that $X$ is defined by a field quantifier free $\\Lebas(A)$-formula.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n(2) $\\Rightarrow$ (3) is clear.\n\n(1) $\\Rightarrow$ (2): Let us assume that (1) holds and let $C = \\{c_1,\\ldots,c_k\\}$, $f(x) := (\\operatorname{rv}_{I}(x-c_1),\\ldots,\\operatorname{rv}_{I}(x-c_k))$ and $Y := f(X) \\subset \\mathrm{RV}_{I}^k$. Then, because $X$ is $I$-prepared by $C$, $X = f^{-1}(Y)$, and this is quantifier free definable in $\\Lbas(A)\\cup\\{Y\\}$.\n\n(3) $\\Rightarrow$ (1): Every field quantifier free $\\Lebas(A)$-formula in a single variable $x$ is equivalent\nto one of the form $\\phi(\\operatorname{rv}_{I}(m_1x + c_1),\\ldots,\\operatorname{rv}_{I}(m_\\ell x + c_\\ell))$, where $\\phi$ is a formula living entirely in the sort $\\mathrm{RV}_I$, $m_i \\ne 0$ are rational numbers and $c_i$ are elements of $A$. Since $\\operatorname{rv}_{I}(x + c_i\/m_i)$ determines\n$m_i\\operatorname{rv}_{I}(x + c_i\/m_i) = \\operatorname{rv}_{I}(m_i x+ c_i)$,\n$X$ is $I$-prepared by $C = \\{- c_1\/m_1, \\dots, -c_\\ell\/m_\\ell\\} \\subset A$.\n\\end{proof}\n\nWe now recall some general model theoretic notions.\n\n\\begin{notn}\nIn the remainder of this subsection, we use the following conventions common in model theory:\n\\begin{itemize}\n \\item\nGiven a set $A$ and a tuple of variables $x$, we write $A^x$ for the Cartesian power of $A$ corresponding to the length of the tuple of variables $x$.\n\\item\nGiven sets $A$ and $B$, we sometimes write $AB$ for their union, and we freely interpret tuples as sets.\n\\end{itemize}\n\\end{notn}\n\n\n\\begin{defn}[Partial (elementary) isomorphisms]\nSuppose that $\\cLarb$ is an arbitrary language, $M$ and $N$ are $\\cLarb$-structures, $A\\subset M$, $B\\subset N$ and $f \\colon A\\to B$ a bijection.\n\\begin{itemize}\n\\item We say that $f$ is a \\emph{partial $\\cLarb$-isomorphism} if for every quantifier free $\\cLarb$-formula $\\phi(x)$ and $a\\in A^{x}$, $M\\models\\phi(a)$ if and only if $N\\models\\phi(f(a))$.\n\\item We say that $f$ is a \\emph{partial elementary $\\cLarb$-isomorphism} if for every $\\cLarb$-formula $\\phi(x)$ and $a\\in A^{x}$, $M\\models\\phi(a)$ if and only if $N\\models\\phi(f(a))$.\n\\end{itemize}\n\\end{defn}\n\nNote that any partial $\\cLarb$-isomorphism has a unique extension to the $\\cLarb$-structure generated by its domain. We implicitly identify $f$ and this extension.\n\n\n\nIn the following, for a subset $A \\subset K$, $\\langle A \\rangle_{\\mathbb Q}$ denotes the ${\\mathbb Q}$-sub-vector space  of $K$ generated by $A$.\n\n\\begin{remark}\\label{rem:join iso}\nFor any set $A \\subset K$, the $\\Lbas$-substructure of $K$ generated by $A$ consists of $\\langle A\\rangle_{{\\mathbb Q}}$\ntogether with its image $\\operatorname{rv}_{I}(\\langle A\\rangle_{{\\mathbb Q}})$ in $\\mathrm{RV}_{I}$.\nIn particular, since $\\Lbas$ has no language on $\\mathrm{RV}_I$ (except for the maps $\\operatorname{rv}_{I}$), we have:\nTo obtain that a sort-preserving map $f\\colon A_1 \\to A_2$ (for some $A_1, A_2 \\subset K \\cup \\mathrm{RV}_I$)\nis a partial $\\Lbas$-automorphism, it suffices to verify that the restriction $f|_{A_1 \\cap K}$ is a partial $\\Lbas$-automorphism and that on $\\tilde A_1 := A_1 \\cap \\operatorname{rv}_I(A_1 \\cap K)$, the map induced by $f|_{A_1 \\cap K}$ agrees with $f|_{\\tilde A_1}$.\n\\end{remark}\n\n\n\n\\begin{lem}[Preparation in terms of partial isomorphisms]\\label{lem:prep eq iso}\nLet $A \\subset K$ be a small ${\\mathbb Q}$-sub-vector space.\nThe following are equivalent:\n\\begin{enumerate}[(i)]\n\\item Any ${\\mathcal L}(A\\cup \\mathrm{RV}_I)$-definable set $X\\subset K$ can be $I$-prepared by some finite set $C\\subset A$.\n\\item For all $A_2\\subset K$, $c_1, c_2 \\in K$ and all (potentially large) $B_1, B_2\\subset \\mathrm{RV}_I$\nwith $\\operatorname{rv}_I(\\langle A,c_1\\rangle_{\\mathbb Q}) \\subset B_1$,\nif $f\\colon A B_1 c_1 \\to A_2 B_2 c_2$ is a partial $\\Lbas$-isomorphism sending $c_1$ to $c_2$\nwhose restriction $\\restr{f}{AB_1}$ is a partial elementary ${\\mathcal L}$-isomorphism, then the entire $f$\nis a partial elementary ${\\mathcal L}$-isomorphism.\n\\item For all $c_1, c_2 \\in K$ and all (potentially large) $B\\subset \\mathrm{RV}_I$ containing $\\operatorname{rv}_I(\\langle A,c_1\\rangle_{\\mathbb Q})$, any partial $\\Lbas(A\\cup B)$-isomorphism $f\\colon\\{c_1\\} \\to \\{c_2\\}$, is a partial elementary ${\\mathcal L}(A\\cup B)$-isomorphism.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{remark}\\label{rem:wlogRVI}\nIf (ii) holds for $B_1 = B_2 = \\mathrm{RV}_I$, then it also holds in general, since by Remark~\\ref{rem:join iso},\nthe partial $\\Lbas$-isomorphism $f\\colon A B_1 c_1 \\to A_2 B_2 c_2$ extends to $f\\colon A c_1 \\cup \\mathrm{RV}_I \\to A_2 c_2 \\cup  \\mathrm{RV}_I$.\nAnalogously, we may assume $B = \\mathrm{RV}_I$ in (iii).\n\\end{remark}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:prep eq iso}]\n(i) $\\Rightarrow$ (iii):\nLet $f$ be as in (iii). We have to check that for every ${\\mathcal L}(A\\cup B)$-definable set $X\\subset K$, $c_1 \\in X$ if and only if $c_2\\in X$. By (i), there exists a finite $C \\subset A$ such that $X$ is $I$-prepared by $C$. Since $f$ is an $\\Lbas(A\\cup B)$-isomorphism and $B$ contains $\\operatorname{rv}_I(\\langle A,c_1\\rangle_{\\mathbb Q})$,\nfor all $a\\in C$ and all $r\\geq 1$, we have\n\\[\n\\operatorname{rv}_{I}(c_2-a) = \\operatorname{rv}_{I}(f(c_1)-f(a)) = f(\\operatorname{rv}_{I}(c_1-a)) = \\operatorname{rv}_{I}(c_1-a).\n\\]\nSince $X$ is $I$-prepared by $C$, it follows that $c_1 \\in X$ if and only if $c_2\\in X$.\n\n\\medskip\n\n(iii) $\\Rightarrow$ (ii):\nLet $f$ be as in (ii). Since $f$ is ${\\mathcal L}$-elementary if and only its restriction to every finite domain is, we may assume $B_i$ small. Using the assumption that $\\restr{f}{AB_1}$ is ${\\mathcal L}$-elementary, we can extend $(\\restr{f}{AB_1})^{-1}$ ${\\mathcal L}$-elementarily to some $g$ defined at $c_2$. Let $c'_1 := g(c_2)$. Then $g\\circ f\\colon \\{c_1\\} \\to \\{c'_1\\}$ is a partial $\\Lbas(A\\cup B_1)$-isomorphism. Since $\\operatorname{rv}_I(\\langle A, c_1\\rangle_{\\mathbb Q})\\subset B_1$, it follows by (iii) that $g\\circ f$ is an elementary ${\\mathcal L}$-isomorphism. As $g$ is also ${\\mathcal L}$-elementary, so is $f$.\n\n\\medskip\n\n(ii) $\\Rightarrow$ (i):\nLet $X$ be as in (i), and let $B \\subset \\mathrm{RV}_I$ be a finite subset such that $X$ is ${\\mathcal L}(A \\cup B)$-definable.\n\nConsider any $c_1, c_2 \\in K$ which have the same qf-$\\LbasLA$-type over $A \\cup B$, where $\\LbasLA$ is the expansion of $\\Lbas$ by the full ${\\mathcal L}(A)$-induced structure on $\\mathrm{RV}_I$.\nThen the map $f\\colon c_1 \\to c_2$ is an $\\LbasLA(A \\cup B)$-isomorphism and extends to $f\\colon A B_1 c_1 \\to A_2 B_2 c_2$, where\n$B_i := B \\cup \\operatorname{rv}_I(\\langle A,c_i\\rangle_{\\mathbb Q})$. By definition of $\\LbasLA$, the restriction $f|_{AB_1}$ is ${\\mathcal L}$-elementary, so by (ii), also the entire $f$ is ${\\mathcal L}$-elementary.\nSince moreover $f$ is the identity on $A \\cup B$, this implies that $c_1$ and $c_2$ have the same ${\\mathcal L}$-type over $A \\cup B$.\n\nWe just proved that the ${\\mathcal L}(A \\cup B)$-type of any element $c \\in K$ is implied by its qf-$\\LbasLA(A \\cup B)$-type.\nBy a classical compactness argument (cf. the proof of \\cite[Theorem 3.2.5]{TentZiegler}), it follows that any ${\\mathcal L}(A\\cup B)$-formula in one valued field variable is equivalent to a quantifier free $\\LbasLA(A\\cup B)$-formula. In particular, this applies to our set $X$. Since $\\LbasLA(B)$ is an $\\mathrm{RV}_I$-expansion of $\\Lbas$, our claim follows from Lemma~\\ref{lem:prep eq qf}.\n\\end{proof}\n\n\\begin{lem}[Back and forth over $\\mathrm{RV}_I$]\\label{lem:b-a-f over RV}\nSuppose that $K$ has $I$-preparation. Then the set of partial elementary ${\\mathcal L}$-isomorphisms $f\\colon \\mathrm{RV}_I\\cup A_1 \\to \\mathrm{RV}_I\\cup A_2$ (where $A_1, A_2$ run over all small subsets of $K$) has the back-and-forth.\n\\end{lem}\n\nRecall that ``having the back and forth'' means: for any such $f$ and any $c_1 \\in K \\setminus A_1$,\n$f$ can be extended to $c_1$ while staying in that set of maps, and similarly for \\(f^{-1}\\).\n\n\\begin{proof}\nLet $f$ be as above. Since partial elementary isomorphisms can always be extended to the algebraic closure of their domain, we may assume that $\\acl_K(A_i) = A_i$.\nBy $I$-preparation, statement (i) of Lemma~\\ref{lem:prep eq iso} now holds for $A = A_i$.\nPick any $c_1\\in K$, set $B_1 := \\operatorname{rv}_I(\\langle A_1, c_1\\rangle_{{\\mathbb Q}})$\nand let $c_2\\models f_{*}\\mathrm{qftp}_{\\Lbas}(c_1\/A_1B_1)$. (Such a $c_2$ exists, since $K$ is $|A_1B_1|^+$-saturated.) Let $g$ extend $f$ by sending $c_1$ to $c_2$. By construction, $\\restr{g}{A_1B_1c_1}$ is a partial $\\Lbas$-isomorphism. This implies that the entire $g$ is a partial $\\Lbas$-isomorphism\n(cf.\\ Remark~\\ref{rem:join iso}). Note also that $\\restr{g}{A_1\\mathrm{RV}_I} = f$ is a partial elementary ${\\mathcal L}$-isomorphism. Hence, by Lemma~\\ref{lem:prep eq iso}~(ii), the entire $g$ is ${\\mathcal L}$-elementary.\n\\end{proof}\n\nWe are now ready for the proof of the central result of this subsection:\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:equiv}]\nSet ${\\mathcal T} := \\operatorname{Th}(K)$. Recall that the implication (ii) to (i) is trivial, so let us assume (i) (namely, ${\\mathcal T}$ has $I$-preparation) and prove (ii) (namely, ${\\mathcal T}$ has resplendent $I$-preparation):\nSuppose that $K$ is an arbitrary model of ${\\mathcal T}$, let $\\cLe$ be an $\\mathrm{RV}_{I}$-expansion of ${\\mathcal L}$, let $A$ be a subset of $K$ and let $X \\subset K$ be an $\\cLe(A)$-definable set;\nwe need to find a finite ${\\mathcal L}(A)$-definable set $C \\subset K$ which $I$-prepares $X$. Since the condition that $C$ $I$-prepares $X$ is first order, we may replace $K$ by a sufficiently saturated elementary extension. For the remainder of the proof, we fix such a $K$.\n\nWe may assume that $A = \\acl_{{\\mathcal L},K}(A)$; then by Lemma~\\ref{lem:prep eq iso} (and Remark~\\ref{rem:wlogRVI}), it suffices to show that every partial $\\Lbas(A\\cup\\mathrm{RV}_I)$-isomorphism $f\\colon\\{c_1\\} \\to \\{c_2\\}$ is an elementary $\\cLe(A\\cup\\mathrm{RV}_I)$-isomorphism.\n\nBy (i) and Lemma~\\ref{lem:prep eq iso} applied in ${\\mathcal L}$, such an $f$ is a partial elementary ${\\mathcal L}(A\\cup\\mathrm{RV}_I)$-isomorphism, and since $f$ is the identity on $\\mathrm{RV}_I$, it is a partial $\\cLe(A\\cup\\mathrm{RV}_I)$-isomorphism. It remains to show that $f$ preserves all $\\cLe$-formulas and not just the quantifier free ones.\n\nLet ${\\mathcal F}$ be the class of partial elementary ${\\mathcal L}(\\mathrm{RV}_I)$-isomorphisms with small domains $A \\subset K$. By Lemma~\\ref{lem:b-a-f over RV}, ${\\mathcal F}$ has the back-and-forth. Moreover, any $g\\in{\\mathcal F}$ is also a partial $\\cLe$-isomorphism, as it fixes $\\mathrm{RV}_I$. Recall that if a class of partial $\\cLarb$-isomorphisms has the back and forth\n(for any given language $\\cLarb$), by an easy induction on the structure of formulas, those partial isomorphisms are automatically $\\cLarb$-elementary.\nThus any $g\\in{\\mathcal F}$, and in particular $f$, is a partial elementary $\\cLe$-isomorphism, which is what we had to show.\n\\end{proof}\n\nWe now mention some easy consequences of Proposition~\\ref{prop:equiv} and its proof.\n\n\\begin{cor}[$\\mathrm{RV}$-expansions preserve $\\acl$]\\label{cor:acl=acl}\nSuppose that $\\operatorname{Th}(K)$ has $I$-preparation, for some $\\emptyset$-definable proper ideal $I \\subset {\\mathcal O}_K$.\nThen for any $\\mathrm{RV}_I$-expansion $\\cLe$ of ${\\mathcal L}$ and any $A \\subset K$, we have $\\acl_{{\\mathcal L}',K}(A) = \\acl_{{\\mathcal L},K}(A)$.\nIn particular:\n\\begin{enumerate}\n\\item If $\\operatorname{Th}(K)$ is $0$-h-minimal, then for any $\\mathrm{RV}$-expansion $\\cLe$ of ${\\mathcal L}$ and any $A \\subset K$, we have $\\acl_{{\\mathcal L}',K}(A) = \\acl_{{\\mathcal L},K}(A)$.\n\\item If $\\operatorname{Th}(K)$ is $\\omega$-h-minimal, then\nfor any $\\lambda\\in\\Gamma_K^\\times$, any $\\mathrm{RV}_\\lambda$-expansion $\\cLe$ of ${\\mathcal L}$ and any $A \\subset K$, we have $\\acl_{{\\mathcal L}',K}(A) = \\acl_{{\\mathcal L},K}(A)$.\n \\end{enumerate}\n\n\\end{cor}\n\\begin{proof}\nAny $b \\in \\acl_{\\cLe,K}(A)$ is an element of a finite $\\cLe(A)$-definable set $X \\subset K$.\nBy Proposition~\\ref{prop:equiv}, $X$ can be $I$-prepared by a finite ${\\mathcal L}(A)$-definable set $C$. This implies $X \\subset C$ and hence $b \\in \\acl_{{\\mathcal L},K}(A)$.\n\\end{proof}\n\n\\begin{remark}\nWe already saw some stable embeddedness results in Proposition~\\ref{prop:stab} and Remark~\\ref{rem:stab}. For similar reasons, $I$-preparation implies stable embeddedness of $\\mathrm{RV}_I$. Alternatively, one may deduce the stable embeddedness from Lemma~\\ref{lem:b-a-f over RV} and \\cite[Appendix, Lemma 1]{ChaHru-ACFA}\\footnote{This uses the existence of fully saturated models; getting rid of those is left as an exercise to the reader.}.\n\\end{remark}\n\n\n\\private{Proof of stab. emb of $\\mathrm{RV}_I$ without using fully saturated models:\n\nLet $K$ be a sufficiently saturated model of ${\\mathcal T}$.\nIt suffices to prove that for every finite $A \\subset K$ and for every tuple $\\xi$ of elements of $\\mathrm{RV}_I$, the type $\\tp(\\xi\/A)$ is already determined by $\\tp(\\xi\/B)$, where $B :=\\operatorname{rv}_I(\\langle A \\rangle_{\\mathbb Q})$.\nIndeed, if $X$ is an $A$-definable subset of a product of some of the sorts of $\\mathrm{RV}_I$, then using compactness, the above implies that each $\\xi \\in X$ is contained in a $B$-definable subset of $X$ and using compactness once more, the union of finitely many of those subsets is equal to $X$.\n\nSo now let $A$ and $B = \\operatorname{rv}_I(\\langle A \\rangle_{\\mathbb Q})$ be as above and suppose that $\\xi_1, \\xi_2$ are two tuples from $\\mathrm{RV}_I$ which have the same ${\\mathcal L}$-type over $B$.\nDefine $f\\colon A \\cup B \\cup \\{\\xi_1\\} \\to A \\cup B \\cup \\{\\xi_2\\}$ to be the identity on $A \\cup B$ and to send $\\xi_1$ to $\\xi_2$.\nThis map is an $\\Lbas$-isomorphism and its restriction $f|_{B\\xi_1}$ is ${\\mathcal L}$-elementary. Applying Lemma~\\ref{lem:prep eq iso} repeatedly\n(once for each of the finitely many elements of $A$), we deduce that the entire $f$ is ${\\mathcal L}$-elementary, hence showing that $\\xi_1$ and $\\xi_2$ have the same type over $A$.\n}\n\nWe conclude this subsection by the result that various notions of Hensel minimality automatically pass to $\\mathrm{RV}$-expansions of the language.\n\n\\begin{thm}[$\\mathrm{RV}$-expansions preserve Hensel minimality]\\label{thm:resp:h}\nLet $K$ be a valued field of equi-characteristic $0$ in a language ${\\mathcal L}$ expanding the language ${\\mathcal L}_\\val$ of valued fields.\nFix $\\ell \\in \\{0, 1, \\omega\\}$ and suppose that $\\operatorname{Th}_{\\mathcal L}(K)$ is $\\ell$-h-minimal.\nLet $\\cLe$ be an arbitrary $\\mathrm{RV}$-expansion of ${\\mathcal L}$ (i.e., an expansion by predicates on Cartesian powers of $\\mathrm{RV}$).\nThen $\\operatorname{Th}_{\\cLe}(K)$ is also $\\ell$-h-minimal.\n\\end{thm}\n\nIt seems plausible that this theorem is valid also for other $\\ell \\in {\\mathbb N}$. However, whereas the cases $0$ and $\\omega$ follow easily from Proposition~\\ref{prop:equiv}, the only proof we have for $\\ell = 1$ makes a detour through the criterion given in Theorem~\\ref{thm:tame2vf}, and we have no such criterion for $\\ell \\ge 2$.\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:resp:h}]\nThe case $\\ell = 0$ follows directly from Proposition~\\ref{prop:equiv}: Since $\\operatorname{Th}_{{\\mathcal L}}(K)$ has ${\\mathcal M}_K$-preparation, it even has resplendent ${\\mathcal M}_K$-preparation, so every $\\cLe(A)$-definable set $X \\subset K$ (for $A \\subset K$) can be prepared by a finite ${\\mathcal L}(A)$-definable (and hence in particular $\\cLe(A)$-definable) subset of $K$.\n\nFor the case $\\ell = \\omega$, we use a similar argument, where ${\\mathcal M}_K$ is replaced by $I = B_{< \\lambda}(0)$ for arbitrary $\\lambda \\le 1$ in $\\Gamma^\\times_K$. Note that indeed, the $\\mathrm{RV}$-expansion $\\cLe$ can also be considered as an $\\mathrm{RV}_\\lambda$-expansion, by pulling back all the new predicates along the canonical map $\\mathrm{RV}_\\lambda \\to \\mathrm{RV}$.\n\nFinally, in the case $\\ell = 1$, we use the $1$-h-minimality criterion given in Theorem~\\ref{thm:tame2vf}: Let $f\\colon K \\to K$ be $\\cLe(A)$-definable, for $A \\subset K \\cup \\mathrm{RV}$. We need to prove the two conditions (T1) and (T2) stated in the theorem.\n\nFirst of all, note that we already know that $\\operatorname{Th}_{\\cLe}(K)$ is $0$-h-minimal.\n\nBy Corollary~\\ref{cor:acl=acl} (applied with $I = {\\mathcal M}_K$), for every $x \\in K$, we have $f(x) \\in \\acl_{{\\mathcal L},K}(A \\cup \\{x\\})$.\nLet $C_x$ be an ${\\mathcal L}(A)$-definable family of finite sets such that $f(x) \\in C_x$ for all $x \\in K$. Lemma~\\ref{lem:average} yields an ${\\mathcal L}(A)$-definable family of injective maps $g_x\\colon C_x \\to \\mathrm{RV}^k$. Using this, we define maps $h\\colon K \\to \\mathrm{RV}^k, x \\mapsto g_x(f(x))$ and $\\tilde f\\colon K \\times \\mathrm{RV}^k \\to K$ with $\\tilde f(x, \\xi) = g_x^{-1}(\\xi)$\nif $\\xi \\in g_x(C_x)$ and $\\tilde f(x, \\xi) = 0$ otherwise.\nNote that we obtain $f(x) = \\tilde f(x, h(x))$ for all $x \\in K$.\nAlso note that $h$ is $\\cLe(A)$-definable and $\\tilde f$ is ${\\mathcal L}(A)$-definable.\n\nApplying Corollary~\\ref{cor:prep} (in $\\cLe$) to the graph of $h$ yields a finite $\\cLe(A)$-definable set $C'$ such that $h$ is constant on every ball $B$ $1$-next to $C'$. Moreover, using Lemma~\\ref{lem:gammaLin} in ${\\mathcal L}$, we find an ${\\mathcal L}(A)$-definable family of sets $C_\\xi$ preparing $x \\mapsto \\tilde f(x, \\xi)$ in the sense of that lemma for every $\\xi \\in \\mathrm{RV}^k$. Now let\n$C$ be the union of $C'$ and all those $C_\\xi$. That union is still finite (using Corollary~\\ref{cor:finiterange:set} in ${\\mathcal L}$), and it prepares $f$ in the way Condition (T1) requires it. Indeed, on each fixed ball $B$ 1-next to $C$, we have $f(x) = \\tilde f(x, \\xi)$ for one fixed $\\xi$, our application of Lemma~\\ref{lem:gammaLin} ensured that $x \\mapsto \\tilde f(x, \\xi)$ is prepared in the desired sense.\n\nCondition (T2) now is also clear: For each $\\xi$, $\\tilde f(\\cdot, x)$ has only finitely many infinite fibers (by Lemma~\\ref{lem:fin-inf} in ${\\mathcal L}$). Taking the union of all those sets (for all $\\xi$) still yields a finite set.\n\\end{proof}\n\n\\subsection{Changing the ideal}\n\nWe now switch back to considering ${\\mathcal L}$ as a single-sorted language.\nIn this subsection, we prove that $\\omega$-h-minimality implies $I$-preparation for most ideals $I$, where ``most'' means ``$\\Gamma_K$-open'' in the following sense:\n\n\\begin{defn}[Notions of openness]\nBy an \\emph{open ball ideal} in ${\\mathcal O}_K$, we mean an ideal of the form $B_{<\\lambda}(0)$ for some\n$\\lambda \\le 1$ in $\\Gamma_K$. By a \\emph{$\\Gamma_K$-open ideal} in ${\\mathcal O}_K$, we mean an ideal which is equal to the union of all open balls ideals it contains.\n\\end{defn}\n\nIn other words, an ideal is $\\Gamma_K$-open if its image in $\\Gamma_K$ is open with respect to the interval topology on $\\Gamma_K$. Note that none of the two above openness notions coincides with being topologically open in the valued field topology.\n\n\n\n\\begin{remark}\nIf the value group is discrete, then every proper ideal is $\\Gamma_K$-open; otherwise, an ideal is $\\Gamma_K$-open if and only if it is not a closed ball.\n\\end{remark}\n\n\n\\begin{thm}[$I$-preparation]\\label{thm:all:I}\nLet $K$ be a valued field of equi-characteristic $0$, considered as a structure in a language ${\\mathcal L} \\supset {\\mathcal L}_\\val$. Suppose that $\\operatorname{Th}(K)$ is $\\omega$-h-minimal.\nThen $K$ has resplendent $I$-preparation (see Definition~\\ref{defn:Iresplendent}) for every proper $\\Gamma_K$-open definable (with parameters) ideal $I$ of ${\\mathcal O}_K$.\n\\end{thm}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:all:I}]\nLet $K$ and $I$ be as in the theorem, let\n$A \\subset K$, let $\\cLe$ be an $\\mathrm{RV}_I$-expansion of ${\\mathcal L}$, and let $X \\subset K$ be $\\cLe(A)$-definable.\nWe need to find a finite ${\\mathcal L}(A)$-definable set $C\\subset K$ such that $X$ is $I$-prepared by $C$. By passing to an elementary extension, we may assume that $K$ is sufficiently saturated as an $\\cLe$-structure.\n\nFix $\\lambda \\in \\Gamma^\\times_K$ such that $B_{<\\lambda}(0)$ is contained in $I$.\nThen up to interdefinability, ${\\mathcal L}'$ is an $\\mathrm{RV}_{\\lambda}$-expansion of ${\\mathcal L}$,\nnamely the expansion by the preimage of each of the\n$\\mathrm{RV}_I$-predicates of ${\\mathcal L}'$ under the map $\\mathrm{RV}_{\\lambda} \\to \\mathrm{RV}_{I}$.\n\nBy Lemma~\\ref{lem:addconst}, adding a $\\lambda$ as a constant to the language preserves $\\omega$-h-minimality, i.e., ${\\mathcal T} := \\operatorname{Th}_{{\\mathcal L}(\\lambda)}(K)$ is still $\\omega$-h-minimal. In that language, $B_{<\\lambda}(0)$ is $\\emptyset$-definable, so Proposition~\\ref{prop:equiv} implies that ${\\mathcal T}$ has resplendent $B_{<\\lambda}(0)$-preparation. In particular, there exists a finite ${\\mathcal L}(A, \\lambda)$-definable set $C$ such that no ball $\\lambda$-next to $C$ ``intersects $X$ properly'', i.e., every such ball is either contained in $X$ or disjoint from $X$.\nUsing Corollary~\\ref{cor:finiterange:set}, we may even assume that $C$ is ${\\mathcal L}(A)$-definable.\n\nThe above set $C$ might depend on the choice of $\\lambda$, so let us denote it by $C_\\lambda$ instead. (Contrary to what the notation might suggest, this dependence is not definable, in general.)\nUsing a similar compactness argument as in Proposition-Definition~\\ref{prop:uniform},\nwe now make $C_\\lambda$ independent of $\\lambda$, as follows:\nThe condition that $X$ intersects no ball $\\lambda$-next to $C_\\lambda$ properly can be expressed by a first order formula in $\\lambda$, and there\nare only a bounded number of choices for $C_\\lambda$. It follows that if no finite ${\\mathcal L}(A)$-definable $C$ works for all $\\lambda$, then by our saturation assumption on $K$, we find a $\\lambda$ such that for every $C$, some ball $\\lambda$-next to $C$ intersects $X$ properly, which is a contradiction. Let now $C$ be a set that works for every $\\lambda$.\n\nTo finish the proof, it remains to prove that every ball $I$-next to $C$ is either contained in $X$ or disjoint from $X$. Let $a, a'\\in K$ be such that $\\operatorname{rv}_{I}(a - c) = \\operatorname{rv}_{I}(a' - c)$ for all $c \\in C$. Since $I$ is the union of the open ball ideals $B_{<\\lambda}(0)$ it contains, $\\frac{a - c}{a'-c} \\in 1 + I$ implies $\\frac{a - c}{a'-c} \\in B_{<\\lambda}(1)$ for one such $\\lambda$. We may moreover choose a single $\\lambda$ such that this holds for all $c \\in C$. Then $\\bigcap_{c\\in C}\\operatorname{rv}_{\\lambda}^{-1}(\\operatorname{rv}_{\\lambda}(a-c))$ is a ball $\\lambda$-next to $C$ containing both $a$ and $a'$. It follows that $a\\in X$ if and only if $a'\\in X$.\n\\end{proof}\n\n\nFrom this, we can now deduce that $\\omega$-h-minimality is preserved under coarsening of the valuation:\n\n\\begin{cor}[Coarsening the valuation]\\label{cor:coarse}\nSuppose that $\\operatorname{Th}_{\\mathcal L}(K)$ is $\\omega$-h-minimal and that $|\\cdot|_c\\colon K \\to \\Gamma_{K,c}$ is a non-trivial coarsening of the valuation $|\\cdot|$ on $K$ with valuation ring ${\\mathcal O}_{K,c}$ (non-trivial meaning ${\\mathcal O}_{K,c} \\ne K$).\nLet ${\\mathcal L}_c$ be the expansion of ${\\mathcal L}$ by a predicate for ${\\mathcal O}_{K,c}$,\nand let us write $K_c$ for $K$ considered as an ${\\mathcal L}_c$-structure, and as a valued field with the valuation $|\\cdot|_c$.\nThen $\\operatorname{Th}_{{\\mathcal L}_c}(K_c)$ is also $\\omega$-h-minimal.\n\\end{cor}\n\n\n\\begin{proof}\nSince ${\\mathcal O}_{K,c}$ is an $\\operatorname{rv}$-pullback (i.e., a preimage in $K$ of a subset of $\\mathrm{RV}$), Theorem~\\ref{thm:resp:h} implies that $\\operatorname{Th}_{{\\mathcal L}_c}(K)$ is $\\omega$-h-minimal for the valuation $|\\cdot|$.\n\nLet $K'$ be ${\\mathcal L}_c$-elementary equivalent to $K$; we need to verify that $K'$ has $B_{<\\lambda}(0)$-preparation for every $\\lambda \\le 1$ in $\\Gamma^\\times_{K',c}$.\nThis follows from Theorem~\\ref{thm:all:I} since $B_{<\\lambda}(0)$ is $\\Gamma_{K'}$-open, which in turn holds because $B_{<\\lambda}(0)$ is the preimage under $K' \\to \\Gamma_{K'}$ of the set of those $\\mu \\in \\Gamma_{K'}$\nwhich get sent to an element less than $\\lambda$ by the map\n$\\Gamma_{K'} \\to \\Gamma_{K', c}$.\n\\end{proof}\n\n\n\n\n\\subsection{Algebraic Skolem functions}\n\\label{sec:alg:skol}\n\nAs usual, $K$ is a valued field of equi-characteristic $0$, in a language ${\\mathcal L} \\supset {\\mathcal L}_\\val$.\n\nThe mere statement of cell decomposition in valued fields is usually very technical, one reason being that one cannot definably pick centers of the cells.\nThis problem is easier to deal with if one adds certain Skolem functions to the language. Usually, one would not want to modify the\nlanguage in such a way. However, in this subsection, we provide tools which make it possible, in many situations, to assume the existence of such Skolem functions (without losing power nor generality).\n\nThe first thing to note is that properly adding the required Skolem functions preserves Hensel minimality; this follows from Theorem~\\ref{thm:resp:h} and the observation that adding ``algebraic'' Skolem functions on $\\mathrm{RV}$ is enough (Lemma~\\ref{lem:alg:skol:K}). This by itself is useful whenever one only wants to prove that every definable set (or function) has some good properties. A similar approach has been recently followed in \\cite{CFL}.\n\nSometimes, one wants to use cell decomposition to prove that every definable object yields some other kind of definable object. If one proves such a result in a language expanded by Skolem functions, one needs to be able to get rid of the Skolem functions again afterwards. There is probably no general recipe for this, but Lemma~\\ref{lem:undo-K-to-RV} is a useful tool; we apply it e.g.\\ in the proof of Theorem~\\ref{thm:T3\/2,mv}.\n\n\\begin{lem}\\label{lem:alg:skol:K}\nSuppose that $\\operatorname{Th}(K)$ is $0$-h-minimal and\nthat for every set $A' \\subset \\mathrm{RV}$, we have $\\acl_{\\mathrm{RV}}(A') = \\dcl_{\\mathrm{RV}}(A')$.\nThen for every set $A \\subset K$, we have $\\acl_K(A) = \\dcl_K(A)$.\n\\end{lem}\n\n\\begin{proof}\nLet $C \\subset K$ be a finite $A$-definable set; we need to prove that $C$ is contained in $\\dcl_K(A)$.\nBy Lemma~\\ref{lem:average}, there exists an $A$-definable bijection $f\\colon C \\to C' \\subset \\mathrm{RV}^k$.\nBy Proposition~\\ref{prop:stab}, $C'$ is definable with parameters from $\\dcl_{\\mathrm{RV}}(A)$, so\n$C' \\subset \\acl_{\\mathrm{RV}}(\\dcl_{\\mathrm{RV}}(A))$. By assumption, this implies $C' \\subset \\dcl_{\\mathrm{RV}}(A)$, which,\nusing $f$, implies $C \\subset \\dcl_K(A)$.\n\\end{proof}\n\n\n\\begin{remark}\\label{rem:alg:skol}\nThe above property ``$\\acl_K(A) = \\dcl_K(A)$'' holds in all models if and only if algebraic Skolem functions exist, i.e.:\nFor all integers $n\\geq 0$ and $m > 0$ and every $\\emptyset$-definable set $X\\subset K^{n+1}$ with the property that the coordinate projection map $p\\colon X \\to K^{n}$ has fibers of cardinality precisely $m$, there is a $\\emptyset$-definable function $f:  K^{n}\\to K$ whose graph is a subset of $X$.\n\\end{remark}\n\n\n\\begin{prop}[Obtaining $\\acl = \\dcl$]\\label{prop:exist:alg:skol}\nGiven a valued field $K$ (in a language ${\\mathcal L}$) whose theory is $\\ell$-h-minimal, for $\\ell \\in \\{0, 1, \\omega\\}$, there exists an\n$\\mathrm{RV}$-expansion ${\\mathcal L}^{\\rm as} \\supset {\\mathcal L}$ of the language such that $\\operatorname{Th}_{{\\mathcal L}^{\\rm as}}(K)$ is $\\ell$-h-minimal and such that for every model $K'\\models \\operatorname{Th}_{{\\mathcal L}^{\\rm as}}(K)$ and for every subset $A \\subset K'$, we have $\\acl_{{\\mathcal L}^{\\rm as},K}(A) = \\dcl_{{\\mathcal L}^{\\rm as},K}(A)$.\n\nMore precisely, ${\\mathcal L}^{\\rm as}$ can be taken to be an $\\mathrm{RV}$-expansion of ${\\mathcal L}$ (i.e., an expansion by predicates on $\\mathrm{RV}^n$).\n\\end{prop}\n\n\n\\begin{proof}\nLet us be lazy and simply define ${\\mathcal L}^{\\rm as}$ to be the language obtained from ${\\mathcal L}$ by adding a predicate for each subset of $\\mathrm{RV}_K^n$ (for every $n$). Then $\\operatorname{Th}_{{\\mathcal L}^{\\rm as}}(K)$ is still $\\ell$-h-minimal by Theorem~\\ref{thm:resp:h}. Since in $\\mathrm{RV}_K^n$, we have algebraic Skolem functions in this language ${\\mathcal L}^{\\rm as}$, we have $\\acl_{\\mathrm{RV}}(A') = \\dcl_{\\mathrm{RV}}(A')$ for every $A' \\subset \\mathrm{RV}_{K'}$ and for every model $K' \\equiv_{{\\mathcal L}^{\\rm as}} K$. Now Lemma~\\ref{lem:alg:skol:K} implies\n$\\acl_K(A) = \\dcl_K(A)$ for every $A \\subset K'$.\n\\end{proof}\n\nOne may use Proposition \\ref{prop:exist:alg:skol} to ensure the condition that $\\acl$ equals $\\dcl$; the next lemma can be useful to get back to the original language.\n\n\n\\begin{lem}[Undoing algebraic skolemization]\\label{lem:undo-K-to-RV}\nSuppose that $\\operatorname{Th}_{{\\mathcal L}}(K)$ is $0$-h-minimal and\nthat ${\\mathcal L}'$ is an $\\mathrm{RV}$-expansion of ${\\mathcal L}$, and let $\\chi'\\colon K^n \\to \\mathrm{RV}^{k'}$ be an ${\\mathcal L}'$-definable map (for some $k' \\ge 0$).\nThen there exists an ${\\mathcal L}$-definable map $\\chi\\colon K^n \\to \\mathrm{RV}^k$ (for some $k \\ge 0$) such that $\\chi'$ factors over $\\chi$, i.e.\n$\\chi' = g\\circ \\chi$, for some function $g\\colon \\mathrm{RV}^k \\to \\mathrm{RV}^{k'}$ (which is automatically ${\\mathcal L}'$-definable).\n\\end{lem}\n\n\\begin{proof}\nWe do an induction over $n$. The case $n = 0$ is trivial, so assume $n \\ge 1$.\n\nFix $a \\in K$ and set $\\chi'_a(\\tilde a) := \\chi'(a, \\tilde a)$, for $\\tilde a \\in K^{n-1}$.\nBy induction, we find an ${\\mathcal L}(a)$-definable map $\\chi_a\\colon K^{n-1} \\to \\mathrm{RV}^k$\nsuch that $\\chi'_a = g_a \\circ \\chi_a$ for some ${\\mathcal L}'(a)$-definable $g_a$.\nBy compactness, we may assume that $\\chi_a$ and $g_a$ are definable uniformly in $a$,\nso that in particular, we obtain an ${\\mathcal L}'$-definable set $W = \\{(a, \\zeta', g_a(\\zeta')) \\mid a \\in K, \\zeta' \\in \\mathrm{RV}^k\\}$.\nUsing Corollary~\\ref{cor:prep}, we find a finite ${\\mathcal L}'$-definable set preparing $W$, and by\nCorollary~\\ref{cor:acl=acl}, that set is contained in a finite ${\\mathcal L}$-definable set $C$.\n\nFinally, let $f\\colon K \\to \\mathrm{RV}^\\ell$ be an ${\\mathcal L}$-definable map as provided by Lemma~\\ref{lem:InextFam}, namely such that each fiber of $f$\nis either a singleton in $C$ or a ball $1$-next to $C$. We claim that the map\n$\\chi(a, \\tilde a) := (f(a), \\chi_a(\\tilde a))$ is as desired.\nIndeed, it is clearly ${\\mathcal L}$-definable, and since $C$ $1$-prepares $W$, the function $g_a$ is determined by $f(a)$,\nso that $f(a)$ and $\\chi_a(\\tilde a)$ together determine $\\chi'(a, \\tilde a) = \\chi'_a(\\tilde a) = g_a(\\chi_a(\\tilde a))$.\n\\end{proof}\n\n\n\n\\section{Geometry in the h-minimal setting}\n\\label{sec:compactn}\n\nIn this section, we deduce geometric results in $K^n$ under the assumption of $1$-h-minimality. Some of those are already known under stronger assumptions,  like $C^k$ properties of definable functions (Subsection \\ref{sec:cont}),\ncell decomposition (Subsection \\ref{sec:cd}), and dimension theory (Subsection \\ref{sec:dim}). Since some proofs are very similar to those in many other papers, we will be somewhat succinct.\n\nAs highlights, we then prove a version of the Jacobian Property in many variables (Subsection \\ref{sec:sjp}) which allows us to get general t-stratifications for definable sets in $1$-h-minimal structures (Subsection \\ref{sec:t-strat}).\nWe also provide a higher-dimension version of the Taylor-approximation result (Subsection \\ref{sec:taylor-box}), the aim being to lay the\nground for the first axiomatic approach in the non-archimedean context of analogues of results by Yomdin--Gromov and Pila--Wilkie on parameterizations and point counting.\n\nThe version of cell decomposition presented in Subsection \\ref{sec:cd} uses a simplified notion of cells, by temporarily adding certain Skolem functions to the language, as detailed in Subsection~\\ref{sec:alg:skol}. Since that approach might not work for all potential applications,\nwe conclude Section~\\ref{sec:compactn} by linking with more classical viewpoints on cells (Subsection \\ref{sec:cd:classical}). \n\n\\subsection{Continuity and Differentiability}\\label{sec:cont}\n\nWe start by proving that $1$-h-minimality implies that definable\nfunctions are almost everywhere continuous and even $C^k$.\n\n\\begin{thm}[Continuity]\\label{thm:C0}\nAssume that $\\operatorname{Th}(K)$ is $1$-h-minimal. For every definable function $f\\colon X \\subset K^n \\to K$, the set $U$ of those $u \\in X$ such that $f$ is continuous on a neighborhood of $u$ is dense in $X$.\n\\end{thm}\n\nFor the moment, we only give the proof in the case where $X$ is open. The general case is proved\nin a joint induction with cell decomposition (more precisely with Addenda~\\ref{add:cd:cont:f} and \\ref{add:cd:cont:c} of Theorem~\\ref{thm:cd:alg:skol}),\nso we postpone that proof until Subsection~\\ref{sec:cd}.\n\nWe here give a different proof than the approach to similar results in \\cite{CCL-PW}, building on the following\ntwo lemmas which might be of independent interest.\n\n\n\\begin{lem}[Continuity on small boxes]\\label{lem:const:type:cont}\nAssume that $\\operatorname{Th}(K)$ is $1$-h-minimal. If $B \\subset K^n$ is a product of balls such that all elements of $B$ have\nthe same type over $\\mathrm{RV}$ and $f\\colon B \\to K$ is a $\\emptyset$-definable function, then $f$ is uniformly continuous on $B$.\n\\end{lem}\n\n\\begin{proof}\nSet $B =: \\hat B \\times B' \\subset K^{n-1} \\times K$.\nUsing Lemma~\\ref{lem:gammaLin}, we deduce that for every fixed $\\hat a \\in \\hat B$, the map $a' \\mapsto f(\\hat a, a')$ is continuous on $B'$.\nMoreover, this continuity is uniform in both, $\\hat a$ and $a'$, since all elements $(\\hat a, a') \\in B$ have the same type over $\\mathrm{RV}$.\n(The type $\\tp(\\hat a a'\/\\mathrm{RV})$ knows which $\\delta$ works for which $\\epsilon$, where $\\epsilon, \\delta>0$ are from the definition of continuity at $(\\hat a, a')$.)\nAfter applying the same argument to all other coordinates, we deduce that $f$ as a whole is uniformly continuous in $B$:\nFor every $\\epsilon > 0$, there exists a $\\delta > 0$ such that if $a_1, a_2 \\in B$ with $|a_1 - a_2| < \\delta$ differ only in one coordinate,\nthen $|f(a_1) - f(a_2)| < \\epsilon$. If $a_1$ and $a_2$ differ in several coordinates, apply this repeatedly.\n\\end{proof}\n\n\\begin{lem}[Balls with constant type]\\label{lem:ex:const:type}\nAssume that $K$ is $|{\\mathcal L}|^+$-saturated and that $\\acl_{K}$ equals $\\dcl_{K}$.\nIf $X \\subset K^n$ is a $\\emptyset$-definable set with non-empty interior,\nthen $X$ contains a ball $B$ such that all elements of $B$ have\nthe same type over $\\mathrm{RV}$ (i.e., $\\tp(a\/\\mathrm{RV}) = \\tp(a'\/\\mathrm{RV})$ for all $a, a' \\in B$).\n\\end{lem}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:ex:const:type}]\nWe may assume that $X$ is of the form $\\hat B \\times B'$ for some balls $\\hat B \\subset K^{n-1}, B' \\subset K$, and by induction, we may assume that all elements of $\\hat B$ have the same type over $\\mathrm{RV}$.\n\nSince any $\\emptyset$-definable function $f\\colon K^{n-1} \\to K$ is continuous on $\\hat B$ (by Lemma~\\ref{lem:const:type:cont}), given such an $f$, we can first shrink $\\hat B$\nso that $B' \\setminus f(\\hat B)$ still contains a ball and then shrink $B'$ so that\n$\\hat B \\times B'$ is disjoint from the graph of $f$. After possibly further shrinking $\\hat B$, we then obtain\n\\begin{equation}\\label{eq:rveq}\n\\operatorname{rv}(a'_1 - f(\\hat a_1)) = \\operatorname{rv}(a'_2 - f(\\hat a_2)).\n\\end{equation}\nfor any $(\\hat a_i, a'_i) \\in \\hat B \\times B'$. By saturation, this shrinking of $\\hat B$ can be done for all $\\emptyset$-definable $f$ simultaneously.\nIn particular, $B'$ is now disjoint from $\\acl_K(\\hat a_1)$ for every $\\hat a_1 \\in \\hat B$, and\none deduces that all elements of $\\hat B \\times B'$ have\nthe same type over $\\mathrm{RV}$, namely as follows:\nGiven $(\\hat a_i, a'_i) \\in \\hat B \\times B'$ for $i=1,2$, we find some $a''_1 \\in B'$ such that $\\tp(\\hat a_1a''_1 \/ \\mathrm{RV}) = \\tp(\\hat a_2a'_2 \/ \\mathrm{RV})$. This implies $\\operatorname{rv}(a''_1 - f(\\hat a_1)) = \\operatorname{rv}(a'_2 - f(\\hat a_2))$ for every $\\emptyset$-definable $f$ and hence (using (\\ref{eq:rveq}) and Lemma~\\ref{lem:type-0-h-min}) $\\tp(\\hat a_1a''_1 \/ \\mathrm{RV}) = \\tp(\\hat a_1a'_1 \/ \\mathrm{RV})$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:C0} when $X$ is open]\nWe may suppose that $f$ is $\\emptyset$-definable, that $K$ is $|{\\mathcal L}|^+$-saturated and that $\\acl_K$ equals $\\dcl_K$ (by Proposition~\\ref{prop:exist:alg:skol}).\nSuppose that the set $U$ from the theorem is not dense in $X$. Choose a ball $B \\subset X \\setminus U$ as provided by Lemma~\\ref{lem:ex:const:type}. By Lemma~\\ref{lem:const:type:cont}, $f$ is continuous on $B$, which is a contradiction to $B \\not\\subset U$.\n\\end{proof}\n\nWe now come to differentiability of definable functions.\nThe definition of $C^k$ is the usual one:\n\n\\begin{defn}[$C^k$]\\label{defn:deriv}\nLet $U \\subset K^m$ be open and let $f\\colon U \\to K^n$ be a map.\nWe say that $f$ is $C^0$ if it is continuous, and that it is $C^k$ for $k \\ge 1$ if there exists a $C^{k-1}$-map $\\operatorname{Jac} f\\colon U \\to K^{n\\times m}$\n(the Jacobian of $f$) such that\nfor every $u \\in U$, we have $\\lim_{x \\to u} \\frac{|f(x) - f(u)-((\\operatorname{Jac} f)(u))(x - u)|}{|x - u|} = 0$.\nIn the case $n = 1$, we also write $\\operatorname{grad} f$ instead of $\\operatorname{Jac} f$ and call it the gradient of $f$.\n\\end{defn}\n\n\\begin{thm}[$C^k$]\\label{thm:Ck}\nAssume that $\\operatorname{Th}(K)$ is $1$-h-minimal and fix $k \\ge 0$.\nFor every definable function $f\\colon K^n \\to K$, the set $U$ of those $u \\in K^n$ such that $f$ is $C^k$ on a neighborhood of $u$ is dense in $K^n$.\n\\end{thm}\n\n\\begin{remark}\\label{rem:Ck}\nClearly, $U$ is open and definable over the same parameters as $f$,\nso $f$ is $C^k$ on a definable dense open subset of $K^n$.\n\\end{remark}\n\n\\begin{remark}\nIn Subsection~\\ref{sec:dim}, we will introduce a notion of dimension, and we will see that $U$ being dense implies that $K^n \\setminus U$ has dimension less than $n$. Thus,  definable functions are almost everywhere $C^k$ in this rather strong sense.\n\\end{remark}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:Ck}]\nThe case $k = 0$ is Theorem~\\ref{thm:Ck}. For $k = 1$,\nnote that as in usual analysis, $f$ is $C^1$ if and only if all partial derivatives $\\partial f\/\\partial x_i$ exist and are continuous. That this is indeed the case almost everywhere follows from Theorems~\\ref{thm:der} and \\ref{thm:C0}.\n\nFor $k \\ge 2$, apply induction.\n\\end{proof}\n\n\n\n\\subsection{Cell decomposition}\\label{sec:cd}\n\nIn this subsection we present a version of cell decomposition results that are simpler than usual, by imposing the following condition on the language:\n\n\\begin{assumption}\\label{ass:acl=dcl} \nWe assume in this subsection that we have algebraic Skolem functions in $K$; or, equivalently, that for every $K' \\equiv K$ and every $A \\subset K'$, we have $\\acl_{K'}(A) = \\dcl_{K'}(A)$. We will abbreviate this by ``$\\acl$ equals $\\dcl$ (in $\\operatorname{Th}(K)$)''.\n\\end{assumption}\n\nThe tools provided in Subsection~\\ref{sec:alg:skol} often make it possible to reduce to the case of languages where this assumption holds, as we will see\nin later subsections of Section~\\ref{sec:compactn}.\nIn fact, Assumption~\\ref{ass:acl=dcl} does more than just simplify the arguments: it also allows one to formulate stronger results like piecewise Lipschitz continuity results as in Theorem \\ref{thm:cd:alg:piece:Lipschitz}. \nThis is similar to \\cite{CFL}, where this condition on $\\acl$ is furthermore used to obtain parametrization results with finitely many maps (in the Pila-Wilkie sense \\cite{PW}), but not under axiomatic assumptions.\n\nClose variants of the results of this subsection appear in \\cite{CFL}; indeed, the tameness condition of \\cite{CFL} is very similar to the\n$1$-h-minimality criteria from Theorem \\ref{thm:tame2vf}; see Remark \\ref{rem::tame2vf}\n\n\nIn the following, for $m \\le n$, we denote the projection $K^n \\to K^{m}$ to the first $m$ coordinates by $\\pi_{\\le m}$, or also by $\\pi_{<m+1}$.\n\n\\begin{defn}[Cells, twisted boxes]\\label{defn:cell}\nFix any parameter set $A \\subset K^{\\mathrm{eq}}$ and consider a non-empty $A$-definable set $X\\subset  K^n$ for some $n$, and, for $i=1,\\ldots, n$, values $j_i$ in $\\{0,1\\}$ and $A$-definable functions $c_i:\\pi_{<i}(X)\\to K$.\nThen $X$ is called an \\emph{$A$-definable cell} with \\emph{center tuple $c=(c_i)_{i=1}^n$} and of \\emph{cell-type $j= (j_i)_{i=1}^n$} if it is of the form\n$$\nX = \\{x\\in K^n\\mid  (\\operatorname{rv}(x_i-c_i(x_{<i})))_{i=1}^n\n   \\in R  \\},\n$$\nfor a (necessarily $A$-definable) set\n$$\nR \\subset  \\prod_{i=1}^n (j_i\\cdot \\mathrm{RV}^\\times),\n$$\nwhere $x_{<i}=\\pi_{<i}(x)$ and where\n$0 \\cdot \\mathrm{RV}^\\times = \\{0\\} \\subset \\mathrm{RV}$, and $1 \\cdot \\mathrm{RV}^\\times = \\mathrm{RV}^\\times \\subset \\mathrm{RV}$.\n\nIf $X$ is such a cell, then, for any $r\\in R$, the subset\n$$\n\\{x \\in K^n\\mid  \\operatorname{rv}(x_i-c_i(x_{<i})))_{i=1}^n   = r  \\},\n$$\nof $X$ is called a \\emph{twisted box} of the cell $X$.\nWe also call $X$ itself a twisted box if it is a cell consisting\nof a single twisted box (i.e., if $R$ is a singleton).\n\\end{defn}\n\n\n\\begin{remark}\\label{rem:cell:homeo}\nGiven a cell $X$ as above, let $\\pi\\colon K^n \\to K^d$ be the projection to those coordinates $i$ for which $j_i = 1$.\nThen $\\pi(X)$ is a cell of cell-type $(1, \\dots, 1)$, and the restriction of $\\pi$ to $X$ is injective. If the components $c_i$ of the center tuple of $X$ are continuous, then $\\pi|_X$ is even a homeomorphism.\n\\end{remark}\n\n\nThere are many variants of cell decomposition results preparing different kinds of data. We first state the simplest version\nand then formulate other versions as addenda:\n\n\\begin{thm}[Cell decomposition]\\label{thm:cd:alg:skol}\nSuppose that $\\acl$ equals $\\dcl$ in $\\operatorname{Th}(K)$ (see Assumption~\\ref{ass:acl=dcl}), and that $\\operatorname{Th}(K)$ is $1$-h-minimal.\nConsider a $\\emptyset$-definable set $X\\subset  K^n$ for some $n$.\nThen there exists a cell decomposition of $X$, more precisely, a partition of $X$ into finitely many $\\emptyset$-definable cells $A_\\ell$.\n\\end{thm}\n\n\\begin{remark}\\label{rem:cd:alg:skol}\nGiven finitely many $\\emptyset$-definable sets $X_i \\subset K^n$, we can find a cell decomposition of $K^n$ such that\neach $X_i$ is a union of cells, namely by applying Theorem~\\ref{thm:cd:alg:skol} to suitable intersections of the sets $X_i$ and $K^n \\setminus X_i$.\n\\end{remark}\n\n\\begin{addendum}[Preparation of $\\mathrm{RV}$-sets]\\label{add:cd:alg:prep}\nOn top of the assumptions from Theorem \\ref{thm:cd:alg:skol}, let also a $\\emptyset$-definable set $P\\subset X\\times \\mathrm{RV}^k$ be given for some $k$. We consider $P$ as the function sending $x\\in X$ to the fiber $P_x := \\{\\xi \\in \\mathrm{RV}^k \\mid (x,\\xi) \\in P\\}$.\n\nThen the cells $A_\\ell$ from Theorem \\ref{thm:cd:alg:skol} can be taken such that moreover $P$ (seen as function)\nis constant on each twisted box of each cell $A_\\ell$.\n\\end{addendum}\n\n\\begin{addendum}[Continuous functions]\\label{add:cd:cont:f}\nOn top of the assumptions from Theorem \\ref{thm:cd:alg:skol} (with Addendum~\\ref{add:cd:alg:prep}, if desired), suppose that finitely many $\\emptyset$-definable functions $f_j\\colon X \\to K$ are given. Then the $A_\\ell$ can be taken such that moreover, the restriction $f_j|_{A_\\ell}$ of each function $f_j$ to each cell $A_\\ell$ is continuous.\n\\end{addendum}\n\nIn the one-dimensional case, we can prepare the domain and the image of the functions in a compatible way:\n\n\\begin{addendum}[Compatible preparation of domain and image]\\label{add:cd:alg:range}\nUnder the assumptions of Addendum~\\ref{add:cd:cont:f}, if the ambient dimension $n$ is equal to $1$,\nwe may moreover impose that for each $\\ell$ and each $j$, $f_j|_{A_\\ell}$ is either constant or injective, $f_j(A_\\ell)$ is a $\\emptyset$-definable cell and $f_{j}$ maps each twisted box of $A_\\ell$ onto a twisted box of $f_j(A_\\ell)$.\n\\end{addendum}\n\n\n\\begin{addendum}[Continuous centers]\\label{add:cd:cont:c}\nIn Theorem~\\ref{thm:cd:alg:skol} (with any of Addenda~\\ref{add:cd:alg:prep}, \\ref{add:cd:cont:f}, if desired), we may assume that for each cell $A_\\ell$, each component $c_i\\colon \\pi_{<i}(A_\\ell) \\to K$ of its center tuple is continuous. In particular, each twisted box of each cell $A_\\ell$ of cell-type $(1, \\dots, 1)$ is an open subset of $K^n$.\n\\end{addendum}\n\n\\begin{remark}\nIn Addenda~\\ref{add:cd:cont:f} and \\ref{add:cd:cont:c}, one can replace continuous by $C^k$, provided that one fixes a reasonable definition of a function being $C^k$ on non-interior points of its domain.\n\\end{remark}\n\n\n\n\nTo formulate the next addendum, we need the notion of Lipschitz continuity, which is defined in the usual way:\n\n\\begin{defn}[Lipschitz continuity]\\label{defn:Lipschitz}\nFor a valued field $K$ and an element $\\lambda$ in its value group $\\Gamma_K^\\times$, a function $f:X\\subset K^n\\to K^m$ is called Lipschitz continuous with Lipschitz constant $\\lambda$ if for all $x$ and $x'$ in $X$ one has\n$$\n|f(x)-f(x')| \\leq \\lambda |x-x'|,\n$$\nwhere the norm of tuples is, as usual, the sup-norm.\nWe call such $f$ shortly $\\lambda$-Lipschitz.\n\nCall $f$ locally $\\lambda$-Lipschitz, if for each $x\\in X$ there is an open neighborhood $U$ of $x$  such that the restriction of $f$ to $U$ is $\\lambda$-Lipschitz.\n\\end{defn}\n\n\n\\begin{addendum}[1-Lipschitz centers]\n\\label{add:cd:Lip:comp}\nTheorem~\\ref{thm:cd:alg:skol} (with any of Addenda~\\ref{add:cd:alg:prep}, \\ref{add:cd:cont:f}, \\ref{add:cd:cont:c}, if desired),\nis also valid in the following variant:\nInstead of imposing that $A_\\ell$ itself is a cell in the sense of Definition~\\ref{defn:cell}, we only impose that $\\sigma_{\\ell}(A_\\ell)$ is a cell, for some coordinate permutation $\\sigma_\\ell\\colon K^n \\to K^n$.\nIn that version, we may moreover impose that $\\sigma_{\\ell}(A_\\ell)$ is of cell-type $(1, \\dots, 1, 0, \\dots, 0)$ and that each component $c_i$ of the center tuple $(c_i)_i$ of $\\sigma_{\\ell}(A_\\ell)$ is $1$-Lipschitz.\n\\end{addendum}\n\nClosely related to that addendum, we also have the following reformulation of the piecewise continuity result of \\cite{CFL} in the Hensel minimal setting.\n\n\\begin{thm}[Piecewise Lipschitz continuity]\\label{thm:cd:alg:piece:Lipschitz}\nSuppose that $\\acl$ equals $\\dcl$ in $\\operatorname{Th}(K)$ (see Assumption~\\ref{ass:acl=dcl}), and that $\\operatorname{Th}(K)$ is $1$-h-minimal.\nConsider a $\\emptyset$-definable set  $X\\subset  K^n$ for some $n$ and a $\\emptyset$-definable function $f:X\\to K$. Suppose that $f$ is locally $1$-Lipschitz.\nThen there exists a finite partition of $X$ into $\\emptyset$-definable sets $A_\\ell$ such that the restriction of $f$ to $A_\\ell$ is $1$-Lipschitz, for each $\\ell$.\n\\end{thm}\n\n\nAll of the above results in this subsection have already been proved under related assumptions. For the convenience of the reader, we nevertheless provide short, self-contained versions of the proofs, with exception of Addendum~\\ref{add:cd:Lip:comp} and Theorem~\\ref{thm:cd:alg:piece:Lipschitz} which will not be used in this paper and where we will only explain how to adapt the proof from \\cite{CFL}. However, we do provide a proof\nof the weak version of Addendum~\\ref{add:cd:Lip:comp} stated as Proposition~\\ref{prop:twibox:1Lip}, since that proposition is used to deduce the existence of t-stratifications.\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:cd:alg:skol} with Addendum \\ref{add:cd:alg:prep}]\nIt suffices to find a cell decomposition adapted to a set $P \\subset K^n \\times \\mathrm{RV}^k$.\n\nCase $n = 1$:\nIn this case,\nCorollary~\\ref{cor:prep} provides\na cell decomposition, namely: Let $C \\subset K$ prepare $P$. Using the assumption that $\\acl$ equals $\\dcl$, each element $c \\in C$ is $\\emptyset$-definable, so we obtain a cell decomposition as follows: For each ball $B$ $1$-next to $C$,\nchoose (in a definable way) a $c(B) \\in C$ such that $B$ is $1$-next to $c(B)$.\nNow the cell decomposition consists of two cells with center $c$ for each $c \\in C$, namely (i) $\\{c\\}$ and (ii) the union of all those $B$ $1$-next to $C$ for which $c(B)$ equals $c$.\n\nCase $n > 1$:\nWe apply the case $n = 1$ to each fiber\n$P_a = \\{(b,\\xi) \\in K \\times \\mathrm{RV}^k \\mid (a,b,\\xi) \\in P\\}$, where $a$ runs over $K^{n-1}$. (Note that by compactness this works uniformly, so that in particular we get definable cell centers $K^{n-1} \\to K$.)\nThen we finish by applying induction to a set $P' \\subset K^{n-1} \\times \\mathrm{RV}^{k'}$ ``describing'' the fibers:\nFor each $0$-cell $\\{c\\} \\subset K$ of the fiber at $a \\in K^{n-1}$,\n$P'_a$ encodes the set $P_{a,c} \\subset \\mathrm{RV}^k$;\nfor each $1$-cell $X \\subset K$ of the fiber at $a \\in K^{n-1}$,\n$P'_a$ encodes (a) the set denoted by $R$ in Definition~\\ref{defn:cell} and (b), for each $\\xi \\in R$, the fiber $P_{a,b} \\subset \\mathrm{RV}^k$,\nwhere $b \\in K$ is an arbitrary element of the twisted box corresponding to $\\xi$ (i.e., $\\operatorname{rv}(b - c) = \\xi$, where $c$ is the center of $X$).\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:C0} and Addenda~\\ref{add:cd:cont:f}, \\ref{add:cd:cont:c}]\nFor $n = 0$, all three results are trivial. We now assume that all three results are already known for $n - 1$ and we deduce them for $n$. Concerning Theorem~\\ref{thm:C0}, note that we may as well\nassume that $\\acl$ equals $\\dcl$ (by Proposition~\\ref{prop:exist:alg:skol}).\n\nAddendum~\\ref{add:cd:cont:c}: First, find a cell decomposition with possibly non-continuous centers. By inductively applying Addendum~\\ref{add:cd:cont:f} to each component of the center tuple of each cell, we may refine the cell decomposition to get continuous centers.\n\nTheorem~\\ref{thm:C0}: We may suppose that $X$ has empty interior, since the proof given at the end of Subsection~\\ref{sec:cont} applies to the interior of $X$.\nChoose a cell decomposition of $X$ with continuous centers. No cell $A_\\ell \\subset X$ is of cell-type $(1, \\dots, 1)$ (since such cells have non-empty interior,\nby the ``in particular'' part of Addendum~\\ref{add:cd:cont:c}). Thus the homeomorphism from Remark~\\ref{rem:cell:homeo} allows us to reduce the problem on $A_\\ell$\nto one of lower ambient dimension. Apply induction.\n\nAddendum~\\ref{add:cd:cont:f}: The above proof of Theorem~\\ref{thm:C0} also yields a finite partition of $X$ such that $f$ is continuous on each piece.\nApply this to each of the given functions $f_j$ and then choose a cell decomposition respecting all pieces from all those partitions.\n\\end{proof}\n\n\\begin{proof}[Proof of Addendum~\\ref{add:cd:alg:range}]\nUsing Lemma~\\ref{lem:fin-inf} and our assumption $\\acl=\\dcl$,\nwe find a partition of $X$ such that on each piece, $f_j$ is continuous and either constant or injective for each $j$; assume without loss that $X$ is a single such piece. In a similar way, assume without loss that $X$ is a cell and that each $f_j$ has the Jacobian Property (Definition~\\ref{defn:JP}) on each twisted box of $X$. Since constant functions pose no problem, we assume that all $f_j$ are injective.\n\nNext, choose a finite $\\emptyset$-definable set $\\tilde C \\subset K$ $1$-preparing $f_j(B)$ for every $j$ and every twisted box $B$ of $X$ (using Corollary~\\ref{cor:prep}) and set $C := \\bigcup_j f^{-1}_j(\\tilde C)$. After a further finite partition of $X$, we may assume that either (i) $X$ consists of a single twisted box or that (ii) $C$ is empty.\n\nIn Case (i), choose any $c \\in C \\subseteq X$ and decompose $X$ as \\(\\{c\\}\\cup (X\\setminus \\{c\\})\\), both of which are \\(\\emptyset\\)-definable cells. Then the desired properties follow from the Jacobian Property, namely the image of both cells are cells with center $f_j(c)$.\n\nIn Case (ii), definably choose, for each twisted box $B$ of $X$ and each $j$, an element $\\tilde c_{j,B} \\in \\tilde C$ in such a way that $f_j(B)$ is $1$-next to $\\tilde c_{j,B}$. By a further finite partition of $X$, we may assume that $\\tilde c_{j,B}$ does not depend on $B$. Then\n$f_j(X)$ is a cell with center $\\tilde c_{j,B}$ and we are done.\n\\end{proof}\n\n\nAs explained above, we will not give detailed proofs of Theorem \\ref{thm:cd:alg:piece:Lipschitz} and its related result from Addendum \\ref{add:cd:Lip:comp} to Theorem \\ref{thm:cd:alg:skol}, and we do not use these results in this paper. We nevertheless specify where this is worked out, under very closely related assumptions.\n\\begin{proof}[Proof of Theorem \\ref{thm:cd:alg:piece:Lipschitz} and Addendum \\ref{add:cd:Lip:comp} to Theorem \\ref{thm:cd:alg:skol}]\nUnder our Assumption \\ref{ass:acl=dcl}  that $\\acl$ equals $\\dcl$, but assuming a notion of tameness with an angular component map ${\\overline{\\rm ac}}$ (instead of $1$-h-minimality with $\\operatorname{rv}$), both results are proved in \\cite{CFL}, and the proof readily adapts. (By Theorem \\ref{thm:tame2vf} and Remark \\ref{rem::tame2vf}, $1$-h-minimality and the tameness notion are very closely related.)\n\\end{proof}\n\n(The proof of Addendum~\\ref{add:cd:Lip:comp} and its variant in \\cite{CFL} essentially comes from \\cite{CCL-PW}, apart from the improvement made possible by the assumption $\\acl_K=\\dcl_K$.) \n\n\n\n\\subsection{Dimension theory}\\label{sec:dim}\n\nUnder the assumption of $1$-h-minimality, there is a good notion of dimension of definable subsets of $K^n$. It can be defined in various equivalent ways; here is one possible definition.\n\n\\begin{defn}[Dimension]\\label{defn:dim}\nWe define the dimension of a non-empty definable set $X \\subset K^n$ as the maximal integer $m$ such that there is a $K$-linear function $\\ell : K^n\\to K^m$ such that $\\ell(X)$ has non-empty interior in $K^m$. If $X$ is empty, we set $\\dim X := -\\infty$.\n\\end{defn}\n\n\\begin{remark}\nIn Proposition~\\ref{prop:dim:basic} (\\ref{dim:1}), we will see that one could equivalently only consider coordinate projections $\\ell\\colon K^n \\to K^m$ (instead of arbitrary linear maps).\n\\end{remark}\n\nMany properties about the dimension of definable sets follow rather easily from\ncell decomposition. A proof of such properties has been carried out in \\cite{CLb} under an axiomatic assumption called $b$-minimality. Instead of repeating that proof, we verify that $1$-h-minimality implies $b$-minimality:\n\n\\begin{prop}[b-minimality]\\label{prop:b-min}\nAssume that $\\operatorname{Th}(K)$ is $1$-h-minimal.\nThen the two sorted structure on $(K,\\mathrm{RV})$ obtained from $K$ by adding the sort $\\mathrm{RV}$ and the map $\\operatorname{rv}$ is $b$-minimal in the sense of Definition 2.1 of \\cite{CLb}, with $K$ as main sort. More specifically, the structure $(K,\\mathrm{RV})$ is $b$-minimal with centers and preserves all balls in the sense of Definitions 5.1 and 6.2 of \\cite{CLb}.\n\\end{prop}\n\\begin{proof}\nThe axioms of Definition 2.1 of \\cite{CLb} clearly hold, and Definition 5.1 of \\cite{CLb} follows from the Jacobian Property as formulated in Corollary \\ref{cor:JP}.\n\\end{proof}\n\nThe definition of dimension given in \\cite[Definition~4.1]{CLb} is different than ours, but the results from \\cite[Section 4]{CLb} imply that the definitions are equivalent:\nIf $X \\subset K^n$ is a finite union of cells, then the dimension of $X$ in our sense equals the dimension of $X$ in the sense of \\cite{CLb}, namely the maximum of the dimensions of the cells, where the\ndimension of a cell of cell-type $(j_i)_{i=1}^n$\nis $\\sum_i j_i$.\n\nThe following proposition summarizes the good properties of dimension; in particular, we have definability of dimension, as in o-minimal structures. Property (\\ref{prop:dim:frontier}) is new.\n\n\\begin{prop}[Dimension theory]\\label{prop:dim:basic}\nAssume that $\\operatorname{Th}(K)$ is $1$-h-minimal. Let $X\\subset K^n$, $Y\\subset K^n$ and $Z\\subset K^{m}$ be non-empty definable sets, and let $f:X\\to Z$ be a definable function. Then the following properties hold.\n\\begin{enumerate}\n \\item\\label{dim:1}\n For any $d \\le n$, we have $\\dim X \\ge d$ if and only if there exists a projection $\\pi\\colon K^n \\to K^d$ to a subset of the coordinates\n such that $\\pi(X)$ has non-empty interior. In particular,\n $\\dim X = 0$ if and only if $X$ is finite.\n \\item\\label{dim:2} $\\dim(X \\cup Y) = \\max\\{\\dim X, \\dim Y\\}$.\n \\item\\label{dim:3}\\label{dim:defble} For any $d \\le n$, the set of $z\\in Z$ such that $\\dim f^{-1}(z) = d$ is definable over the same parameters as $f$.\n \\item\\label{dim:4} If all fibers of $f$ have dimension $d$, then $\\dim X = d + \\dim Z$.\n \\item\\label{dim:5}\\label{prop:dim:local} There exists an $x \\in X$ such that the local dimension of $X$ at $x$ is equal to the dimension of $X$, i.e.,\n   such that for every open ball $B \\subset K^n$ around $x$, we have $\\dim (X \\cap B) = \\dim X$.\n \\item\\label{dim:6}\\label{prop:dim:frontier} One has $\\dim (\\overline X\\setminus X) < \\dim X$, where $\\overline X$ is the topological closure of $X$, for the valuation topology.\n\\end{enumerate}\n\\end{prop}\n\n\nAlthough most likely, property (\\ref{prop:dim:frontier}) can be proved in a similar way as Theorem (1.8) of \\cite{vdD}, we postpone that proof until the end of Subsection~\\ref{sec:t-strat}, where we will have t-stratifications at our disposal, which will make the proof much simpler.\nWe do however right away prove the ``easy'' case of Property (\\ref{prop:dim:frontier}), namely when $\\dim X$ is equal to the ambient dimension $n$.\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:dim:basic}, except for (\\ref{prop:dim:frontier}) when $\\dim X < n$]\nProperties (\\ref{dim:1}) to (\\ref{dim:4}) follow from Proposition \\ref{prop:b-min} and \\cite[Section 4]{CLb}, except for the ``in particular'' part of (\\ref{dim:1}).\n\nConcerning that ``in particular'' part:\nIt is clear that a finite set has dimension $0$. If $X$ is infinite, then there exists a coordinate projection $\\pi\\colon K^n \\to K$ such that $\\pi(X)$ is infinite. By Lemma~\\ref{lem:finite}, $\\pi(X)$ contains a ball, so by Definition~\\ref{defn:dim}, $X$ has dimension at least $1$.\n\nProperty (\\ref{prop:dim:local}) is proved in \\cite{FornHal} in a much more general context; here is a much shorter proof in the present setting:\nWe may assume that $\\acl$ equals $\\dcl$ in $\\operatorname{Th}(K)$ (using Proposition~\\ref{prop:exist:alg:skol}), so that we can apply cell decomposition (Theorem~\\ref{thm:cd:alg:skol}) to $X$; we also use Addendum~\\ref{add:cd:cont:c} to get continuous centers.\n\nChoose a cell $A_\\ell \\subset X$ of maximal dimension (of cell-type $(j_i)_{i=1}^n$, with $\\sum_i j_i = \\dim X =: d$). Then for any $x \\in A_\\ell$, the local dimension of $X$ at $x$ is $d$. Indeed,\nthe projection $\\pi(A_\\ell) \\subset K^d$ to the coordinates $\\{i \\le n \\mid j_i = 1\\}$ is a cell of cell-type $(1, \\dots, 1)$ with continuous center, and hence open. Thus for every sufficiently small ball $B \\subset K^n$ around $x$, we have $\\pi(X \\cap B) = \\pi(B)$, witnessing $\\dim (X \\cap B) \\ge d$.\n\n\nTo prove Property (\\ref{prop:dim:frontier}) in the case $\\dim X = n$, we again\nfirst expand the language so that $\\acl$ equals $\\dcl$ and then find a cell decomposition of $\\overline X \\setminus X$. Since every $n$-dimensional cell has non-empty interior, no such cells can be contained in $\\overline X \\setminus X$. This implies\n$\\dim (\\overline X\\setminus X) < n$.\n\\end{proof}\n\n\n\n\\subsection{Jacobian Properties in many variables}\n\\label{sec:sjp}\n\nThere are different ways to generalize the Jacobian Property (Definition~\\ref{defn:JP})\nto functions $f$ in several variables. The one presented in this subsection (which we now call the Supremum Jacobian Property) has been introduced in \\cite{Halup} and is used to obtain t-stratifications. (To be precise, \\cite[Definition~2.19]{Halup} is a bit weaker than Definition~\\ref{defn:sup-prep} below).\n\nFirst of all, we need a specific higher-dimensional version of the $\\operatorname{rv}$-map.\n\n\\begin{notn}[$\\mathrel{<\\joinrel\\llap{\\raisebox{-1ex}{$\\scriptstyle{0}\\mkern8mu$}}}$]\nGiven $\\lambda, \\mu \\in \\Gamma_K$, we define $\\lambda \\mathrel{<\\joinrel\\llap{\\raisebox{-1ex}{$\\scriptstyle{0}\\mkern8mu$}}} \\mu$ as $\\lambda < \\mu  \\vee \\lambda = \\mu = 0$.\n\\end{notn}\n\n\\begin{defn}[Higher-dimensional $\\mathrm{RV}$]\nFor every $n \\ge 1$, we define $\\mathrm{RV}^{(n)}$ as the quotient $K^n\/\\mathord{\\sim}$, where $x \\sim x' \\iff |x - x'| \\mathrel{<\\joinrel\\llap{\\raisebox{-1ex}{$\\scriptstyle{0}\\mkern8mu$}}} |x|$.\nWe write $\\operatorname{rv}^{(n)}$ for the canonical map $K^n \\to\\mathrm{RV}^{(n)}$. (For matrices $M \\in K^{n \\times m}$, we will use the more suggestive notation\n$\\operatorname{rv}^{(n \\times m)}(M)$ instead of $\\operatorname{rv}^{(n \\cdot m)}(M)$.)\n\\end{defn}\n\n(Recall that for $x \\in K^n$, $|x|$ denotes the maximum norm of $x$.)\n\n\\begin{remark}\nNote that $\\mathrm{RV}^{(1)}$ is just the usual $\\mathrm{RV}$.\nFor $n \\ge 2$, $\\mathrm{RV}^{(n)}$ is not the same as $\\mathrm{RV}^n$, but $\\operatorname{rv}^{(n)}$ factors over coordinate-wise $\\operatorname{rv}$, so that we have a natural surjection $\\mathrm{RV}^n \\to \\mathrm{RV}^{(n)}$. Moreover, the maximum norm on $K^n$ factors over $\\mathrm{RV}^{(n)}$.\n\\end{remark}\n\n\\begin{remark}\\label{rem:rvn:GLn}\nAs explained in \\cite[Section~2.2]{Halup}, $\\operatorname{rv}^{(n)}$ interacts well with ${\\rm GL}_n({\\mathcal O}_K)$; in particular, given $M \\in {\\rm GL}_n({\\mathcal O}_K)$ and $x \\in K^n$, $\\operatorname{rv}^{(n)}(Mx)$ is determined by $\\operatorname{rv}^{(n \\times n)}(M)$ and $\\operatorname{rv}^{(n)}(x)$.\n\\end{remark}\n\n\\begin{defn}[Sup-Jac-prop, sup-preparation]\\label{defn:sup-prep}\nFor $X \\subset K^n$ open and $f\\colon X \\to K$, we say that $f$ has the\n\\emph{Supremum Jacobian Property} (\\emph{sup-Jac-prop} for short) on $X$ if\n$f$ is $C^1$ on $X$, $\\operatorname{rv}^{(n)}(\\operatorname{grad} f)$ is constant on $X$, and for every $x_0$ and $x$ in $X$ we have:\n\\begin{equation}\\label{eq:T3\/2,mv}\n|f(x) - f(x_0) -  ((\\operatorname{grad} f)(x_0))\\cdot(x - x_0) | \\mathrel{<\\joinrel\\llap{\\raisebox{-1ex}{$\\scriptstyle{0}\\mkern8mu$}}}  |\\operatorname{grad} f |\\cdot |x-x_0|.\n\\end{equation}\nAs usual, we consider $(\\operatorname{grad} f)(x_0)$ as a matrix with a single row, which we multiply with the column vector $x - x_0$ in the usual way.\nWe say that a map $\\chi\\colon X \\to \\mathrm{RV}^k$ \\emph{sup-prepares} $f$ (for some $k \\ge 0$) if each $n$-dimensional fiber $F \\subset K^n$ of $\\chi$ is open and $f$ has the sup-Jac-prop on each such $F$.\n\\end{defn}\n\n\n\n\\begin{remark}\\label{rem:move:x_0}\nOne easily checks that the validity of (\\ref{eq:T3\/2,mv}) does not depend on the precise value of $(\\operatorname{grad} f)(x_0)$, but only on $\\operatorname{rv}^{(n)}((\\operatorname{grad} f)(x_0))$, so it does not play a role whether we evaluate $\\operatorname{grad} f$ at $x_0$ or at any other point of $X$.\n\\end{remark}\n\n\n\n\\begin{remark}\nIn the case $n = 1$, (\\ref{eq:T3\/2,mv}) is equivalent to $\\operatorname{rv}(f(x) - f(x_0)) = \\operatorname{rv}(f'(x_0))\\cdot \\operatorname{rv}(x - x_0)$ (which is exactly the main condition of the one-dimensional Jacobian Property; see Definition~\\ref{defn:JP}).\nFor $n\\ge 2$ however, (\\ref{eq:T3\/2,mv}) does not always determine\n$\\operatorname{rv}(f(x) - f(x_0))$.\nIndeed, if e.g.\\ $x - x_0$ is orthogonal to $(\\operatorname{grad} f)(x_0)$, then (\\ref{eq:T3\/2,mv}) only imposes an upper bound on $|f(x) - f(x_0)|$.\n\\end{remark}\n\n\\begin{remark}\nOne cannot expect to be able to sup-prepare definable functions in the stronger sense that $\\operatorname{rv}(f(x) - f(x_0))$ is equal to\n$\\operatorname{rv}(((\\operatorname{grad} f)(x_0))\\cdot(x - x_0))$ within fibers of $\\chi$ (which would correspond to replacing the right hand side of (\\ref{eq:T3\/2,mv}) by $|((\\operatorname{grad} f)(x_0))\\cdot(x - x_0)|$). Indeed, consider for example $f(x, y) = y - x^2$, fix any $(x_0, y_0) \\in K^2$ and any $\\epsilon \\in K^\\times$, and set $(x,y) := (x_0 + \\epsilon, y_0 + 2x_0\\epsilon)$. Then\n$((\\operatorname{grad} f)(x_0,y_0))\\cdot((x,y) - (x_0,y_0)) = 0$ but\n$f(x,y) - f(x_0,y_0) \\ne 0$. For any $\\chi$ potentially preparing $f$, we can make such choices such that $(x_0,y_0)$ and $(x, y)$ lie in the same fiber.\n\\end{remark}\n\n\nThe following lemma states that the sup-Jac-prop is preserved by certain transformations.\n\n\n\\begin{lem}[Preservation of sup-Jac-prop]\\label{lem:liptrans}\nLet $X \\subset K^m$ and $Y \\subset K^n$ be open subsets\nand let $\\alpha\\colon X \\to Y$ be a $C^1$-map.\nSuppose that $\\operatorname{rv}^{(n \\times m)}(\\operatorname{Jac} \\alpha)$ is constant on $X$ and that\nfor every $x_1, x_2 \\in X$ we have\n\\begin{equation}\\label{eq:scaling}\n|\\alpha(x_2) - \\alpha(x_1)| = |\\operatorname{Jac} \\alpha|\\cdot |x_2 - x_1|\n\\end{equation}\nand\n\\begin{equation}\\label{eq:alpha}\n\\operatorname{rv}^{(n)}(\\alpha(x_2) - \\alpha(x_1)) = \\operatorname{rv}^{(n)}((\\operatorname{Jac} \\alpha)(x_1)\\cdot (x_2 - x_1))\n\\end{equation}\nFinally, suppose that $f\\colon Y \\to K$ is a $C^1$-map such that $f \\circ \\alpha$ has the sup-Jac-prop (on $X$).\nThen $f$ satisfies (\\ref{eq:T3\/2,mv}) for all $x_0, x \\in \\alpha(X)$.\n\\end{lem}\n\n\\begin{proof}\nLet $x_1, x_2 \\in X$ be given and set $y_i := \\alpha(x_i)$ and $z_i := f(y_i)$.\nIn the following, gradients and Jacobians will always be computed at $x_1$ or $y_1$; we\nwill omit those points from the notation.\n\nWhat we need to show is:\n\\begin{equation}\\label{eq:lt-want}\n|z_2 - z_1 - (\\operatorname{grad} f)\\cdot (y_2 - y_1)| \\mathrel{<\\joinrel\\llap{\\raisebox{-1ex}{$\\scriptstyle{0}\\mkern8mu$}}}  |\\operatorname{grad} f| \\cdot |y_2 - y_1|.\n\\end{equation}\nBy assumption, we have\n\\begin{equation}\\label{eq:lt-have}\n|z_2 - z_1 - (\\operatorname{grad} (f\\circ \\alpha))\\cdot (x_2 - x_1)| \\mathrel{<\\joinrel\\llap{\\raisebox{-1ex}{$\\scriptstyle{0}\\mkern8mu$}}}  |\\operatorname{grad} (f\\circ \\alpha)| \\cdot |x_2 - x_1|.\n\\end{equation}\nApplying $\\operatorname{grad} (f\\circ \\alpha) = (\\operatorname{grad} f)\\cdot (\\operatorname{Jac} \\alpha)$\nto the right hand side of (\\ref{eq:lt-have}) gives\n\\[\n|\\operatorname{grad} (f\\circ \\alpha)| \\cdot |x_2 - x_1| \\le |\\operatorname{grad} f| \\cdot |\\operatorname{Jac} \\alpha| \\cdot |x_2 - x_1|\n\\overset{(\\ref{eq:scaling})}{=} |\\operatorname{grad} f| \\cdot |y_2 - y_1|.\n\\]\nOn the left hand side of (\\ref{eq:lt-have}), we do the following:\n\\[\n(\\operatorname{grad} (f\\circ \\alpha))\\cdot (x_2 - x_1) = (\\operatorname{grad} f)\\cdot (\\operatorname{Jac} \\alpha)\\cdot (x_2 - x_1) \\approx\n(\\operatorname{grad} f)\\cdot (y_2 - y_1),\n\\]\nwhere in the ``$\\approx$'', we use (\\ref{eq:alpha}) to get an error $e$ with  $e \\mathrel{<\\joinrel\\llap{\\raisebox{-1ex}{$\\scriptstyle{0}\\mkern8mu$}}} |\\operatorname{grad} f| \\cdot |y_1 - y_2|$ (and this is what ``$\\approx$'' means here).\nPutting things together yields (\\ref{eq:lt-want}), as desired.\n\\end{proof}\n\nThe main result of this subsection is that every definable function on $K^n$ can be sup-prepared:\n\n\\begin{thm}[Sup-preparation]\\label{thm:T3\/2,mv}\nSuppose that $\\operatorname{Th}(K)$ is $1$-h-minimal.\nFor every $\\emptyset$-definable function $f\\colon K^n \\to K$, there exists a $\\emptyset$-definable map $\\chi\\colon K^n\\to \\mathrm{RV}^k$ (for some $k\\geq 0$) sup-preparing $f$\n(in the sense of Definition~\\ref{defn:sup-prep}).\n\\end{thm}\n\nThe proof needs some kind of cell decomposition with 1-Lipschitz centers, as e.g.\\ provided by Theorem~\\ref{thm:cd:alg:skol}, Addendum~\\ref{add:cd:Lip:comp}. To keep this paper more self-contained (since we did not give the proof of Addendum~\\ref{add:cd:Lip:comp} in full detail), we will instead prove and use the following weaker version of the addendum; more precisely, this proposition is proved in a joint induction with Theorem~\\ref{thm:T3\/2,mv}.\n\n\\begin{prop}[Twisted boxes with $1$-Lipschitz centers]\\label{prop:twibox:1Lip}\nAssume $1$-h-minimality and that $\\acl$ equals $\\dcl$ (in the sense of Assumption~\\ref{ass:acl=dcl}).\nThen, for every $\\emptyset$-definable set $X \\subset K^n$, there exists a\n$\\emptyset$-definable map $\\chi\\colon X \\to \\mathrm{RV}^{k'}$ such that each fiber $F$ of $\\chi$ is, up to permutation of coordinates,\nan $\\mathrm{RV}$-definable twisted box of cell-type $(1, \\dots, 1, 0, \\dots, 0)$ with $1$-Lipschitz center (i.e., each component $c_i\\colon \\pi_{<i}(F) \\to K$ of the center tuple is $1$-Lipschitz).\n\\end{prop}\n\nHere, by ``twisted box of cell-type etc.'', we mean: cell of cell-type etc., consisting of a single twisted box; see Definition~\\ref{defn:cell}.\n\nA key step in the proof of the proposition consists in swapping\ntwo coordinates, to make the derivative of some center smaller.\nThis uses the following lemma, which has a similar but simpler proof than Cases 1 and 2 of the proof of Proposition 2.4 of \\cite{CCLLip}.\nBy a ``genuine box'' in $K^n$, we mean a Cartesian product of balls in $K$ (as opposed to the more general notion of ``twisted box'').\n\n\\begin{lem}\\label{lem:swap:twibox}\nAssuming $1$-h-minimality and that $\\acl$ equals $\\dcl$:\nSuppose that \\[X = \\{(x_1,x_2) \\in K^2 \\mid \\operatorname{rv}(x_1 - c_1) = \\xi_1,\\ \\operatorname{rv}(x_2 - c_2(x_1)) = \\xi_2\\}\\]\nis an $A$-definable twisted box, for some set of parameters $A \\subset K$ and that $c_2$ has the Jacobian Property. Then either $X$ is a genuine box (i.e., a Cartesian product of two balls), or,  we have $X = \\{(x_1,x_2) \\in K \\times Y  \\mid  \\operatorname{rv}(x_1 - c_2^{-1}(x_2)) = \\xi_3\\}$ for some suitable constant $\\xi_3 \\in \\dcl_\\mathrm{RV}(A)$ and where $Y$ is the projection of $X$ to the second coordinate.\n\\end{lem}\n\n\\begin{proof}\nThe image $c_2(\\pi_{\\le 1}(X))$ is an open ball of radius $\\rho := |\\xi_1|\\cdot |c_2'|$ (where $c_2'$ is the derivative of $c_2$). If $\\rho \\le |\\xi_2|$, then whether \\(\\operatorname{rv}(x_2 - c_2(x_1)) = \\xi_2\\) holds does not depend on the choice on \\(x_1\\in \\pi_{\\le 1}(X)\\) and \\(X\\) is a genuine box.\nOtherwise, if $\\rho >  |\\xi_2|$, then \\(c_2^{-1}\\) is defined on the whole of \\(Y\\) and the Jacobian property implies that \\(\\operatorname{rv}(x_2 -c_2(x_1)) = \\xi_2\\) if and only if \\(\\operatorname{rv}(x_1 - c_2^{-1}(x_2)) = -\\operatorname{rv}(c_2')^{-1}\\cdot\\xi_2\\).\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:twibox:1Lip} in the case $n = 1$]\nPrepare $X$ by a finite set $C$ and let $\\chi$ be the map given by Lemma~\\ref{lem:InextFam}.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:twibox:1Lip},\nassuming Proposition~\\ref{prop:twibox:1Lip} and Theorem~\\ref{thm:T3\/2,mv} for $n - 1$]\n\nBy an ``$\\mathrm{RV}$-partition'' of $X$, we mean a partition into fibers of an $\\mathrm{RV}$-definable\nmap $X \\to \\mathrm{RV}^k$. Note that if we are already given an $\\mathrm{RV}$-partition of $X$, it suffices to prove the proposition for each fiber individually. (Then put everything together using compactness.)\n\nWe say that $X$ is a ``thick graph'' (of the map $c_n$) if it is of the form $\\{(y, x_n) \\in Y \\times K\\mid \\operatorname{rv}(x_n - c_n(y)) = \\xi\\}$ for some $Y \\subset K^{n-1}$, some $c_n \\colon Y \\to K$, and some $\\xi \\in \\mathrm{RV}$. (Note that we allow $\\xi = 0$, which means that $X$ is just the graph of $c_n$.)\n\nNote that it suffices to obtain the claim in the last coordinate, i.e., to $\\mathrm{RV}$-partition $X$ into sets that are,\nup to permutation of coordinates, thick graphs of $1$-Lipschitz functions $c_n\\colon Y \\to K$.\nAfter that, the proposition follows by applying induction to $Y$.\n\nUsing cell decomposition, we reduce to the case where $X$ is a thick graph of a function $c_n\\colon Y \\to K$ and $Y$ is a twisted box. In particular, $X$ is a twisted box. We may assume that $Y$ is either open or has empty interior (by treating the interior separately).\n\nStep 1: If $Y$ has empty interior, we apply induction to $Y$ to reduce to the case that $Y$ is a twisted box with $1$-Lipschitz centers and we translate the centers away so that $X$ lives in a subspace of $K^n$ where some of the coordinates are $0$; then apply induction once more to finish. Note that, translating the variables of a $1$-Lipschitz function by $1$-Lipschitz functions yields a  $1$-Lipschitz function and hence the original $X$ does have the required properties.\n\nSo suppose from now on that $Y$ is open. By partitioning $Y$, we may assume that $c_n$ is $C^1$ and that $|\\partial c_n\/\\partial x_{i}|$ is constant on $Y$ for each $i$. (Lower-dimensional pieces are treated as in Step 1.)\n\nStep 2: Assume $|\\operatorname{grad} c_n| \\le 1$.  Using the $n-1$ case of Theorem~\\ref{thm:T3\/2,mv}, we may assume that $c_n$ has the sup-Jac-prop. This, together with\n$|\\operatorname{grad} c_n| \\le 1$, implies that $c_n$ is $1$-Lipschitz, and hence we are done.\n\nStep 3: So now suppose $|\\operatorname{grad} c_n| > 1$. We do an induction on the number of partial derivatives of $c_n$ satisfying $|\\partial c_n\/\\partial x_{i}| > 1$. We suppose without loss that\n$|\\partial c_n\/\\partial x_{n-1}| = |\\operatorname{grad} c_n|$. Let $Z$ be the projection of $Y$ to the first $n-2$ coordinates.\nBy further partitioning, we reduce to the case where, for each individual $a \\in Z$,\nthe function $c_n(a, \\cdot)$ has the Jacobian Property (using Corollary~\\ref{cor:JP} and compactness) and\nhas an open ball as domain. (Again, lower-dimensional pieces are treated as in Step 1.)\n\nNote that for each $a \\in Z$, the fiber $X_{a} \\subset K^2$ is a twisted box. By further partitioning $Z$, we may assume that either all of them or none of them are genuine boxes.\n\nStep 3.a: If all fibers are genuine boxes: by induction on $n$, we may assume that the projection $\\tilde X$ of $X$ to the coordinates $1, \\dots, n-2, n$ is a thick graph of a $1$-Lipschitz function $\\tilde c_n\\colon Z \\to K$.\n(This involves permuting the coordinates $1, \\dots, n-2, n$.) Then  $X$ is a thick graph of $c_n(z, x_{n-1}) := \\tilde c_n(z)$, which is $1$-Lipschitz, so we are done.\n\nStep 3.b: If no fiber is a genuine box, we apply the map $\\sigma\\colon K^n \\to K^n$ swapping the coordinates $n-1$ and $n$. By compactness and Lemma~\\ref{lem:swap:twibox},\n$\\sigma(X)$ is the thick graph of the function $c_{n,\\mathrm{new}}$ sending $(x_1, \\dots, x_{n-2}, c_n(x_{n-1}))$ to $x_{n-1}$. Since $|\\partial c_{n,\\mathrm{new}}\/\\partial x_i|\\le |\\partial c_{n}\/\\partial x_i|$ for $i \\le n - 2$ and\n$|\\partial c_{n,\\mathrm{new}}\/\\partial x_n|= 1\/ |\\partial c_{n}\/\\partial x_{n-1}| < 1$, $c_{n,\\mathrm{new}}$ has fewer partial derivatives bigger than $1$, so we can finish by the induction from Step~3.\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:T3\/2,mv} in the case $n = 1$]\nThis follows directly from Corollary~\\ref{cor:JP}\nand Lemma~\\ref{lem:InextFam}: The corollary yields a finite $\\emptyset$-definable set $C$ such that (\\ref{eq:T3\/2,mv}) holds on every ball $1$-next to $C$, and Lemma~\\ref{lem:InextFam} then yields a map $\\chi\\colon K \\to \\mathrm{RV}^k$ whose $1$-dimensional fibers are exactly those balls.\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:T3\/2,mv}, assuming Proposition~\\ref{prop:twibox:1Lip} for $n$ and Theorem~\\ref{thm:T3\/2,mv} for $n - 1$]\nThe proof consists of three parts.\n\n\\medskip\n\nPart 1: Some preliminaries:\n\n\\medskip\n\n\\begin{claim}\\label{cl:3\/2-acldcl}\nIt suffices to prove the theorem under the assumption that $\\acl$ equals $\\dcl$.\n\\end{claim}\n\n\\begin{proof}\nLet ${\\mathcal L}^{\\rm as} \\supset {\\mathcal L}$ be as given by Proposition~\\ref{prop:exist:alg:skol}, i.e., $\\operatorname{Th}_{{\\mathcal L}^{\\rm as}}(K)$ is still $1$-h-minimal, and in $\\operatorname{Th}_{{\\mathcal L}^{\\rm as}}(K)$, we have $\\acl$ equals $\\dcl$. Assuming that Theorem~\\ref{thm:T3\/2,mv} holds for this language, we find an ${\\mathcal L}^{\\rm as}$-definable $\\chi'\\colon K^n\\to \\mathrm{RV}^{k'}$ sup-preparing $f$.\nSince ${\\mathcal L}^{\\rm as}$ is an $\\mathrm{RV}$-expansion of ${\\mathcal L}$, Lemma~\\ref{lem:undo-K-to-RV} provides an ${\\mathcal L}$-definable $\\chi\\colon K^n\\to \\mathrm{RV}^k$ such that each fiber $F$ of $\\chi$\nis contained in a fiber of $\\chi'$; in particular, (\\ref{eq:T3\/2,mv}) holds whenever $(\\operatorname{grad} (f|_F))(x_0)$ is defined. It remains to refine $\\chi$ in such a way that each of its $n$-dimensional fibers is open.\nWe do this by splitting each $n$-dimensional fiber $F$ into its interior $\\mathring F$ and the remainder. Then indeed $\\mathring F$ is open, and\n$F \\setminus \\mathring F$ has dimension less than $n$,\nby Proposition~\\ref{prop:dim:basic} (\\ref{prop:dim:frontier}) applied to\n$K^n \\setminus F$. (Note that we only use the case of Proposition~\\ref{prop:dim:basic} (\\ref{prop:dim:frontier}) which we already proved right after the proposition.)\n\\qedhere(\\ref{cl:3\/2-acldcl})\n\\end{proof}\n\nSo for the remainder of the proof we assume that $\\acl$ equals $\\dcl$ (so that we can apply Cell Decomposition and Proposition~\\ref{prop:twibox:1Lip}).\n\nRecall that we inductively assume that the theorem holds up to dimension $n-1$. From this, we deduce the following for functions defined on some $n$-dimensional neighborhoods of lower-dimensional subsets of $K^n$.\n\n\\begin{claim}\\label{cl:on-graphs}\nGiven any $(n-1)$-dimensional $\\emptyset$-definable $Z \\subset K^{n}$ and any $\\emptyset$-definable $C^1$-function $f$ to $K$ defined on an open neighborhood of $Z$, there exists a $\\emptyset$-definable map $\\chi\\colon Z \\to \\mathrm{RV}^k$ (for some $k \\ge 0$) such that if $F \\subset Z$ is an $(n-1)$-dimensional fiber of $\\chi$, then\n(\\ref{eq:T3\/2,mv}) holds for every pair $x_0, x \\in F$.\n\\end{claim}\n\n\\begin{proof}\nWe can (and will repeatedly) partition $Z$ into fibers of a $\\emptyset$-definable map $Z \\to \\mathrm{RV}^k$. (If the claim holds for each fiber of such a partition, we then obtain the desired map $Z \\to \\mathrm{RV}^{k'}$ using compactness.)\nBy partitioning $Z$ into the twisted boxes of a cell decomposition, we may assume that $\\operatorname{rv}^{(n)}(\\operatorname{grad} f)$ is constant. By Proposition~\\ref{prop:twibox:1Lip}, we may assume that $Z$ is a twisted box of cell-type $(1, \\dots, 1, 0)$ with\n$1$-Lipschitz center.\n\nLet $\\hat Z$ be the projection of $Z$ to the first $n-1$ coordinates, so that $Z$ is the graph of a $1$-Lipschitz function $c \\colon \\hat Z \\to K$. Apply Theorem~\\ref{thm:T3\/2,mv} to $c$ and to $f \\circ \\alpha$, where $\\alpha\\colon \\hat Z \\to Z, x \\mapsto (x, c(x))$, and partition $\\hat Z$ and $Z$ accordingly, i.e., so that after the partition, $c$ and $f \\circ \\alpha$ have the sup-Jac-prop on $\\hat Z$. Using that $c$ is $1$-Lipschitz, one obtains that the map $\\alpha\\colon \\hat Z \\to Z, x \\mapsto (x, c(x))$ satisfies the assumptions of Lemma~\\ref{lem:liptrans}. Therefore, the fact that $f \\circ \\alpha$ satisfies (\\ref{eq:T3\/2,mv}) on $\\hat Z$ implies that $f$ satisfies (\\ref{eq:T3\/2,mv}) on $Z$, as desired.\n\\qedhere(\\ref{cl:on-graphs})\n\\end{proof}\n\nLet now a $\\emptyset$-definable function $f\\colon K^n \\to K$ be given (with $n \\ge 2$); we need to find a $\\emptyset$-definable map $K^n \\to \\mathrm{RV}^k$ sup-preparing $f$. We more generally allow the domain of $f$ to be any $\\emptyset$-definable set $X \\subset K^n$.\nAs in the proof of Claim~\\ref{cl:on-graphs}, if we have a $\\emptyset$-definable map $\\chi\\colon X \\to \\mathrm{RV}^k$, it suffices to sup-prepare the restrictions of $f$ to each fiber of $\\chi$. Moreover, fibers of dimension less than $n$ can always be neglected. This argument will be applied repeatedly.\n\nWe write elements of $K^n$ as $(x, y)$, with $x\\in K^{n-1}$ and $y \\in K$.\n\n\\medskip\n\nPart 2: Reducing to the case where\n$X$ is of the form $\\bar X\\times B$ for some $\\bar X \\subset K^{n-1}$ and some ball $B \\subset K$, $f$ is $C^1$, $\\operatorname{rv}^{(n)}(\\operatorname{grad} f)$ is constant, and\n\\begin{enumerate}\n\\item[(SJP1)] for each fixed $x \\in \\bar X$, the function $f(x, \\cdot)$ has the sup-Jac-prop.\n\\end{enumerate}\n\n\\medskip\n\nWe start by partitioning $K^n$ as follows:\n\n\\begin{enumerate}[(a)]\n \\item By repeatedly applying the case $n = 1$ (and using compactness), we may assume that $f$ has the sup-Jac-prop fiberwise: for every fixed $x \\in K^{n-1}$ and every coordinate permutation $\\sigma\\colon K^n \\to K^n$, the map $y \\mapsto f(\\sigma(x,y))$ has the sup-Jac-prop.\n \\item We moreover assume that $f$ is $C^1$ (using Theorem~\\ref{thm:Ck}) and that\n $\\operatorname{rv}^{(n)}(\\operatorname{grad} f)$ is constant.\n\\end{enumerate}\n\nBy applying Proposition~\\ref{prop:twibox:1Lip} and permuting coordinates, we may assume that $X$ is a twisted box with $1$-Lipschitz center:\n\\begin{equation}\\label{eq:F1}\nX = \\{(x, y) \\in K^{n-1} \\times K \\mid x \\in \\bar X,\\ \\operatorname{rv}(y -c(x)) = \\rho\\}\n\\end{equation}\nfor some $\\rho \\in \\mathrm{RV}^\\times$, some definable $1$-Lipschitz $c\\colon \\bar X \\to K$ and where $\\bar X := \\pi_{\\le n-1}(X) \\subset K^{n-1}$ is the projection to the first $n-1$ coordinates. We may assume that $c$ is $C^1$, and applying the Theorem inductively to $c$ allows us to moreover assume that $c$ has the sup-Jac-prop.\n\nSet $X' := \\bar X \\times \\operatorname{rv}^{-1}(\\rho) \\subset K^{n-1} \\times K$. The bijection $\\alpha\\colon X' \\to X, (x,y) \\mapsto (x, y + c(x))$\nsatisfies the assumptions of Lemma~\\ref{lem:liptrans}, so to prove that $f$ has the sup-Jac-prop on some set $F \\subset X$, it suffices\nto verify that $f' := f \\circ \\alpha$ has the sup-Jac-prop on $\\alpha^{-1}(F)$.\nIn other words, it remains to sup-prepare $f'\\colon X' \\to K$.\n\nBy (b), $f'$ is $C^1$ and $\\operatorname{rv}^{(n)}(\\operatorname{grad} f') = \\operatorname{rv}^{(n)}((\\operatorname{grad} f)(\\operatorname{Jac} \\alpha))$ is constant (using Remark~\\ref{rem:rvn:GLn});\nby (a),\n$f'(x,\\cdot)$ has the sup-Jac-prop for each $x \\in \\bar X$; thus we are done with Part~2.\n\n\\medskip\n\nPart 3: Finishing under the assumptions obtained in Part~2.\n\n\\medskip\n\nRecall that $X = \\bar X \\times B \\subset K^{n-1} \\times K$.\n\n\\begin{enumerate}\n \\item[(SJP2)] By induction, we find a map $\\chi\\colon X \\to \\mathrm{RV}^k$ such that for each fixed $y \\in B$, $g(\\cdot, y)$ is sup-prepared by $\\chi(\\cdot, y)$.\n\\end{enumerate}\nWe choose a cell decomposition of $X$ with continuous centers and we refine $\\chi$ in such a way that the fibers of $\\chi$ are exactly the twisted boxes of the cells. Given such a cell $A_\\ell$, let $\\bar A_\\ell := \\pi_{\\le n-1}(A_\\ell)$ be its projection and let\n$c_\\ell\\colon \\bar A_\\ell \\to K$ be the last component of its center tuple.\n\nLet $Z_\\ell$ be the intersection of the graph of $c_\\ell$ with $X$.\nWe apply Claim~\\ref{cl:on-graphs} to $Z_\\ell$ and $f$, yielding a map $\\chi_\\ell\\colon Z_\\ell \\to \\mathrm{RV}^k$,\nwe extend $\\chi_\\ell$ by $0$ to the whole graph of $c_\\ell$,\nand we replace $\\chi$ by the refinement $(x, y) \\mapsto (\\chi(x, y), (\\chi_\\ell(x, c_\\ell(x)))_\\ell)$.\nIn this way, we achieved the following:\n\\begin{enumerate}\n\\item[(SJP3)]\nGiven any $(x_1, y_1), (x_2, y_2)$ in the same $n$-dimensional fiber of $\\chi$, and given any $\\ell$, the pair of points $(x_i, c_\\ell(x_i))$ ($i =1,2$) satisfies (\\ref{eq:T3\/2,mv}), provided that both of those points $(x_i, c_\\ell(x_i))$ lie in $X$.\n\\end{enumerate}\nNote that since this refinement of $\\chi$ only depends on $x$,\neach fiber $F$ of $\\chi$ is still of the form\n\\begin{equation}\\label{eq:fiber}\nF = \\{(x, y) \\in K^{n-1} \\times K \\mid x \\in \\bar F,\\operatorname{rv}(y -c_\\ell(x)) = \\xi\\} \\subset \\bar X \\times B,\n\\end{equation}\nfor some $\\ell$, some $\\xi \\in \\mathrm{RV}$ and where $\\bar F = \\pi_{\\le n-1}(F)$.\nUsing one last refinement of $\\chi$ (also depending only on $x$), we may assume that $\\bar F$ is either open or has dimension less than $n-1$,\nso that if $F$ is $n$-dimensional, it is open.\nTo finish the proof of the theorem, we will prove that $f$ has the sup-Jac-prop on each such $n$-dimensional fiber $F$.\n\nWe already know that $f$ is $C^1$ on $F$ and that $\\operatorname{rv}^{(n)}(\\operatorname{grad} f)$ is constant on $F$, so it remains to verify (\\ref{eq:T3\/2,mv}); thus\nlet $(x_1,y_1), (x_2, y_2) \\in F$ be given.\n\nRecall (Remark~\\ref{rem:move:x_0}) that\nin (\\ref{eq:T3\/2,mv}), it does not matter at which point of $F$ we evaluate the gradient $\\operatorname{grad} f$. Using this, an easy computation\nshows that (\\ref{eq:T3\/2,mv}) can be verified\nin several steps, jumping to certain intermediate points $(x_3,y_3) \\in F$ first, namely:\nIf (\\ref{eq:T3\/2,mv}) holds for $(x_1,y_1), (x_3, y_3)$ and also for $(x_3,y_3), (x_2, y_2)$, and if moreover\n$|(x_1,y_1) - (x_3, y_3)| \\le |(x_1,y_1) - (x_2, y_2)|$, then (\\ref{eq:T3\/2,mv}) follows for $(x_1,y_1), (x_2, y_2)$.\nIn a similar way, we can also jump through several intermediate points.\nNote also that the intermediate points can be arbitrary points of $X$ (and do not need to lie in $F$), since $\\operatorname{rv}^{(n)}(\\operatorname{grad} f)$ is constant on all of $X$.\n\nWe use the notation from (\\ref{eq:fiber}) and distinguish three cases:\n\n\\medskip\n\nCase 1: $|c_\\ell(x_1) - y_1| > |y_2 - y_1|$.\nThen we have $(x_1, y_2) \\in F$, so we can jump from  $(x_1, y_1)$ to $(x_1, y_2)$ by (SJP1) and from $(x_1, y_2)$ to $(x_2, y_2)$ by (SJP2).\n\n\n\\medskip\n\nCase 2: $|c_\\ell(x_2) - y_2| > |y_2 - y_1|$: analogous to Case 1.\n\n\\medskip\n\nCase 3: $|c_\\ell(x_i) - y_i| \\le |y_2 - y_1|$ for $i = 1,2$:\nFrom $y_1, y_2 \\in B$, we deduce $c_\\ell(x_i) \\in B$,\nso that $(x_i, c_\\ell(x_i))$ lies in $X$. Moreover, we have $|(x_i,y_i) - (x_i, c_\\ell(x_i))| \\le |(x_1, y_1) - (x_2, y_2)|$.\nThis means that we can jump from $(x_1,y_1)$ to $(x_1, c_\\ell(x_1))$ by (SJP1), then to $(x_2, c_\\ell(x_2))$ by (SJP3), and then to $(x_2,y_2)$ by (SJP1) again.\n\\end{proof}\n\n\\subsection{t-stratifications}\n\\label{sec:t-strat}\n\nIn \\cite{Halup}, a notion of stratifications in valued fields has been introduced, called ``t-stratifications''. Intuitively, given a definable set $X \\subset K^n$, a t-stratification captures, for every ball $B \\subset K^n$, the dimension of the space of directions in which $X \\cap B$ is ``roughly translation invariant''. This strengthens classical notions of stratifications (like Whitney or Verdier stratifications), which capture rough translation invariance only locally.\n\nThe existence proof of t-stratifications given in \\cite{Halup} is carried out under some axiomatic assumptions, namely \\cite[Hypothesis~2.21]{Halup}. Those assumptions hold in valued fields with analytic structure (in the sense of \\cite{CLip}) by\n\\cite[Proposition~5.12]{Halup} and in power-bounded T-convex structures by \\cite{Gar.powbd}. We will now show that the assumptions hold in any 1-h-minimal theory of equi-characteristic $0$, hence implying that t-stratifications exist\nin this generality. By the examples of 1-h-minimal theories given in Section~\\ref{sec:examples}, this generalizes\nboth of the above results.\n\nIn this entire section, let $K$ be an equi-characteristic $0$ valued field with $1$-h-minimal theory.\n\nWe quickly recall the necessary definitions related to t-stratifications. First, here is the precise notion of ``roughly translation invariant'':\n\n\\begin{defn}[Risometries, translatability]\\label{defn:trble}\nLet $B \\subset K^n$ be a ball.\n\\begin{enumerate}\n\\item A bijection $f\\colon B \\to B$ is a \\emph{risometry}\n if for every $x_1, x_2 \\in B$, we have $\\operatorname{rv}^{(n)}(f(x_1) - f(x_2)) = \\operatorname{rv}^{(n)}(x_1 - x_2)$.\n\\item A map $\\chi\\colon K^n \\to Q$ (for some arbitrary set $Q$)\nis \\emph{$d$-translatable} on $B$, for some $0 \\le d \\le n$, if there exists a definable (with parameters) risometry $f\\colon B \\to B$ and a $d$-dimensional sub-vector space $V \\subset K^n$ such that for every $x, x' \\in B$ satisfying $x - x' \\in V$, we have $\\chi(f(x)) = \\chi(f(x'))$.\n\\item\nA subset $X \\subset K^n$ is called $d$-translatable if its characteristic function $1_X\\colon K^n \\to \\{0,1\\}$ is $d$-translatable.\n\\end{enumerate}\n\\end{defn}\n\n\n\n\n\\begin{defn}[t-stratifications]\nLet $\\chi\\colon K^n \\to Q$ be a map (for some arbitrary set $Q$) and let $A$ be a set of parameters. An \\emph{$A$-definable t-stratification reflecting} $\\chi$ is a partition of $K^n$ into $A$-definable sets $S_0, \\dots, S_n$ with the following properties:\n\\begin{enumerate}\n \\item $\\dim S_d \\le d$.\n \\item\n Set $\\chi'(x) := (\\chi(x), d(x)) \\in Q \\times \\{0,\\dots, n\\}$, where $d(x)$ is defined such that $x \\in S_d$ for every $x \\in K^n$.\n For each $d \\le n$ and each open or closed ball $B \\subset S_d \\cup \\dots \\cup S_n$,\nthis map $\\chi'$ is $d$-translatable on $B$.\n\\end{enumerate}\n\\end{defn}\n\n\n\n\\begin{thm}[t-stratifications]\\label{thm:t-strats}\nLet $K$ be a valued field of equi-characteristic $0$ with $1$-h-minimal theory, and\nlet $\\chi \\colon K^n \\to Q$ be a $\\emptyset$-definable map, where $Q$ is a sort of $\\mathrm{RV}^{\\mathrm{eq}}$.\nThen there exists a $\\emptyset$-definable t-stratification $(S_i)_{i \\le n}$ reflecting $\\chi$.\n\\end{thm}\n\n\n\\begin{proof}\nAccording to \\cite[Theorem~4.12]{Halup}, the existence of t-stratifications follows from \\cite[Hypothesis~2.21]{Halup}, so the theorem follows from the following lemma.\n\\end{proof}\n\n\\begin{lem}\nHypothesis~2.21 of \\cite{Halup} holds in $1$-h-minimal theories.\n\\end{lem}\n\n\\begin{proof}\nThe hypothesis consists of the following four parts.\n\\begin{enumerate}\n \\item $\\mathrm{RV}$ is stably embedded:\n\n This is Proposition~\\ref{prop:stab}.\n\\item  Definable maps from $\\mathrm{RV}$ to $K$ have finite image:\n\nThis is (a special case of) Corollary~\\ref{cor:finiterange}.\n\n\\item\nFor every $A \\subset K \\cup \\mathrm{RV}^{\\mathrm{eq}}$, every $A$-definable set $X \\subset K$ can be $1$-prepared by a finite $A$-definable set $C \\subset K$:\n\nThis is clear from the definition of $1$-h-minimality and\nLemma~\\ref{lem:addconst}.\n\n\\item\nThe theory has the Jacobian Property in the sense of \\cite[Theorem~2.19]{Halup}, namely: For every $A \\subset K \\cup \\mathrm{RV}^{\\mathrm{eq}}$, every $A$-definable function $f\\colon K^n \\to K$ can be sup-prepared (in the sense of Definition~\\ref{defn:sup-prep}) by an $A$-definable map $\\xi\\colon K^n \\to Q$, where $Q$ is a sort of $\\mathrm{RV}^{\\mathrm{eq}}$:\n\nAdd $A$ as constants to the language. Then (4) is just Theorem~\\ref{thm:T3\/2,mv}.\\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\nNote that the proof we gave here also simplifies the proofs from \\cite{Halup} (in the case of fields with analytic structure) and \\cite{Gar.powbd} (in the case of $T$-convex structures): In those papers, the proof of (4) was done using a complicated inductive argument using the existence\nof t-stratifications in lower dimension. This has been replaced by the more direct proof of our Theorem~\\ref{thm:T3\/2,mv}.\n\n\\medskip\n\nWe end this subsection with the promised proof of the missing part of Proposition \\ref{prop:dim:basic}, namely that for definable sets $X \\subset K^n$, the frontier\n $\\overline X\\setminus X$ has lower dimension than $X$:\n\n\\begin{proof}[Proof of Proposition \\ref{prop:dim:basic} (\\ref{prop:dim:frontier})]\nChoose a t-stratification reflecting\nthe Cartesian product $\\chi(x) := (1_X(x), 1_Y(x))$ of the characteristic functions of $X$ and of the frontier $Y := \\bar X \\setminus X$, and set $d := \\dim Y$.\nFor dimension reasons, $Y$ contains at least one point $y \\in S_d$.\n(Note that the definition of t-stratification implies $Y \\cap S_j = \\emptyset$ for $j > \\dim Y$; see \\cite[Lemma~3.10]{Halup}.) Assuming $\\dim X \\le d$, we will show that $y$ cannot be contained in the topological closure of $X$.\n\nSince $S_{\\le d-1} := S_0 \\cup \\dots \\cup S_{d-1}$ is closed (by \\cite[Lemma~3.17 (a)]{Halup}), there exists a ball $B \\subset S_d \\cup \\dots \\cup S_n$ containing $y$.\nLet $f\\colon B \\to B$ be a risometry and $V \\subset K^n$ be a vector space witnessing $d$-translatability of $\\chi$ on this $B$, as in Definition~\\ref{defn:trble}. Since $f$ is a homeomorphism (and $y \\notin X$), to obtain $y \\notin \\bar X$, it suffices to show that $X' := f^{-1}(X \\cap B)$ is closed in $B$.\nLet $\\pi \\colon K^n \\to K^n\/V$ be the canonical map. The definition of $d$-translatability implies that $X' = B \\cap \\pi^{-1}(\\pi(X'))$. Now the assumption $\\dim X \\le d$ implies $\\dim \\pi(X') = 0$, so indeed $X'$ is closed in $B$.\n\\end{proof}\n\n\n\n\\subsection{Taylor approximation on boxes disjoint from a lower dimensional set}\n\\label{sec:taylor-box}\n\n\nWe prove a higher-dimensional version of the Taylor approximation Theorem~\\ref{thm:high-ord}.\nBy a \\emph{box} in $K^n$, we mean a Cartesian product of balls in $K$.\n\n\\begin{thm}[Taylor approximations on boxes]\\label{thm:t-high-high}\nGiven a $\\emptyset$-definable function $f\\colon K^n\\to K$,\nthere exists a $\\emptyset$-definable set $C\\subset K^n$ of dimension at most $n-1$ such that for any box $B \\subset K^n \\setminus C$, $f$ is $(r+1)$-fold differentiable on $B$, for each $i\\in{\\mathbb N}^n$ with $|i|= r+1$ one has that  $|f^{(i)}|$ is constant on $B$, and we have\n\\begin{equation}\\label{eq:t-high-high}\n|f(x) -  T^{\\le r}_{f,x'}(x)| \\le \\max_{|i| = r+1} |f^{(i)}(x')(x - x')^i|\n\\end{equation}\nfor every $x, x' \\in B$. (Here, $i$ runs over $n$-tuples and we use the usual multi-index notation.)\n\\end{thm}\n\n\nOne may investigate whether one can obtain (\\ref{eq:t-high-high}) to not only holds on boxes disjoint from $C$, but also on fibers of a map $K^n \\to \\mathrm{RV}^k$, as in Theorem~\\ref{thm:T3\/2,mv}; however, we do not know how to prove this in general. See Question~\\ref{qu:t-high-high} for more discussion around this.\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:t-high-high}]\nWe do an induction over $n$, the case $n = 1$ being Theorem~\\ref{thm:high-ord}. Applying the induction hypothesis fiberwise (and using compactness) allows us to find a $C \\subset K^n$ ($\\emptyset$-definable, of dimension less than $n$) such that for every box $B = \\hat B \\times B_n \\subset K^{n-1} \\times K$ disjoint from $C$, for $x_n \\in B_n$ and for every $\\hat x, \\hat x' \\in \\hat B$, we have:\n\\begin{equation}\\label{eq:hh:1}\n |f(x) -\\sum_{|\\hat i| \\le r} \\frac{f^{(\\hat i, 0)}(\\hat x', x_n)}{\\hat i!}(\\hat x - \\hat x')^{\\hat i}| \\le \\max_{|\\hat i| = r+1} |f^{(\\hat i, 0)}(\\hat x', x_n)(\\hat x - \\hat x')^{\\hat i}|\n\\end{equation}\nFor each $\\hat i \\in {\\mathbb N}^{n-1}$ with $|\\hat i| \\le r$,  we moreover apply the $n = 1$ case of the theorem to $f^{(\\hat i, 0)}(\\hat x, \\cdot)$ (for each fixed $\\hat x$) and $r' := r - |\\hat i|$, so that for $\\hat x' \\in \\hat B$ and $x_n, x'_n \\in B'$, we have:\n\\begin{equation}\\label{eq:hh:2}\n|f^{(\\hat i, 0)}(\\hat x', x_n) - \\sum_{i_n = 0}^{r'} \\frac{f^{(\\hat i, i_n)}(\\hat x', x'_n)}{i_n!}(x_n - x_n')^{i_n}| \\le | f^{(\\hat i, r' + 1)}(\\hat x', x'_n)(x_n - x'_n)^{r' + 1}|.\n\\end{equation}\nUsing (\\ref{eq:hh:2}) to estimate the $f^{(\\hat i, 0)}(\\hat x', x_n)$ from the left hand side of\n(\\ref{eq:hh:1}) yields (\\ref{eq:t-high-high}), as desired.\n\\end{proof}\n\n\n\n\\subsection{Classical cells}\\label{sec:cd:classical}\n\nWe end this section by recalling the older, more classical notion of cells which has to be used in the absence of the condition that $\\acl$ equals $\\dcl$ in $\\operatorname{Th}(K)$, and by stating the corresponding classical cell decomposition results under the assumption of $1$-h-minimality.\n\n\n\\begin{defn}[Reparameterized cells]\\label{defn:reparamcell}\nConsider integers $n,k\\geq 0$, a $\\emptyset$-definable set $X\\subset  K^n$, and a $\\emptyset$-definable function\n$$\n\\sigma : X\\to \\mathrm{RV}^k.\n$$\nThen $(X,\\sigma)$ is called a $\\emptyset$-definable reparameterized cell (reparameterized by $\\sigma$),\nif, for each $\\xi\\in \\mathrm{RV}^k$, the set $\\sigma^{-1}(\\xi)$, when non-empty, is a $\\xi$-definable cell with some center tuple $c_\\xi$ (see Definition \\ref{defn:cell}), such that moreover $c_\\xi$ depends definably on $\\xi$ and such that the cell-type of $\\sigma^{-1}(\\xi)$ is independent of $\\xi$.  If $(X,\\sigma)$ is such a reparameterized cell\nthen, by a twisted box of $X$ we mean a twisted box of $\\sigma^{-1}(\\xi)$ for some $\\xi$ as in Definition \\ref{defn:cell}, and similarly, by the center tuple and the cell-type of $(X,\\sigma)$, we mean the definable family of the center tuples of the cells $\\sigma^{-1}(\\xi)$ (with family parameter $\\xi$), and the cell-type of the $\\sigma^{-1}(\\xi)$, respectively.\n\\end{defn}\n\n\\begin{remark}\nIn the above definition, one can always modify $\\sigma$ in such a way that afterwards,\neach $\\sigma^{-1}(\\xi)$ is either empty or a single twisted box. In a different direction, if the language ${\\mathcal L}$ has an angular component map ${\\overline{\\rm ac}}$ sending $K$ to its residue field,\nthen one can take $\\sigma$ from Definition \\ref{defn:reparamcell} to be residue field valued (instead of $\\mathrm{RV}$-valued).\nEither of those additional assumptions on $\\sigma$ can in particular be imposed on the cells appearing in the following theorem.\n\\end{remark}\n\n\\begin{thm}[Reparameterized cell decomposition]\\label{thm:rcd}\nSuppose that $\\operatorname{Th}(K)$ is $1$-h-minimal.\nConsider $n,k\\geq 0$ and $\\emptyset$-definable sets  $X\\subset  K^n$ and $P\\subset X\\times \\mathrm{RV}^k$. We consider $P$ as the function sending $x\\in X$ to the fiber $P_x := \\{\\xi \\in \\mathrm{RV}^k \\mid (x,\\xi) \\in P\\}$.\nThen there exists a finite decomposition of $X$ into $\\emptyset$-definable reparameterized cells $(A_i,\\sigma_i)$ such that moreover $P$ (considered as a function) is constant on each twisted box of each $A_i$.\n\nThe other addenda of Theorem \\ref{thm:cd:alg:skol} can be adapted in a similar way.\nIn particular, for the analogue of Addendum \\ref{add:cd:alg:range} of \\ref{thm:cd:alg:skol}, given finitely many $\\emptyset$-definable  $f_j:X\\to K$ and assuming $n=1$, we can moreover assume that there are $\\emptyset$-definable reparameterized cells $(B_{ij},\\tau_ {ij})$  such that $f_j(A_i)=B_{ij}$ and such that any twisted box of $A_i$ is mapped by $f_j$ onto a twisted box of $B_{ij}$.\n\nFor the analogue of Addendum \\ref{add:cd:Lip:comp} of \\ref{thm:cd:alg:skol}, up to allowing a well-chosen coordinate permutation for each for each $i$, we can moreover for each $\\xi$ obtain that the center of $\\sigma_i^{-1}(\\xi)$ is $1$-Lipschitz.\n\\end{thm}\n\\begin{proof}\nThe proof is analogous to the proof of Theorem \\ref{thm:cd:alg:skol} and its addenda.\n\\end{proof}\nNote that the above version of Addendum \\ref{add:cd:Lip:comp} is weaker than the original Addendum \\ref{add:cd:Lip:comp} of Theorem \\ref{thm:cd:alg:skol}, since instead of obtaining finitely many $1$-Lipschitz centers,\nwe now only obtain that for each $\\xi$ separately, the center of\n$\\sigma_i^{-1}(\\xi)$ is $1$-Lipschitz;\nthis corresponds to an infinite partition.\n\nA similar phenomenon also arises with Theorem \\ref{thm:cd:alg:piece:Lipschitz} in the absence of the condition that $\\acl=\\dcl$, as in Theorem ~2.1.7 of \\cite{CCL-PW}: $f$ being locally $1$-Lipschitz implies globally $1$-Lipschitz only after some infinite partition of the domain.\n\n\n\n\\section{Mixed characteristic and non-complete theories}\n\\label{sec:mixed}\n\nSo far in this paper, we introduced Hensel minimality in equi-characteristic $0$. It seems plausible that the entire paper can be adapted to include mixed characteristic, but\nthis would require a considerable amount of work, so we leave this for the future. Instead, in Subsection~\\ref{ssec:mixed}, we present a simple method to deduce\nvariants of most of our results in mixed characteristic directly from the equi-characteristic $0$ version, using coarsenings of the valuation.\nWe focus on a few sample results; transferring other results in a similar way is left to the reader as an exercise.\n\nEven though this kind of transfer could be applied to $\\ell$-h-minimality for any $\\ell \\in {\\mathbb N} \\cup \\{\\omega\\}$,\nonly the case $\\ell = \\omega$ seems natural, (see Remark~\\ref{rem:1heqc}) so we focus on this case, introducing a notion of ``$\\omega$-h$^\\eqc$-minimality''\nwhich makes sense in any residue field characteristic.\n\nThe different notions of Hensel minimality naturally extend to non-complete theories, where preparation works uniformly for all models.\nIn Subsection \\ref{ssec:non-comp}, we illustrate this on some sample results.\n\n\\subsection{Mixed characteristic}\\label{ssec:mixed}\nAs usual, we fix a complete theory ${\\mathcal T}$ of valued fields in a language ${\\mathcal L}$ expanding the language ${\\mathcal L}_\\val$ of valued fields. We do require the characteristic of models to be $0$, but in Subsection~\\ref{ssec:mixed}, we allow the models to have arbitrary residue field characteristic.\n\n\\begin{notn}[Equi-characteristic $0$ coarsening]\nGiven a model $K \\models {\\mathcal T}$,\nwe write ${\\mathcal O}_{K,\\eqc}$ for the smallest subring of $K$ containing ${\\mathcal O}_K$ and ${\\mathbb Q}$ and we let $|\\cdot|_{\\eqc}\\colon K \\to \\Gamma_{K,\\eqc}$ be the corresponding valuation. (Thus, $|\\cdot|_{\\eqc}$ is the finest coarsening of $|\\cdot|$ which has equi-characteristic $0$; note that $|\\cdot|_{\\eqc}$ can be a trivial valuation on $K$.) If $|\\cdot|_{\\eqc}$ is a nontrivial valuation (i.e., ${\\mathcal O}_{K,\\eqc} \\ne K$), then we\nalso use the following notation: $\\operatorname{rv}_{\\eqc}\\colon K \\to \\mathrm{RV}_\\eqc$ is the leading term map with respect to $|\\cdot|_{\\eqc}$;\ngiven $\\lambda \\in \\Gamma_{K,\\eqc}$, $\\operatorname{rv}_{\\lambda}\\colon K \\to \\mathrm{RV}_\\lambda$ is the leading term map with respect to $\\lambda$;\nand ${\\mathcal L}_{\\eqc}$ for the expansion of ${\\mathcal L}$ by a predicate for ${\\mathcal O}_{K,\\eqc}$.\n\\end{notn}\n\n\\begin{defn}[$\\omega$-h$^\\eqc$-minimality]\\label{defn:mixed}\nLet ${\\mathcal T}$ be a complete theory\nof valued fields of characteristic $0$ (and arbitrary residue field characteristic) in a language ${\\mathcal L}$ expanding the language ${\\mathcal L}_\\val$ of valued fields. We say that ${\\mathcal T}$ is \\emph{$\\omega$-h$^\\eqc$-minimal}\nif for every model $K \\models {\\mathcal T}$ the following holds:\nIf the valuation $|\\cdot|_{\\eqc}$ on $K$\nis non-trivial, then\nthe ${\\mathcal L}_{\\eqc}$-theory of $K$, when considered\nas a valued field with the valuation $|\\cdot|_{\\eqc}$, is $\\omega$-h-minimal in the sense of Definition~\\ref{defn:hmin:intro}.\n\\end{defn}\n\nWe will see in Section~\\ref{sec:analyt} that $\\omega$-h$^\\eqc$-minimality is satisfied in many mixed characteristic settings of interest.\n\n\\begin{remark}\nFor theories of fields of equi-characteristic $0$, $\\omega$-h$^\\eqc$-minimality is clearly equivalent to $\\omega$-h-minimality.\n\\end{remark}\n\n\\begin{remark}\\label{rem:1heqc}\nA reason why $\\omega$-h$^\\eqc$-minimality seems to be a natural notion is that $\\omega$-minimality is preserved under coarsening the valuation (by Corollary~\\ref{cor:coarse}).\nIndeed, we could equivalently say that ${\\mathcal T}$ is $\\omega$-h$^\\eqc$-minimal if every model, with every equi-characteristic $0$ coarsening of the valuation, satisfies Definition~\\ref{defn:hmin:intro}.\nWe do not know whether something similar would be true for $1$-h$^\\eqc$-minimality.\n\\end{remark}\n\nResults which simply state that every definable set\/function of a certain kind has some nice (language-independent) properties\nimmediately follow for valued fields of mixed characteristic under the above assumption; in particular, we have the following:\n\n\\begin{prop}[Language-independent properties]\\label{prop:eqc-easy}\nSuppose that ${\\mathcal T}$ is $\\omega$-h$^\\eqc$-minimal as in Definition~\\ref{defn:mixed}. Then the conclusions of the following results hold for any model $K$ of ${\\mathcal T}$:\nLemma~\\ref{lem:finite} ($\\exists^\\infty$-elimination),\nLemma~\\ref{lem:intersection} (definable spherical completeness),\nTheorem~\\ref{thm:C0} (almost everywhere continuity), Theorem~\\ref{thm:Ck} (almost everywhere $C^k$), Proposition~\\ref{prop:dim:basic} (basic properties of dimension).\n\\end{prop}\n\n\\begin{proof}\nWe may work in a model $K \\models {\\mathcal T}$ such that $|\\cdot|_{\\eqc}$ is non-trivial. Then every ${\\mathcal L}$-definable object is in particular ${\\mathcal L}_{\\eqc}$-definable, so all of the above equi-characteristic $0$ results apply to the definable objects in question and yield the desired mixed-characteristic result, except in the case of Proposition~\\ref{prop:dim:basic} (\\ref{dim:defble}).\n(Concerning \\ref{thm:C0} and \\ref{thm:Ck}, note that $|\\cdot|$ and $|\\cdot|_{\\eqc}$\ninduce the same topology and hence equivalent notions of continuity and derivatives.)\n\nProposition~\\ref{prop:dim:basic} (\\ref{dim:defble}) (definability of dimension) can easily be reproved directly in ${\\mathcal L}$, using Lemma~\\ref{lem:finite} and that $0$-dimensional is equivalent to finite.\n\\end{proof}\n\n\nWe now provide the tools necessary to transfer preparation results to mixed characteristic. As sample results, we then\nstate mixed characteristic versions of Corollary~\\ref{cor:prep} about preparation of families (as a warm-up) and of Theorem~\\ref{thm:high-ord} about Taylor approximations.\n\nIn general, whenever something can be $\\lambda$-prepared in equi-characteristic $0$,\nin mixed characteristic, one should expect to obtain $\\lambda\\cdot |m|$-preparation for some integer $m \\ge 1$,\nas the following example illustrates.\n\n\\begin{example}\nThe set $X$ of cubes in the $3$-adic numbers ${\\mathbb Q}_3$ cannot be $1$-prepared by any finite set $C$, since each of the infinitely many disjoint balls $27^r(1+3{\\mathbb Z}_p)$, $r \\in {\\mathbb Z}$, contains both, cubes and non-cubes. However, $X$ is a union of fibers of the map $\\operatorname{rv}_{|3|}\\colon {\\mathbb Q}_3 \\to \\mathrm{RV}_{|3|}$, so it is $|3|$-prepared by the set $\\{0\\}$.\n\\end{example}\n\n\n\\begin{notn}\nDefinition~\\ref{defn:next} about balls $\\lambda$-next to a finite set $C \\subset K$ now has different meanings for $|\\cdot|$ and for $|\\cdot|_\\eqc$. To make clear which of the valuations we mean, we either write $|1|$-next or $|1|_\\eqc$-next (instead of just $1$-next).\n\\end{notn}\n\n\\begin{remark}\nSuppose that $|\\cdot|_{\\eqc}$ is non-trivial on $K$.\nFor any $x, x' \\in K$, we have\n\\begin{equation}\\label{eq:coarse}\n|x|_\\eqc \\le |x'|_\\eqc \\quad \\iff \\quad \\exists m \\in {\\mathbb N}_{\\ge 1}\\colon  |m\\cdot x| \\le |x'|,\n\\end{equation}\nand given a finite set $C \\subset K$, the points $x$ and $x'$ lie in the same ball $|1|_\\eqc$-next to $C$ if and only if for every integer $m \\ge 1$, they lie in the same ball\n$|m|$-next to $C$.\n\\end{remark}\n\nUsing $\\omega$-h-minimality of $K$ as an ${\\mathcal L}_\\eqc$-structure, we will be able to find finite ${\\mathcal L}_\\eqc$-definable sets. To get back to the smaller language ${\\mathcal L}$, we will use the following lemma:\n\n\\begin{lem}[From ${\\mathcal L}_\\eqc$-definable to ${\\mathcal L}$-definable]\\label{lem:mix}\nSuppose that ${\\mathcal T}$ is $\\omega$-h$^\\eqc$-minimal. Let $K$ be a model of ${\\mathcal T}$ and suppose\nthat $K$ is $\\aleph_0$-saturated and strongly $\\aleph_0$-homogeneous as an ${\\mathcal L}^{\\mathrm{eq}}$-structure. (Note that this in particular implies that $|\\cdot|_\\eqc$ is non-trivial.)\nThen, any finite ${\\mathcal L}_\\eqc$-definable set $C \\subset K$ is already ${\\mathcal L}$-definable.\n\\end{lem}\n\n\\begin{remark}\\label{rem:homog}\nNote that such models as in the lemma exist: Any model which is special in the sense of \\cite[Definition~6.1.1]{TentZiegler} is strongly $\\aleph_0$-homogeneous by\n\\cite[Theorem~6.1.6]{TentZiegler}, and it is easy to construct $\\aleph_0$-saturated special models.\n \\end{remark}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:mix}]\nIt suffices to prove that for any $a \\in C$, all realizations of $p:=\\tp_{{\\mathcal L}}(a\/\\emptyset)$ lie in $C$; indeed, by $\\aleph_0$-saturation, this then implies that $p$ is algebraic (using that $C$ is finite), and hence isolated by some formula $\\phi_p(x)$, and hence $C$ is defined by the disjunction of finitely many such $\\phi_p(x)$.\n\nSo now suppose for contradiction that there exist $a \\in C$ and $a' \\in K \\setminus C$ which have the same ${\\mathcal L}$-type over $\\emptyset$. Then by our homogeneity assumption, there exists an ${\\mathcal L}$-automorphism of $K$ sending $a$ to $a'$ (and hence not fixing $C$ setwise). Such an automorphism also preserves ${\\mathcal O}_{\\eqc}$ and hence is an ${\\mathcal L}_\\eqc$-automorphism, but this contradicts $C$ being ${\\mathcal L}_\\eqc$-definable.\n\\end{proof}\n\n\nNow we are ready to deduce preparation results in mixed characteristic:\n\n\n\\begin{cor}[of Corollary~\\ref{cor:prep}; preparing families]\\label{cor:prep:mix}\nAssume that $\\operatorname{Th}(K)$ is $\\omega$-h$^\\eqc$-minimal.\nFor any $k > 0$\nand any ${\\mathcal L}$-definable set\n$$\nW\\subset K\\times \\mathrm{RV}_{\\lambda}^k,\n$$\nthere exists a finite non-empty ${\\mathcal L}$-definable set $C$ and an integer $m\\ge1$ such that for every ball $B$ $\\lambda\\cdot|m|$-next to $C$, the fiber $W_{x} := \\{\\xi \\in \\mathrm{RV}_{\\lambda}^k \\mid (x,\\xi)\\in W\\}$\ndoes not depend on $x$ when $x$ runs over $B$.\n\\end{cor}\n\n\\begin{remark}\nFor simplicity we omitted the $\\mathrm{RV}$-coordinates (since assuming $\\omega$-h-minimality, we are allowed arbitrarily many $\\mathrm{RV}_\\lambda$-coordinates anyway)\nand we require $W$ to be definable without parameters (since as explained in Remark \\ref{rem:acl:mixed} below, we can add parameters for free).\n\\end{remark}\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:prep:mix}]\nBy Remark~\\ref{rem:homog}, we may assume that $K$ is a sufficiently saturated and sufficiently homogeneous model of ${\\mathcal T}$ (as in Lemma~\\ref{lem:mix}).\n\nLet $\\lambda_\\eqc$ be the image of $\\lambda$ in $\\Gamma_{K,\\eqc}$. Since $B_{<\\lambda_\\eqc}(0) \\subset B_{<\\lambda}(0)$, we have a canonical surjection\n$\\mathrm{RV}_{\\lambda_\\eqc} \\to \\mathrm{RV}_{\\lambda}$; let\n$W_\\eqc \\subset K \\times  \\mathrm{RV}_{\\lambda_\\eqc}^k$ be the corresponding preimage of $W$.\nBy Corollary~\\ref{cor:prep}, there exists a finite ${\\mathcal L}_\\eqc$-definable set $C$ such that for every pair $x, x'$ in the same ball $\\lambda_\\eqc$-next to $C$, we have $W_{\\eqc,x} = W_{\\eqc,x'}$.\nBy Lemma~\\ref{lem:mix}, $C$ is already ${\\mathcal L}$-definable; we claim that it is as desired.\n\nSuppose for contradiction that there exists no $m$ as in the corollary, i.e., for every integer $m \\ge 1$, there exists a pair of points $(x, x') \\in K^2$ which lie in the same ball $\\lambda\\cdot|m|$-next to $C$ such that $W_x \\ne W_{x'}$.\nBy $\\aleph_0$-saturation (in the language ${\\mathcal L}$), we find a single pair $(x, x') \\in K^2$ of points with $W_x \\ne W_{x'}$ (which implies $W_{\\eqc,x} \\ne W_{\\eqc,x'}$)\nand which lie in the same ball $\\lambda\\cdot |m|$-next to $C$ for every $m \\ge 1$. The latter implies that $x$ and $x'$\nlie in the same ball $\\lambda_\\eqc$-next to $C$, so we get a contradiction to our choice of $C$.\n\\end{proof}\n\n\n\\begin{cor}[of Theorem~\\ref{thm:high-ord}; Taylor approximation]\\label{cor:high-ord:mix}\nSuppose that  ${\\mathcal T}$ is $\\omega$-h$^\\eqc$-minimal  and let  $K$ be a model of ${\\mathcal T}$.\nLet $f\\colon K \\to K$ be an ${\\mathcal L}$-definable function and let $r \\in {\\mathbb N}$ be given. Then there exists a finite ${\\mathcal L}$-definable set $C$ and an integer $m \\ge 1$ such that for every ball $B$ $|m|$-next to $C$, $f$ is $(r+1)$-fold differentiable on $B$,\n$|f^{(r+1)}|$ is constant on $B$,\nand we have:\n\\begin{equation}\\label{eq:t-higher:mix}\n |f(x) -  T^{\\le r}_{f,x_0}(x) | \\leq \\left|\\frac1m\\cdot f^{(r+1)}(x_0)\\cdot (x-x_0)^{r+1}\\right|\n\\end{equation}\nfor every $x_0, x \\in B$.\n\\end{cor}\n\n\\begin{proof}\nProceeding as in the proof of Corollary~\\ref{cor:prep:mix}, we assume that\n$K$ is sufficiently saturated and sufficiently homogeneous and we use\nTheorem~\\ref{thm:high-ord} and Lemma~\\ref{lem:mix} to find a finite ${\\mathcal L}$-definable set $C$ such that for $x_0, x$ in the same ball $|1|_\\eqc$-next to $C$, we have\n\\begin{equation}\\label{eq:t-higher:coarse}\n|f(x) -  T^{\\le r}_{f,x_0}(x) |_\\eqc \\leq |f^{(r+1)}(x_0)\\cdot (x-x_0)^{r+1}|_\\eqc.\n\\end{equation}\nDifferentiability of $f$ away from $C$ is clear.\nNow suppose that there exists no $m \\ge 1$ satisfying the condition involving (\\ref{eq:t-higher:mix}). As before, we then use $\\aleph_0$-saturation to find\na pair $(x_0, x) \\in K^2$ of points\nwhich lie in the same ball $|1|_\\eqc$-next to $C$ and such that (\\ref{eq:t-higher:mix}) fails for every $m$.\nThe latter means that (\\ref{eq:t-higher:coarse}) fails for $x_0, x$, so we have a contradiction to our choice of $C$.\n\nIt remains to ensure that $|f^{(r+1)}|$ is constant on balls $|m|$-next to $C$. By applying Corollary~\\ref{cor:prep:mix} to the graph of $x \\mapsto \\operatorname{rv}(f^{(r+1)}(x))$, we can\nenlarge $C$ so that this works for balls $|m'|$-next to $C$, for some $m' \\ge 1$. Now the corollary holds using $m \\cdot m'$ (since $|m \\cdot m'| \\le |m|, |m'|$).\n\\end{proof}\n\n\n\\begin{remark}\nIf $K$ has equi-characteristic $0$, the conclusions of Corollaries~\\ref{cor:prep:mix} and \\ref{cor:high-ord:mix} are just the same as our original equi-characteristic $0$\nconclusions, since $|m| = 1$ for ever integer $m \\ge 1$.\n\\end{remark}\n\n\n\\begin{remark}\nWithout the additional enlargement of $C$ at the end of the proof, instead of $|f^{(r+1)}|$ being constant on $B$, we only would have obtained\n$|f^{(r+1)}(x)| \\le |\\frac1m f^{(r+1)}(x')|$ for all $x, x' \\in B$.\n\\end{remark}\n\nWe leave it to the reader to transfer further preparation results to the mixed characteristic setting in a similar way.\nOn the other hand, for some higher dimensional results (in particular Theorem~\\ref{thm:T3\/2,mv} about sup-preparation),\nit is not so clear how to transfer them.\n\nWe finish this subsection by showing that\nwe can add certain parameters for free, similarly to the equi-characteristic $0$ case.\n\n\\begin{notn}\nWe set $\\mathrm{RV}_\\star := \\bigcup_{m \\ge 1} \\mathrm{RV}_{|m|}$, and we write $\\mathrm{RV}_\\star^{\\mathrm{eq}}$ for the union of all quotients of the form $(\\prod_{i=1}^n \\mathrm{RV}_{|m_i|})\/\\mathord{\\sim}$, for $m_i \\ge 1$ and for ${\\mathcal L}$-definable equivalence relations $\\sim$.\n\\end{notn}\n\n\n\\begin{lem}[Adding constants]\\label{lem:addconst:mix}\nLet ${\\mathcal T}$ be $\\omega$-h$^\\eqc$-minimal and let $K$ be a model of ${\\mathcal T}$.\nLet ${\\mathcal L}'$ be an expansion of ${\\mathcal L}$ by constants from $K \\cup \\mathrm{RV}_\\star^{\\mathrm{eq}}$ and let ${\\mathcal T}'$ be the corresponding (complete) ${\\mathcal L}'$-theory of $K$. Then ${\\mathcal T}'$ is also $\\omega$-h$^\\eqc$-minimal.\n\\end{lem}\n\n\\begin{proof}\nSuppose that $K$ is a model of ${\\mathcal T}'$ on which $|\\cdot|_\\eqc$ is non-trivial; we need to verify that the ${\\mathcal L}'_\\eqc$-theory of $K$, with the valuation $|\\cdot|_\\eqc$,\nis $\\omega$-h-minimal. This follows from the equi-characteristic $0$ version of the lemma (namely Lemma~\\ref{lem:addconst}), since ${\\mathcal L}'_\\eqc$ is obtained from ${\\mathcal L}_\\eqc$\nby adding constants from $K \\cup \\mathrm{RV}_\\star^{\\mathrm{eq}}$ and $\\mathrm{RV}_\\star^{\\mathrm{eq}} \\subset \\mathrm{RV}_\\eqc^{\\mathrm{eq}}$. (Note that $\\mathrm{RV}_{|m|}$ is a quotient of $\\mathrm{RV}_\\eqc$.)\n\\end{proof}\n\n\n\\begin{cor}[Removing $\\mathrm{RV}$-parameters]\\label{cor:acl:mix}\nLet ${\\mathcal T}$ be $\\omega$-h$^\\eqc$-minimal and let $K$ be a model of ${\\mathcal T}$.\nFor any $A \\subset K$ and any finite ${\\mathcal L}(A \\cup \\mathrm{RV}_\\star)$-definable set $C \\subset K$, there exists a finite ${\\mathcal L}(A)$-definable set $C' \\subset K$ containing $C$.\nIn other words, $\\acl_{{\\mathcal L},K}(A \\cup \\mathrm{RV}_\\star) = \\acl_{{\\mathcal L},K}(A)$.\n\\end{cor}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:addconst:mix}, it is enough to consider the case $A = \\emptyset$.\nFirst assume that $K$ is a sufficiently saturated and sufficiently homogeneous model of ${\\mathcal T}$ (as in Lemma~\\ref{lem:mix}).\nOur ${\\mathcal L}(\\mathrm{RV}_\\star)$-definable definable set $C$ is then ${\\mathcal L}_\\eqc(\\mathrm{RV}_\\eqc)$-definable (since each $\\mathrm{RV}_{|m|}$ is a quotient of $\\mathrm{RV}_\\eqc$).\nBy Corollary~\\ref{cor:acl}, we find a finite ${\\mathcal L}_\\eqc$-definable set $C'$ containing $C$, and by Lemma~\\ref{lem:mix}, $C'$ is ${\\mathcal L}$-definable.\n\nNow let \\(K\\) be general. Let \\(K'\\) be a sufficiently saturated and sufficiently homogeneous elementary extension of \\(K\\). Let \\(\\mathrm{RV}_{K,\\star}\\), respectively \\(\\mathrm{RV}_{K',\\star}\\), the interpretation of \\(\\mathrm{RV}_\\star\\) in \\(K\\), respectively \\(K'\\). We have \\(\\acl_{{\\mathcal L},K}(A\\cup\\mathrm{RV}_{K,\\star}) = \\acl_{{\\mathcal L},K'}(A\\cup\\mathrm{RV}_{K,\\star}) \\subset \\acl_{{\\mathcal L},K'}(A\\cup\\mathrm{RV}_{K',\\star}) = \\acl_{{\\mathcal L},K'}(A) = \\acl_{{\\mathcal L},K}(A)\\).\n\\end{proof}\n\n\n\\begin{remark}\\label{rem:acl:mixed}\nUsing Lemma~\\ref{lem:addconst:mix} and Corollary~\\ref{cor:acl:mix},\nwe obtain a mixed characteristic analogue of Remark~\\ref{rem:acl}: Firstly, if we can prepare $\\emptyset$-definable objects by $\\emptyset$-definable $C$,\nthen we can also prepare $A$-definable objects by $A$-definable $C$, for $A \\subset K \\cup \\mathrm{RV}_\\star$; and secondly, we can then enlarge $C$ to make it even\n$(A \\cap K)$-definable.\n\\end{remark}\n\n\\subsection{Tameness for non-complete theories}\\label{ssec:non-comp}\n\nThe various notions of Hensel minimality naturally extend to non-complete theories:\n\n\\begin{defn}[non-complete theories]\nLet ${\\mathcal T}$ be a (not necessarily complete) theory in a language ${\\mathcal L}$ expanding ${\\mathcal L}_\\val$, whose models are non-trivially valued fields of characteristic $0$. Say that ${\\mathcal T}$ is X-minimal if each model $K$ of ${\\mathcal T}$\nis X-minimal. Here, X can be ``$\\omega$-h$^\\eqc$'', or, if all models of ${\\mathcal T}$\nare of equi-characteristic $0$, ``$\\ell$-h'' for any $\\ell \\in {\\mathbb N} \\cup \\{\\omega\\}$.\n\\end{defn}\n\nBy the usual play of compactness, preparation results that hold in each model of ${\\mathcal T}$ also hold uniformly for all models of ${\\mathcal T}$.\nWe give two examples to illustrate how this works.\n\nIn this entire subsection, we allow ${\\mathcal T}$ to be non-complete.\n\n\\begin{cor}[of Corollary~\\ref{cor:prep:mix}; preparing families]\\label{cor:prep:nc}\nAssume that ${\\mathcal T}$ is $\\omega$-h$^\\eqc$-minimal, and suppose that $\\phi$ is an ${\\mathcal L}$-formula such that for every model $K \\models {\\mathcal T}$, $W_K := \\phi(K)$\nis a subset of $K\\times \\mathrm{RV}_{\\lambda_K}^k$ for some $k > 0$ and some $\\lambda_K \\le 1$ in $\\Gamma_K$.\nThen there exists an ${\\mathcal L}$-formula $\\psi$ and an integer $m \\ge 1$ such that for every model $K \\models {\\mathcal T}$,\n$C_K := \\psi(K)$ is a finite subset of $K$ which $\\lambda_K\\cdot |m|$-prepares $W_K$ in the following sense:\nFor every ball $B \\subset K$ which is $\\lambda_K\\cdot |m|$-next to $C_K$,\nthe fiber $W_{K,x} := \\{\\xi \\in \\mathrm{RV}_{K,\\lambda_K}^k \\mid (x,\\xi)\\in W_K\\}$\ndoes not depend on $x$ when $x$ runs over $B$.\n\\end{cor}\n\n\\begin{proof}\nFix $\\phi$ as in the Corollary~\\ref{cor:prep:nc}. Whether a pair $(m, \\psi)$ works as desired in a model $K$ can be expressed by\nan ${\\mathcal L}$-sentence. (This uses $\\exists^\\infty$-elimination, as provided by Proposition~\\ref{prop:eqc-easy}.) By compactness,\nwe deduce that there exist finitely many pairs $(m_i, \\psi_i)$ which cover all models. Let $m$ be the least common multiple of the $m_i$\n(so that $|m| \\le |m_i|$ for each $i$) and let $\\psi$ be the disjunction of the $\\psi_i$.\n\\end{proof}\n\n\nIn the following, recall that $\\mathrm{RV}_\\star := \\bigcup_{m \\ge 1} \\mathrm{RV}_{|m|}$ and note that each $\\mathrm{RV}_{|m|}$ is an imaginary sort.\nIf we do not consider $\\mathrm{RV}_{|m|}$ as an actual sort, then by ``adding a constant symbol from $\\mathrm{RV}_{|m|}$ to the language'',\nwe mean: adding a predicate on the valued field, and imposing (in the theory) that the predicate\nholds for exactly for one fiber of the map $\\operatorname{rv}_{|m|}$.\n\n\n\\begin{cor}[of Corollary~\\ref{lem:addconst:mix}; adding constants]\\label{cor:addconst:nc}\nSuppose that ${\\mathcal T}$ is $\\omega$-h$^\\eqc$-minimal. Let ${\\mathcal L}'$ be an expansion of ${\\mathcal L}$ by constant symbols from $K$ and from $\\mathrm{RV}_\\star$, and let ${\\mathcal T}'$ be the same theory as ${\\mathcal T}$, but\nconsidered as an ${\\mathcal L}'$-theory. Then ${\\mathcal T}'$ is also $\\omega$-h$^\\eqc$-minimal.\n\\end{cor}\n\n\\begin{proof}\nA model $K$ of ${\\mathcal T}'$ is a model of ${\\mathcal T}$ with some constants from $K$ and $\\mathrm{RV}_\\star$ added to the language. Thus $K$ is $\\omega$-h$^\\eqc$-minimal by Corollary~\\ref{lem:addconst:mix}.\n\\end{proof}\n\n\n\\begin{remark}\nObviously, Corollaries~\\ref{cor:prep:nc} and \\ref{cor:addconst:nc} also work with $1$-h-minimality instead of $\\omega$-h$^\\eqc$-minimality if all models of $K$ are of equi-characteristic $0$;\nand in that case, the $m$ in Corollary~\\ref{cor:prep:nc} can be omitted and $\\mathrm{RV}_\\star$ becomes $\\mathrm{RV}$.\n\\end{remark}\n\nWe leave it to the reader to formulate other results for non-complete theories. This is usually straight-forward but sometimes a bit technical.\n\n\n\n\n\\section{Examples of Hensel minimal structures}\\label{sec:examples}\n\nIn this section, we provide many examples of Hensel minimal valued fields of equi-characteristic $0$, with various kinds of languages.\nIn some cases, we prove $\\omega$-h-minimality, in others, we only obtain $1$-h-minimality (namely for power-bounded T-convex structures). We also provide examples of valued fields of mixed characteristic which are $\\omega$-h$^\\eqc$-minimal, so that our results and methods from\nSubsection~\\ref{ssec:mixed} apply.\n\n\n\\subsection{Valued fields with or without analytic structure}\n\\label{sec:analyt}\n\nIn this subsection, we prove $\\omega$-h-minimality of arbitrary Henselian valued fields $K$  of equi-characteristic $0$ with analytic structure in the sense of \\cite{CLip}. If $K$ has mixed characteristic and analytic structure, we will show that it is $\\omega$-h$^\\eqc$-minimal, which means that its equi-characteristic $0$ coarsenings are $\\omega$-h-minimal.\nAs so often, obtaining results in the positive equi-characteristic case seems completely out of reach at present.\n\nThe pure valued field language is a special case of an analytic structure on $K$. Nevertheless, we will treat this case separately in this section, avoiding the machinery of analytic structures and instead building only on classical quantifier elimination results. Note that before, the only known proofs of the Jacobian Property (Corollary~\\ref{cor:JP}) either went via analytic structures (as in \\cite{CLoes, CLip}) or were restricted to algebraically closed valued fields (as in \\cite{HK}).\n\nThe proofs of $\\omega$-h-minimality (i) in the pure field language and (ii) in fields with analytic structure are very similar, the main differences being that, in case (ii) we require additional input from \\cite{CLip}. We therefore formulate both proofs simultaneously, tagging differences with (i) and (ii).\n\nWe fix the following language and structure for the remainder of this subsection.\nNote that in the mixed characteristic case, the Definition~\\ref{defn:mixed} of  $\\omega$-h$^\\eqc$-minimality requires us to work with two different valuations simultaneously (though in Case~(i), just understanding the coarser valuation is enough for the proof of $\\omega$-h-minimality).\n\\begin{enumerate}[(i)]\n \\item Let ${\\mathcal L} := {\\mathcal L}_\\val \\cup \\{0,1,-\\}  = \\{+,-,0,1,\\cdot,{\\mathcal O}_K\\}$ be the pure valued field language together with $0,1,-$ with their natural meaning, and let $K$ be a Henselian valued field of equi-characteristic $0$, considered as an ${\\mathcal L}$-structure.\n\\item\n Let ${\\mathcal A}=(A_{m,n})_{m,n}$ be a separated Weierstrass system in the sense of \\cite{CLip}. Let $K$ be a characteristic zero valued field (possibly of positive residue field characteristic) with a separated analytic ${\\mathcal A}$-structure, as in \\cite[Definitions 4.1.5 and 4.1.6]{CLip}. Note that by \\cite[Proposition 4.5.10 (i)]{CLip}, any such $K$ is Henselian.\n\n We denote the valuation ring of $K$ by ${\\mathcal O}_{K,\\fine}$, and we fix an equi-characteristic $0$ coarsening\n ${\\mathcal O}_K \\supset {\\mathcal O}_{K,\\fine}$. (If ${\\mathcal O}_{K,\\fine}$ itself is of equi-characteristic $0$, one can as well\n  choose ${\\mathcal O}_K = {\\mathcal O}_{K,\\fine}$.) We write ${\\mathcal M}_{K,\\fine}$, ${\\mathcal M}_{K}$ for the corresponding maximal ideals and $|\\cdot|_{\\fine}\\colon K \\to \\Gamma_{K,\\fine}$, $|\\cdot|\\colon K \\to \\Gamma_K$\n for the corresponding valuations.\n\nWe let, still in this case (ii), ${\\mathcal L}$ be the expansion of the language from (i) by a\nfunction symbol for field division (extended by zero on zero), by a predicate for ${\\mathcal O}_{K,\\fine}$ and\nby one function symbol for each $f$ in $A_{m,n}$, interpreted as a function ${\\mathcal O}_{K,\\fine}^m\\times{\\mathcal M}_{K,\\fine}^n\\to K$ via the analytic ${\\mathcal A}$-structure on $K$ (and extended by $0$ outside of its domain).\n\\end{enumerate}\n\n\\begin{thm}[Fields with analytic structure]\\label{thm:hmin analytic}\nLet $(K, |\\cdot|)$ be an equi-characteristic $0$ valued field in a language ${\\mathcal L}$ as above in (i) or (ii). (Recall that in particular, in Case~(ii) there is a valuation ring ${\\mathcal O}_{K,\\fine}$ of $K$ which may differ from ${\\mathcal O}_K$.)\nThen the ${\\mathcal L}$-theory of $K$ with the valuation $|\\cdot|$ is $\\omega$-h-minimal.\n\\end{thm}\n\nIn one word, the idea of the proof of Theorem \\ref{thm:hmin analytic} goes via quantifier elimination of valued field quantifiers, in a language to which\nthe sort $\\mathrm{RV}_\\lambda$ and the map $\\operatorname{rv}_\\lambda$ has been added, for some given $\\lambda \\le 1$ in $\\Gamma_K^\\times$. The conditions of $\\omega$-h-minimality are then easily checked. Similar quantifier elimination results have been proved but not yet with such an $\\mathrm{RV}_\\lambda$, so we take care to give sufficient details.\n\nThe first technical ingredient, inspired by \\cite{vdDHM}, is the following:\n\n\\begin{lem}\\label{lem:lin an}\nLet \\(F\\leq K\\) be a subfield which is moreover an ${\\mathcal L}$-substructure of $K$, let $\\lambda \\le 1$ be an element of $\\Gamma_K^\\times$ and let \\(\\tau(x)\\) be an \\({\\mathcal L}(F)\\)-term, with \\(x\\) a single variable. Then there exists a finite set $C \\in K$ consisting only of algebraic elements over $F$ such that\n\\(\\operatorname{rv}_\\lambda(\\tau(x))\\) only depends on $(\\operatorname{rv}_{\\lambda}(x-c))_{c \\in C}$.\n\nMoreover, if we restrict (in our desired property for $C$) to the \\(x\\) that are solutions of a given degree \\(d\\) polynomial equation $P \\in F[X]$, (i.e., if\nwe want to obtain the implication $(\\operatorname{rv}_{\\lambda}(x-c))_{c \\in C} = (\\operatorname{rv}_{\\lambda}(x'-c))_{c \\in C} \\Rightarrow \\operatorname{rv}_\\lambda(\\tau(x)) = \\operatorname{rv}_\\lambda(\\tau(x'))$ only under the assumption $P(x) = P(x') = 0$), then the elements of \\(C\\) can all be assumed to have degree strictly less than \\(d\\) over \\(F\\).\n\\end{lem}\n\n\\begin{proof}[Proof in Case (i)]\nNote that our ${\\mathcal L}(F)$-term $\\tau$ is simply a polynomial in $F[x]$, so the first part of the lemma is immediate from \\cite[Proposition 3.6]{Flen}, and its proof, namely using for $C$ the set of all roots of all derivatives of $\\tau$ (including the roots of $\\tau$ itself). Note that the proof of \\cite[Proposition 3.6]{Flen} yields that the ``Swiss cheeses'' $U_i$ appearing in the statement of the proposition are\n$1$-prepared by $C$.\n\nFor the second part, choose a polynomial $Q$ of degree less than ${\\rm deg} (P)$ such that the polynomial $\\tau$ is congruent to $Q$ modulo $P$. Then for every $x \\in K$ which is a zero of $P$, we have $\\tau(x) = Q(x)$, so we may as well replace $\\tau$ by $Q$. Then choosing $C$ as in the proof of the first part yields the claim.\n\\end{proof}\n\n\\begin{proof}[Proof in Case (ii)]\nWe apply \\cite{CLip} to $K$ with the valuation $|\\cdot|_{\\fine}$.\nThe main idea is to use \\cite[Theorem 5.5.3]{CLip} to reduce to Case (i).\nAs a preparation, note that it is sufficient to obtain the conclusion of the lemma for $x \\in {\\mathcal O}_K$. Indeed,\nthe $x$ outside of ${\\mathcal O}_K$ can be treated by applying the lemma separately to the ${\\mathcal L}(F)$-term $\\tau(1\/x)$.\n\nBy \\cite[Theorem 5.5.3 and Remark 5.5.4]{CLip}, there is a cover of the valuation ring ${\\mathcal O}_{K_{\\mathrm{alg}}}$ of the algebraic closure of $K$ by finitely many $F$-annuli ${\\mathcal U}_i$ (cf.\\ \\cite[Definition~5.1.1]{CLip}) and there is a finite \\(F\\)-definable set \\(S \\subset K\\), such that, on each \\({\\mathcal U}_i \\setminus S\\), we have \\[\\tau(x) = G_i(x)\/H_i(x)\\cdot E_i(x),\\] where \\(G_i, H_i\\in F[x]\\) are polynomials and where \\(E_i\\) is an \\({\\mathcal L}(F)\\)-term which is a strong unit on ${\\mathcal U}_i$ (cf.\\ \\cite[Definition~5.1.4]{CLip}).\nIndeed, that these data are defined over $F$ follows from \\cite[Remark 5.5.4]{CLip} with $K'=F$.\n\nLet\\(P_S \\in F[x]\\) be the polynomial whose set of roots is \\(S\\) and let \\(P_j \\in F[x]\\) be the collection of polynomials appearing in the definition of the $F$-annuli \\({\\mathcal U}_i\\). (In particular, each ${\\mathcal U}_i$ is\ndefined by a boolean combination of inequalities between the valuations of the \\(P_j\\).) Since $|\\cdot|_{\\fine}$ factors over $\\operatorname{rv}_\\lambda$, whether an element $x \\in Z$ lies in ${\\mathcal U}_i$ is determined by $(\\operatorname{rv}_\\lambda(P_{j}(x)))_j$.\nMoreover, by \\cite[Lemma 6.3.12]{CLip} and \\cite[Remark A.1.12]{CLips}, $\\operatorname{rv}_\\lambda(E_i)$ only depends on $(\\operatorname{rv}_\\lambda(P_{j}(x)))_j$.\nThus to prove the lemma for $\\tau$, it suffices to prove it for the polynomials $G_i$, $H_i$, $P_j$ and \\(P_S\\) (considered as functions on $K$). But this has already been done in Case (i).\n\\end{proof}\n\nThe second ingredient is a quantifier elimination result. We\nfix $\\lambda \\le 1$ in $\\Gamma_K^\\times$ and consider the following expansion $\\Lqe$ of ${\\mathcal L}$:\nWe add $\\mathrm{RV}_\\lambda$ as a new sort, together with the map $\\operatorname{rv}_\\lambda$ and with the $\\emptyset$-induced structure on $\\mathrm{RV}_\\lambda$,\n i.e., one predicate for each $\\emptyset$-definable subset of $\\mathrm{RV}_\\lambda^n$, for every $n$.\n\n\\begin{prop}\\label{prop:QE analytic}\nThe $\\Lqe$-theory of $K$ eliminates field quantifiers.\n\\end{prop}\n\nThe particularity of Proposition \\ref{prop:QE analytic} is not only that it has $\\operatorname{rv}_\\lambda$ and $\\mathrm{RV}_\\lambda$ on top of the analytic structure, but also that there are two valuation rings at play, namely ${\\mathcal O}_{K,\\fine}$ (which may be of mixed characteristic) and ${\\mathcal O}_K$ (which is of equi-characteristic zero).\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:QE analytic}]\nFirst note that in the case $\\lambda = 1$, this result is known:\nin Case (i), this is \\cite[Proposition~4.3]{Flen} (or also \\cite[Theorem~B]{Basarab})\nand in Case (ii), it is \\cite[Theorem 3.10]{Rid}. This already implies a partial result for arbitrary $\\lambda \\le 1$, namely:\n\\begin{condition}\\label{cond:field-vars}\nEvery $\\Lqe$-formula $\\phi(x)$ having only $K$-variables is equivalent to a field quantifier free formula.\n\\end{condition}\n Indeed, $\\phi(x)$ is equivalent to an $\\Lqe[1](\\lambda)$-formula $\\psi(x)$ without $K$-quantifiers, and each $\\mathrm{RV}$-quantifier of $\\psi(x)$ can easily be replaced by some $\\mathrm{RV}_\\lambda$-quantifiers.\n\nTo prove the general case, we need the following variants of results from \\cite{Flen}.\n\n\nFix a non-zero polynomial $P \\in K[x]$ and an element $a_0 \\in K$.\nLet $c_i \\in K$ be the coefficients of $P$ when developed around $a_0$, i.e., $P(x) = \\sum_i c_i (x-a_0)^i$.\n\nGiven $b \\in K$, we say, cf. \\cite[Definition 3.1]{Flen}, that $P$ has a \\(\\lambda\\)-collision at $b$ around \\(a_0\\) if \\(|P(b)| < \\lambda \\max_i |c_i(b-a_0)^i|\\). Note that whether \\(P\\) has a \\(\\lambda\\)-collision at \\(b\\) around \\(a_0\\) only depends on $\\operatorname{rv}_\\lambda(c_i)$ (for all $i$) and $\\operatorname{rv}_\\lambda(b - a_0)$. (This will be useful later.)\n\n\\begin{claim}\\label{cl:col root}\nSuppose that the above $P \\in K[x]$ has a \\(\\lambda\\)-collision at \\(b\\) around \\(a_0\\). Then there exists an integer $n\\geq 0$ with $n < {\\rm deg} P$ and an element $b'\\in K$ ``close to $b$'' such that \\(P^{(n)}(b') = 0\\). Here,  ``close to $b$'' means $\\operatorname{rv}_\\lambda(b'-a_0) = \\operatorname{rv}_\\lambda(b-a_0)$ if $n = 0$ and\n$\\operatorname{rv}(b'-a_0) = \\operatorname{rv}(b-a_0)$ if $n \\ge 1$, and $P^{(n)}$ stands for the $n$-th derivative, with $P^{(0)}=P$.\n\\end{claim}\n\n\\begin{proof}\nWithout loss, we may assume that \\(a_0 = 0\\), \\(b=1\\) and \\(\\max_i |c_i| = 1\\). Let \\(Q = \\operatorname{res}(P)\\), the reduction of $P$ modulo ${\\mathcal M}_{K}$. We have \\(Q(1) = 0\\). Let \\(n\\) be such that $1$ is a root of multiplicity $1$ of \\(Q^{(n)}\\). Then Hensel's Lemma yields a root \\(b'\\) of \\(P^{(n)}\\) such that \\(|b'-1| \\leq |P^{(n)}(1)|\\). Since $|P^{(n)}(1)| \\le 1$, this implies $\\operatorname{rv}(b') = \\operatorname{rv}(b)$. In the case $n = 0$, that $1$ is not a root of $Q'$ implies $|P'(1)| = 1$.\nTogether with $|P(1)| \\le \\lambda$, Hensel's Lemma implies $|b' - b| < \\lambda$ and hence $\\operatorname{rv}_\\lambda(b') = \\operatorname{rv}_\\lambda(b)$.\n\\qedhere(\\ref{cl:col root})\n\\end{proof}\n\n\\begin{claim}\\label{cl:exists root}\nSuppose that the above $P \\in K[x]$ has no common zero with any of its proper derivatives and fix $\\xi\\in\\mathrm{RV}^\\times_\\lambda$. The following are equivalent:\n\\begin{enumerate}\n\\item There exists a root $b \\in K$ of \\(P\\) with \\(\\operatorname{rv}_\\lambda(b-a_0) = \\xi\\);\n\\item there exists \\(b\\in K\\) such that\\\\\n(2a) \\(\\operatorname{rv}_\\lambda(b-a_0) = \\xi\\), \\(P\\) has a \\(\\lambda\\)-collision at \\(b\\) around \\(a_0\\), and\\\\\n(2b) for every root $a \\in K$ of every proper derivative of $P$,\n\\(P\\) has a \\(\\lambda\\)-collision at \\(b\\) around $a$.\n\\end{enumerate}\n\\end{claim}\n\n\\begin{proof}\nIf \\(b\\) is a root of \\(P\\), then \\(P\\) has a \\(\\lambda\\)-collision at \\(b\\) around any \\(a\\neq b\\). So (1) implies (2). Let us now assume that we have \\(b\\) such that (2) holds. If (1) does not hold, then by Claim~\\ref{cl:col root} (and (2a)) there exists a root \\(a\\) of some proper derivative of $P$ with \\(\\operatorname{rv}(a-a_0) = \\operatorname{rv}(b-a_0)\\). Pick the closest such \\(a\\) to \\(b\\). By Claim~\\ref{cl:col root}, around \\(a\\) this time (and using (2b)), there exists a root \\(c\\) of some \\(P^{(m)}\\) with \\(\\operatorname{rv}(c-a) = \\operatorname{rv}(b-a)\\); in particular, $|b-c| < |b-a|$, a contradiction to our choice of $a$.\n\\qedhere(\\ref{cl:exists root})\n\\end{proof}\n\nWe now come back to the actual proof of field quantifier elimination. We abbreviate ``field quantifier free'' by ``fqf''.\nIt suffices to prove the following: Suppose\nthat $K' \\equiv K$ is $|K|^+$-saturated, that $A \\subset K \\cup \\mathrm{RV}_{K,\\lambda}$ and $A' \\subset K' \\cup \\mathrm{RV}_{K',\\lambda}$ are substructures and that\n$\\alpha\\colon A \\to A'$ is an fqf-elementary bijection, i.e., that it preserves the validity of fqf formulas.\nThen for any $a \\in K$, there exists an $a'$ such that $\\alpha$ extends to an fqf-elementary map sending $a$ to $a'$.\n\nFor $\\alpha$ to be fqf-elementary, it suffices that it is an isomorphism of substructures and that $\\alpha|_{A \\cap \\mathrm{RV}_{K,\\lambda}}$ is fqf-elementary. Indeed, suppose that $\\phi$ is an\nfqf $\\Lqe(A)$-sentence that holds in $K$. Then without loss,\n$\\phi = \\psi((\\operatorname{rv}_\\lambda(\\tau_i))_i)$ for some $\\Lqe(A\\cap K)$-terms $\\tau_i$ and for $\\psi(y)$ an fqf $\\Lqe(A \\cap \\mathrm{RV}_{K,\\lambda})$-formula.\nLet $\\xi_i \\in A \\cap \\mathrm{RV}_{K,\\lambda}$ be the interpretation of $\\operatorname{rv}_\\lambda(\\tau_i)$. Then $\\phi$ follows from $\\bigwedge_i \\tau_i = \\xi_i$ (which is quantifier free) and\n$\\psi((\\xi_i)_i)$ (which is an fqf $\\Lqe(A \\cap \\mathrm{RV}_{K,\\lambda})$-sentence).\n\nWe may assume $\\mathrm{RV}_{K,\\lambda} \\subset A$, since $\\alpha_{A \\cap \\mathrm{RV}_{K,\\lambda}}$ extends to an fqf-elementary map on $\\mathrm{RV}_{K,\\lambda}$ and the union of this extension with the original $\\alpha$ is an isomorphism of substructures. In particular, when further extending $\\alpha$, we now only need to make sure that it remains an isomorphism of substructures.\nIn terms of formulas, this means (by a usual compactness argument) that given a quantifier free $\\Lqe(A)$-formula $\\phi(x)$ with $x$ a valued field variable, we need to check that $K \\models \\exists x\\,\\phi(x)$ implies $K' \\models \\exists x\\,\\phi^\\alpha(x)$ (where ``$\\phi^\\alpha$'' is the $\\Lqe(A')$-formula obtained from $\\phi$ by applying $\\alpha$ to the parameters from $A$).\n\nWe may assume that $F := K \\cap A$ is a subfield. In Case (ii), this is automatic, since $\\Lqe$ contains field division. In Case (i), the ring homomorphism $\\alpha_{K \\cap A}$ uniquely extends to the fraction field $F$ of $K \\cap A$, and extending $\\alpha$ in this way yields an isomorphism of substructures, since\nfor $\\frac{b}{b'} \\in F$ ($b, b' \\in K \\cap A$), we have\n$\\operatorname{rv}_\\lambda(\\frac{b}{b'}) = \\frac{\\operatorname{rv}_\\lambda(b)}{\\operatorname{rv}_\\lambda(b')}$.\n\nFrom now on, we identify $A$ with its image $\\alpha(A)$.\nFix $a \\in K$ and fix a quantifier free $\\Lqe(A)$-formula $\\phi(x)$ such that $K \\models \\phi(a)$ holds.\nWe need to show that $K' \\models \\exists x\\colon \\phi(x)$. To do so, we will successively reduce to simpler formulas, until we can get rid of the $K$-quantifier $\\exists x$.\n\n\nLet $P \\in F[x]$ be the minimal polynomial of $a$ over $F$ and set $d := {\\rm deg} P$. If \\(a\\) is transcendental over $F$, we set \\(P := 0\\) and \\(d := \\infty\\).\nBy induction on $d$, we may assume:\n\\begin{condition}\\label{cond:F}\n$F$ contains all roots $b$ in \\(K\\) of polynomials over $F$ of degree strictly less than \\(d\\).\n\\end{condition}\n\nAs before, we can assume that the above formula $\\phi$ is of the form $\\phi(x) = \\psi((\\operatorname{rv}_\\lambda(\\tau_i(x)))_i)$ for some ${\\mathcal L}(F)$-terms $\\tau_i$.\nBy Lemma~\\ref{lem:lin an} (and (\\ref{cond:F}))\n$\\phi(x)$ is equivalent, in the structure $K$, to\na formula of the form \\[\\phi'(x) = \\psi'((\\operatorname{rv}_\\lambda(x - c_j)_j) \\wedge P(x) = 0\\] for some $c_j \\in F$, where\n$\\psi'(y) = \\psi((\\eta_i(y))_i)$ for suitable $\\Lqe(F)$-definable functions $\\eta_i$.\nWe claim that this equivalence also holds in $K'$, so that we can without loss replace $\\phi$ by $\\phi'$.\n\nTo prove the claim, note that the\nequivalence $\\phi \\leftrightarrow \\phi'$ follows from an $\\Lqe(F)$-sentence $\\chi$, namely\n$\\chi = \\bigwedge_i \\forall x \\in K\\colon \\tau_i(x) = \\eta_i((\\operatorname{rv}_\\lambda(x - c_j))_j)$.\nSince $\\chi$ only uses $K$-parameters, we already know (by (\\ref{cond:field-vars})) that it is equivalent to a fqf $\\Lqe(F)$-sentence $\\chi'$ (modulo only the $\\Lqe$-theory of $K$, i.e., without using the specific embedding of $F$ into $K$). Thus the truth of $\\chi'$ is preserved by $\\alpha$, so that we obtain the desired equivalence in $K'$.\\footnote{In Case (i), we do not need to invoke another field quantifier elimination result. As in Lemma~\\ref{lem:lin an}, we assume that each \\(\\tau_i\\) has degree smaller than \\(P\\) and we choose, as $c_j$, all the roots of derivatives of all \\(\\tau_i\\), including \\(\\tau_i\\) itself. Then, by \\cite[Proposition 3.6]{Flen}, for every \\(x\\) and $i$, there exists a $j$ such that if we write \\(\\tau_i(x) = \\sum_k a_{k} (x-c_{j})^k\\), the sum\n\\(\\sum_k \\operatorname{rv}_\\lambda(a_{k})\\operatorname{rv}_\\lambda(x-c_{j})^k\\) is well-defined and hence equal to \\(\\operatorname{rv}_\\lambda(\\tau_i(x))\\). Using that the well-definedness of the sum is an fqf condition, we can define $\\eta_i \\colon (\\operatorname{rv}_\\lambda(x-c_{j}))_j \\mapsto \\operatorname{rv}_\\lambda(\\tau_i(x))$ without field quantifiers, so that the equality $\\eta_i(\\operatorname{rv}_\\lambda(x-c_{j}))_j = \\operatorname{rv}_\\lambda(\\tau_i(x))$ is preserved by $f$.}\n\nNext, note that we can get rid of all the $c_j$ appearing in $\\phi'$ except for the one closest to $a$, i.e., denoting that closest $c_j$ by $c$, we can replace $\\phi'$ by\n\\[\\phi''(x) = \\psi''(\\operatorname{rv}_\\lambda(x - c)) \\wedge P(x) = 0.\\] Indeed, one can easily choose $\\psi''$ in such a way that $K \\models \\phi''(a)$ and that\n$\\operatorname{Th}_{\\Lqe}(K)$ implies $\\phi'' \\to \\phi'$.\n\nNow $\\exists x \\colon \\phi''(x)$ is equivalent to\n\\(\\exists\\xi \\in \\mathrm{RV}_\\lambda\\colon \\psi''(\\xi)\\wedge(\\exists x\\colon  \\operatorname{rv}_\\lambda(x-c) = \\xi \\wedge P(x) = 0)\\),\nand it remains to get rid of the $\\exists x$ in that formula.\nIf $P$ is the zero polynomial, then the $\\exists x$ part is trivially true and we are done.\nOtherwise, by Claim \\ref{cl:exists root}, the existence of such an $x$ is equivalent to the existence of an $x$ with $\\operatorname{rv}_\\lambda(x-c) = \\xi$ such that $P$ has a $\\lambda$-collision at $x$ around certain points $b_j$ from $F$ (namely around $c$ and around the roots of the derivatives of $P$, which are in $F$ by (\\ref{cond:F})). For fixed $P$ and $b_i$, the existence of such a collision is determined by $\\operatorname{rv}_\\lambda(x - b_j)$, so\nit remains to eliminate the $\\exists x$ from an $\\Lqe(B)$-formula of the form\n$\\exists x\\colon \\psi'''((\\operatorname{rv}_\\lambda(x - b_j))_j)$ (with $\\psi'''$ fqf, expressing that the collisions exist). This can then be further simplified to a formula of the form $\\exists x\\colon \\bigwedge_j \\operatorname{rv}_\\lambda(x - b_j) = \\xi_j$ (where we take $\\xi_j := \\operatorname{rv}_\\lambda(a - b_j))$).\nThis formula now expresses that the intersection of certain balls is non-empty, a condition which can easily be seen to only depend on $\\operatorname{rv}_\\lambda(b_j - b_{j'})$ (see \\cite[Proposition~4.1]{Flen} for details). Thus we are done.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:hmin analytic}]\nWe need to show that for every $K' \\equiv K$ and every $A \\subset K'$, every $(A \\cup \\mathrm{RV}_{K',\\lambda})$-definable set $X = \\phi(K') \\subset K'$ can be $\\lambda$-prepared by a finite $A$-definable set $C$.\n\nBy Proposition \\ref{prop:QE analytic}, we may assume that $\\phi$ contains no field quantifiers, so that it\nsuffices to $\\lambda$-prepare the graph of functions of the form \\(\\operatorname{rv}_\\lambda(\\tau(x))\\), where \\(\\tau\\) is an \\(\\Lqe(A)\\)-term. (Here, ``preparing a graph'' is in the sense of Corollary \\ref{cor:prep}.)\n\nBy the first half of Lemma \\ref{lem:lin an}, such a graph can be prepared by a finite set $C$ of elements that are algebraic over\nthe field $F \\le K'$ generated by $A$. Since such a set $C$ is contained in a finite $A$-definable set $C'$, we are done.\n\\end{proof}\n\n\n\\begin{cor}[Equi-characteristic $0$ examples]\\label{cor:ex:equi}\nLet $K$ be a Henselian valued field of equi-characteristic $0$ in a language ${\\mathcal L}$. Then in each of the following cases, $\\operatorname{Th}(K)$ is $\\omega$-h-minimal.\n\\begin{enumerate}\n \\item ${\\mathcal L}$ is the pure valued field language ${\\mathcal L}_\\val$.\n \\item\\label{ex:analytic:00} ${\\mathcal L}$ is the valued field language expanded by function symbols from a separated Weierstrass system ${\\mathcal A}$ and $K$ is equipped with analytic ${\\mathcal A}$-structure in the sense of \\cite{CLip}.\n\\end{enumerate}\n\\end{cor}\n\n\\begin{proof}\nThese are just examples of Theorem~\\ref{thm:hmin analytic}, namely with ${\\mathcal O}_K = {\\mathcal O}_{K,\\fine}$.\n\\end{proof}\n\n\n\\begin{cor}[Mixed characteristic examples]\\label{cor:ex:mixed}\nLet $K$ be a Henselian valued field of mixed characteristic in a language ${\\mathcal L}$. Then in each of the following cases, $\\operatorname{Th}(K)$ is $\\omega$-h$^\\eqc$-minimal (as in Definition~\\ref{defn:mixed}).\n\\begin{enumerate}\n \\item ${\\mathcal L}$ is the pure valued field language ${\\mathcal L}_\\val = \\{+,\\cdot,{\\mathcal O}_K\\}$.\n \\item $K$ is a finite field extension of  ${\\mathbb Q}_p$ and ${\\mathcal L}$ is the sub-analytic language from \\cite{vdDHM} (which is a variant on the language from \\cite{DvdD}).\n\\item ${\\mathcal L}$ is the valued field language expanded by function symbols from a separated Weierstrass system ${\\mathcal A}$ and $K$ is equipped with analytic ${\\mathcal A}$-structure in the sense of \\cite{CLip}.\n\\end{enumerate}\n\\end{cor}\n\n\n\\begin{proof}\nGiven $K' \\equiv_{{\\mathcal L}} K$, let $|\\cdot|_{\\eqc}$ be the finest equi-characteristic $0$ coarsening\nof the valuation $|\\cdot|$, and let ${\\mathcal L}_{\\eqc}$ be the expansion of ${\\mathcal L}$ by a predicate for the valuation ring ${\\mathcal O}_{K,\\eqc}$ corresponding to $|\\cdot|_{\\eqc}$.\nSuppose that $|\\cdot|_{\\eqc}$ is non-trivial.\nUnder those assumptions, we need to show that $\\operatorname{Th}_{{\\mathcal L}_{\\eqc}}(K'_{\\eqc})$ is $\\omega$-h-minimal, where $K'_{\\eqc}$ is the field $K'$ considered as a valued field\nwith the valuation $|\\cdot|_{\\eqc}$.\n\n(1)\nIf ${\\mathcal L}$ is the pure valued field language, we consider ${\\mathcal L}_{\\eqc}$ as an expansion of ${\\mathcal L}_{{\\mathrm{val}},\\eqc} := \\{+,\\cdot,{\\mathcal O}_{K',\\eqc}\\}$ (which is also the pure valued field language) by a predicate\nfor ${\\mathcal O}_{K'}$. By Theorem~\\ref{thm:hmin analytic}, $\\operatorname{Th}_{{\\mathcal L}_{{\\mathrm{val}},\\eqc}}(K')$ is $\\omega$-h-minimal.\nSince the map $K' \\to  \\Gamma_{K'}$ factors over $\\mathrm{RV}_\\eqc$ (where $\\mathrm{RV}_\\eqc$ denotes the leading term structure with respect to $|\\cdot|_\\eqc$),\n${\\mathcal L}_\\eqc$ is an $\\mathrm{RV}_\\eqc$-expansion of ${\\mathcal L}_{{\\mathrm{val}},\\eqc}$, so Theorem~\\ref{thm:resp:h} implies that $\\operatorname{Th}_{{\\mathcal L}_{\\eqc}}(K')$ is also $\\omega$-h-minimal.\n\n(2)\nThis language ${\\mathcal L}$ defines an analytic structure on $K$ (by \\cite[Section~4.4]{CLip} (2)) and hence also on $K'$. Hence, it suffices to prove (3).\n\n(3)\nThe language ${\\mathcal L}_{\\eqc}$ has the shape of the language called ${\\mathcal L}$ in the above Case~(ii) of Section \\ref{sec:analyt}, so\nby Theorem~\\ref{thm:hmin analytic}, $\\operatorname{Th}_{{\\mathcal L}_{\\eqc}}(K')$ is $\\omega$-h-minimal.\n\\end{proof}\n\n\n\\subsection{${\\mathcal T}_\\omin$-convex valued fields}\n\\label{sec:Tcon}\n\nFix a language ${\\mathcal L}_\\omin$ expanding the language of ordered rings\nand fix a complete o-minimal ${\\mathcal L}_\\omin$-theory ${\\mathcal T}_\\omin$ containing the theory RCF of real closed fields.\nGiven a pair of models $K_0 \\prec K$ of ${\\mathcal T}_\\omin$, we can turn $K$\ninto a valued field by using the convex closure of $K_0$ in $K$ as the valuation ring ${\\mathcal O}_K$.\nWe suppose that ${\\mathcal O}_K \\ne K$ and we let ${\\mathcal L}$ be the expansion of ${\\mathcal L}_\\omin$ by a predicate for ${\\mathcal O}_K$.\nIn \\cite{DL.Tcon1,Dri.Tcon2} van den Dries--Lewenberg obtained various results about the model theory of such valued fields $K$ as ${\\mathcal L}$-structures.\nIn particular, the theory ${\\mathcal T} := \\operatorname{Th}_{{\\mathcal L}}(K)$ only depends on ${\\mathcal T}_\\omin$ (and not on the choice of $K$ and $K_0$, provided that ${\\mathcal O}_K \\ne K$) \\cite[Corollary~3.13]{DL.Tcon1}. (Van den Dries--Lewenberg call such a ring ${\\mathcal O}_K$ a ``${\\mathcal T}_\\omin$-convex subring of $K$''. Accordingly, and following other subsequent literature, we call $K$ a ``${\\mathcal T}_\\omin$-convex valued field''.)\n\nWe will prove that this theory ${\\mathcal T}$ is\n$1$-h-minimal, under the assumption that no fast-growing functions are definable in ${\\mathcal T}_\\omin$. In ${\\mathbb R}$, ``no fast-growing functions'' means that every definable function is eventually bounded by a function of the form $x \\mapsto x^n$.\nIn arbitrary real closed fields, the right generalization is\npower-boundedness; see \\cite{Mil.powBd}:\n\n\\begin{defn}[Power-bounded]\nA \\emph{power function} in $K$ is an ${\\mathcal L}_\\omin$-definable function $g\\colon K^\\times \\to K^\\times$ which is an endomorphism of the multiplicative group $K^\\times$. We call the ${\\mathcal L}_\\omin$-structure $K$ (and its theory ${\\mathcal T}_\\omin$) \\emph{power-pounded}, if for every ${\\mathcal L}(K)$-definable function $f\\colon K \\to K$, there exists a\npower function $g$ such that $|f(x)| \\le g(x)$ for all sufficiently large $x \\in K$.\n\\end{defn}\n\nFrom now on, we will assume that ${\\mathcal T}_\\omin$ is power-bounded.\n\nThe proof that $\\operatorname{Th}(K)$ is $1$-h-minimal is essentially contained in the existing literature: Using the criteria given in Theorem~\\ref{thm:tame2vf}, this can be deduced from \\cite[Theorems~2.1 and 2.9]{Gar.powbd}. However, Theorem~2.9 is a lot deeper than what we really need, so we give a more direct proof below (mainly following the ideas from \\cite{Gar.powbd}).\n\n\n\\begin{lem}\\label{lem:tcon-0-h}\nThe theory of $K$ (as an ${\\mathcal L}$-structure) is $0$-h-minimal.\n\\end{lem}\n\n\\begin{proof}\nWe assume that $K$ is sufficiently saturated and use the criterion given by Lemma~\\ref{lem:type-0-h-min}: Given a parameter set $A \\subset K$ and a ball $B \\subset K \\setminus \\acl_K(A)$, we need to verify that all elements of $B$ have the same type over $A\\cup \\mathrm{RV}$. We may assume $A = \\acl_K(A)$.\n\nSince $B \\cap A = \\emptyset$ and since ${\\mathcal L}_\\omin$-types over $A$ correspond to cuts in $A$,\nall elements of $B$ have the same ${\\mathcal L}_\\omin$-type over $A$.\nThus \\cite[Lemma~3.15]{Yin.tcon} applies and tells us that for any $x, x' \\in B$, there exists an automorphism of $K$ fixing $A$ and $\\mathrm{RV}$ but sending $x$ to $x'$. This shows that $x$ and $x'$ have the same type over $A \\cup \\mathrm{RV}$.\n\\end{proof}\n\n\n\nThe following lemma states that ${\\mathcal L}$-definable functions are piecewise ${\\mathcal L}_\\omin$-definable. This is already stated in \\cite[Lemma~2.6]{Dri.Tcon2}, but we shall use a variant from \\cite{Yin.tcon}:\n\n\\begin{lem}[{\\cite[Lemma~3.3]{Yin.tcon}}]\\label{lem:piecewise-omin}\nLet $f\\colon K \\to K$ be an ${\\mathcal L}(A)$-definable function, for some\n$A \\subset K \\cup \\mathrm{RV}$. Then there exists a partition of $K$ into finitely many ${\\mathcal L}(A)$-definable sets $X_i$ and ${\\mathcal L}_\\omin(A \\cap K)$-definable functions $g_i\\colon K \\to K$ such that $f|_{X_i} = g_i|_{X_i}$ for each $i$.\n\\end{lem}\n\nIt might be a bit unclear from the formulation of that lemma in \\cite{Yin.tcon} whether it is intended that parameters from $\\mathrm{RV}$ are allowed. In any case, the proof given in \\cite{Yin.tcon} goes through with parameters from $\\mathrm{RV}$.\n\n\n\n\\begin{thm}[${\\mathcal T}_\\omin$-convex examples]\\label{thm:Tcon}\nLet ${\\mathcal T}_\\omin$ be a power-bounded o-minimal theory containing the theory of real closed fields, in a language ${\\mathcal L}_\\omin$ expanding the language of rings.\nLet ${\\mathcal T}$ be the theory of ${\\mathcal T}_\\omin$-convex valued fields, in the language ${\\mathcal L}$ which is obtained by expanding ${\\mathcal L}_\\omin$ by a predicate for the valuation ring (as explaind at the beginning of this subsection).\nThen ${\\mathcal T}$ is $1$-h-minimal.\n\\end{thm}\n\n\n\\begin{proof}\nWe use the criteria from Theorem~\\ref{thm:tame2vf} to prove $1$-h-minimality, so let an $A$-definable map $f\\colon K \\to K$ be given, for some $A \\subset K \\cup \\mathrm{RV}$.\nLet $X_i \\subset K$ and $g_i\\colon K \\to K$ be as obtained from Lemma~\\ref{lem:piecewise-omin}.\n\nCondition~(T2) of Theorem~\\ref{thm:tame2vf} holds for each $g_i$ by o-minimality (namely, the set $\\{d \\in K \\mid g_i^{-1}(d)$ is infinite$\\}$ is finite), so it also holds for $f$.\n\nCondition~(T1), too, follows for $f$ if we can prove it for each $g_i$. Indeed, take the union of the sets $C$ obtained for all the $g_i$, and further enlarge $C$ so that it $1$-prepares each $X_i$. (This is possible, since by Lemma~\\ref{lem:tcon-0-h}, we already have $0$-h-minimality.) So it remains to prove Condition~(T1) for an ${\\mathcal L}_\\omin(A \\cap K)$-definable function $g_i$.\n\nBy o-minimality, we find a finite $(A \\cap K)$-definable $C \\subset K$ such that $g_i$ is continuously differentiable on $K \\setminus C$. Further enlarge $C$ (using $0$-h-minimality and Corollary~\\ref{cor:prep}) so that it prepares the map $K \\to \\Gamma_K, x \\mapsto |g_i'(x)|$.\n\nLet $B \\subset K$ be a ball $1$-next to $C$. We claim that Condition~(T1) is satisfied with $\\mu_B := |g_i'(x)|$ for any $x \\in B$. Indeed, let $x_1, x_2 \\in B$ be given, with $x_1 \\ne x_2$. By the Mean Value Theorem for o-minimal fields,\nthere exists an $x_3$ in-between (and hence also in $B$) such that $g_i(x_1) - g_i(x_2) = g_i'(x_3)\\cdot (x_1 - x_2)$. Taking valuations on both sides implies $|g_i(x_1) - g_i(x_2)| = \\mu_B\\cdot |x_1 - x_2|$, as desired.\n\\end{proof}\n\n\n\\begin{remark}\nThe assumption that ${\\mathcal T}_\\omin$ is power-bounded is necessary to obtain $1$-h-minimality of ${\\mathcal T}$. Indeed, in the presence of an exponential map, we can define $K \\to \\mathrm{RV}, x \\mapsto \\operatorname{rv}(e^x)$, whose fibers are exactly the translates of the maximal ideal $B_{<1}(0)$ and which hence cannot be prepared in the sense of Corollary~\\ref{cor:prep}.\n\\end{remark}\n\n\\begin{remark}\nWe were not able to prove that power-bounded ${\\mathcal T}_\\omin$-convex valued fields are $\\omega$-h-minimal. This is one of the main reasons why we only assume $1$-h-minimality in most of the paper.\n\\end{remark}\n\nUsing methods from non-standard analysis, results in a ${\\mathcal T}_\\omin$-convex valued field $K$ can often be translated into results about $K$ as an ${\\mathcal L}_\\omin$-structure.\nWe finish this subsection by stating what our Taylor approximation result (Theorem~\\ref{thm:high-ord}) becomes under such a translation, namely a version of Taylor approximation which has some uniformity even when one approaches a bad point.\n\n\n\\begin{cor}\\label{cor:arch}\nLet ${\\mathcal T}_\\omin$ be a power-bounded o-minimal theory containing the theory of real closed fields, in a language ${\\mathcal L}_\\omin$ expanding the language of rings.\nLet $K \\models {\\mathcal T}_\\omin$ be a model, let $f\\colon K \\to K$ be an ${\\mathcal L}_\\omin$-definable function and let $r \\in\\ {\\mathbb N}$ be given.\nThen there exists a finite ${\\mathcal L}_\\omin$-definable set $C \\subset K$ and a constant $c \\in K_{>0}$ such that for every pair $x_0, x \\in K$ satisfying\n\\begin{equation}\\label{eq:arch.ass}\nc\\cdot |x - x_0| < \\min_{a \\in C}|x_0 -  a|,\n\\end{equation}\nwe have\n\\begin{equation}\\label{eq:arch.imp}\n|f(x) -  T^{\\le r}_{f,x_0}(x) | \\leq c\\cdot |f^{(r+1)}(x_0)\\cdot (x-x_0)^{r+1}|\n\\end{equation}\n(where $|\\cdot|$ denotes the usual absolute value and $T^{\\le r}_{f,x_0}$ is the Taylor polynomial of $f$ around $x_0$ of degree $r$; see Definition~\\ref{defn:taylor}).\n\nMore generally, if $(f_q)_{q \\in K^m}$ is an ${\\mathcal L}_\\omin$-definable family of functions $K \\to K$, then\nwe obtain an ${\\mathcal L}_\\omin$-definable family of sets $(C_q)_{q \\in K^m}$ and a constant $c \\in K_{>0}$ which is independent of $q$ such\nthat the above holds for every $q \\in K^m$.\n\\end{cor}\n\n\\begin{remark}\nNote that this result would be false without the assumption on power-boundedness. Indeed, one can check that it fails near $0$ for the function $x \\mapsto e^{1\/x}$.\nOn the other hand, it should be rather easy to obtain for sub-analytic functions, so this is another instance (along with the Jacobian Property)\nof a generalization of a result from an analytic setting to an axiomatic one.\n\\end{remark}\n\n\\private{\nSet $r = 0$ and $x = x_0 + d$, with $dc < x_0$. We consider $|e^{1\/x} - e^{1\/x_0}| < |c \\cdot (e^{1\/x_0})'\\cdot d| = |c \\cdot e^{1\/x_0} \\cdot x_0^{-2}\\cdot d|$.\nAfter dividing by $e^{1\/x_0}$, we obtain $|e^{1\/x-1\/x_0}| = |e^{-d\/(x\\cdot x_0)} - 1| < cdx_0^{-2}$. In the LHS, we approximate $x$ by $x_0$, and then we approximate\nthe exponential by the degree 3 Taylor approx. Then one sees that the corollary fails.\n}\n\n\\begin{remark}\nIt seems that this result should be rather easy to obtain for sub-analytic functions.\n\\end{remark}\n\n\nIn the proof, we use the following lemma: \n\n\\begin{lem}\\label{lem:L2Lomin}\nFor any $A \\subset K$, every finite ${\\mathcal L}(A)$-definable set $C \\subset K$ is already ${\\mathcal L}_\\omin(A)$-definable.\n\\end{lem}\n\n\\begin{proof}\nUsing the order, we reduce to the case where $C = \\{a\\}$ is a singleton.\n${\\mathcal L}(A)$-definability means that $a = f(0)$, for some ${\\mathcal L}(A)$-definable function $f\\colon K \\to K$.\nBy Lemma~\\ref{lem:piecewise-omin}, $f(0) = g(0)$ for some ${\\mathcal L}_\\omin(A)$-definable $g\\colon K \\to K$; this implies that $C$ is ${\\mathcal L}_\\omin(A)$-definable.\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:arch}]\nFix a $|K|^+$-saturated elementary extension $K' \\succ K$ and let ${\\mathcal O}_{K'}$ be the convex closure of $K$ in $K'$. By Theorem~\\ref{thm:Tcon}, $K'$ is $1$-h-minimal as an ${\\mathcal L}$-structure, for ${\\mathcal L}$ as in the theorem.\n(Note that the saturation assumption implies ${\\mathcal O}_{K'} \\ne K'$.)\nIn the following, we denote the valuation on $K'$ by $|\\cdot|_v$, to distinguish it from the absolute value $|\\cdot|$. We suppose that a family of functions $(f_q)_{q \\in K^m}$ is given as in the corollary, and by abuse of notation, we also write $(f_q)_{q \\in (K')^m}$ for the corresponding family in $K'$ (defined by the same formula).\n  \nSuppose that the corollary fails, i.e., that for every $c \\in K_{>0}$ and for every formula $\\psi$ that could potentially define the family $(C_q)_{q \\in K^m}$, there exists a $q \\in K^m$ and a pair $x_0, x$ for which the implication (\\ref{eq:arch.ass}) $\\Rightarrow$ (\\ref{eq:arch.imp}) fails.\nBy our saturation assumption, there exist $q \\in (K')^m$ and $x_0, x \\in K'$ such that the implication fails for every $c \\in K_{>0}$ and every $\\psi$.\nUsing that ${\\mathcal O}_{K'}$ is the convex closure of $K$, this failure for every $c \\in K_{>0}$ is equivalent to the conjunction\n\\begin{equation}\\label{eq:v.ass}\n|x - x_0|_v < \\min_{a \\in C_q}|x_0 -  a|_v \n\\end{equation}\nand\n\\begin{equation}\\label{eq:v.imp}\n|f(x) -  T^{\\le r}_{f_q,x_0}(x) |_v >  |f_q^{(r+1)}(x_0)\\cdot (x-x_0)^{r+1}|_v.\n\\end{equation}\nSo we obtained: For every ${\\mathcal L}_\\omin(q)$-definable set $C_q$, there exist $x_0, x \\in K'$ in the same ball $1$-next to $C_q$ (by (\\ref{eq:v.ass})) such that (\\ref{eq:v.imp}) holds.\nThis contradicts Theorem~\\ref{thm:high-ord}: \\emph{A priori}, that theorem only provides a finite ${\\mathcal L}(q)$-definable set $C$, but ${\\mathcal L}_\\omin(q)$-definability of that set then follows using Lemma~\\ref{lem:L2Lomin}.\n\\end{proof}\n\n\nOne can expect that similar results in higher dimension can be obtained (maybe building on Question~\\ref{qu:t-high-high} below), and that they can lead to finer versions of the preparation results in power-bounded real closed fields from \\cite{DS.ominPrep,NguyenValette}, which are used to deduce the existence of Mostowski's Lipschitz stratifications.\n\n\n\n\\subsection{Comparison to V-minimality}\\label{sec:comparison}\n\nIn \\cite{HK}, Hrushovski--Kazhdan introduced the notion of $V$-minimal theories, which at first sight has the same goal as Hensel minimality, namely: to provide a powerful axiomatic framework for geometry in valued fields. However,\nthe relation between $V$-minimality and Hensel minimality is similar to the relation between strong minimality and o-minimality:\nBy working in a strongly minimal theory of (algebraically closed) fields, one obtains many useful results about geometry in real closed fields, but one cannot treat genuinely o-minimal languages like ${\\mathbb R}_{\\mathrm{exp}}$. In a similar way, working in a $V$-minimal theory of (algebraically closed) valued fields does provide many useful insights about Henselian valued fields (as explained in \\cite[Section~12]{HK}), but there are examples of Hensel minimal theories that cannot be treated in this way.\n\nConcretely, since a V-minimal theory has not more structure on $\\mathrm{RV}$ than a pure valued field, every definable function $K \\to K$ ultimately grows like $x \\mapsto x^r$ for some rational number $r$,\nand this remains true if we expand the language by predicates on $\\mathrm{RV}$ (following \\cite[Section~12]{HK}). In contrast, Section~\\ref{sec:Tcon} provides examples of $1$-h-minimal structures without this property, for example the $T$-convex structure obtained from the (power-bounded o-minimal) expansion of ${\\mathbb R}$ by one function $x \\mapsto x^r$ for every real number $r$.\n\nOn the other hand, if we restrict to the context for which V-minimality has been designed, then it agrees with Hensel minimality. Moreover, in this case, $0$-h-minimality\nand $1$-h-minimality agree. Let us recall the definition of V-minimality.\n\n\\begin{defn}[V-minimality; {\\cite[Section~3.4]{HK}}]\\label{defn:Vmin}\nFix a language ${\\mathcal L} \\supset {\\mathcal L}_\\val$ and a complete theory ${\\mathcal T}$ containing the theory $\\operatorname{ACVF}_{0,0}$ of algebraically closed fields of equi-characteristic $0$.\nThe theory ${\\mathcal T}$ is called \\emph{$V$-minimal} if for every model $K \\models {\\mathcal T}$, we have the following:\n\\begin{enumerate}\\setcounter{enumi}{-1}\n \\item Every definable (with parameters) subset of $K$ is a finite boolean combination of points, open balls, and closed balls.\n \\item Every ${\\mathcal L}(K)$-definable subset of $\\mathrm{RV}^n$ is already\n   ${\\mathcal L}_\\val(K)$-definable (where ${\\mathcal L}_\\val$ is the pure valued field language).\n \\item Every definable (with parameters) family of nested closed balls in $K$ has non-empty intersection.\n \\item For every $A \\subset K$, if $X \\subset K$ is an $A$-definable set which is the union of finitely many disjoint closed balls $B_i$, then $\\acl_K(A) \\cap B_i \\ne \\emptyset$ for every $i$.\n\\end{enumerate}\n\\end{defn}\n\n\\private{In the last item, HK write that $X$ should be an ``almost $A$-definable closed ball''; this is defined as: there exists an $A$-definable equivalence relation with finitely many classes such that $X$ is a union of equivalence classes....}\n\n\n\\begin{prop}[V-minimality]\\label{prop:vmin}\nSuppose that ${\\mathcal T}$ is a complete theory containing $\\operatorname{ACVF}_{0,0}$, in a language ${\\mathcal L} \\supset {\\mathcal L}_\\val$, and suppose moreover that every ${\\mathcal L}(K)$-definable subset of $\\mathrm{RV}^n$ is already ${\\mathcal L}_\\val(K)$-definable.\nThen the following are equivalent:\n\\begin{enumerate}[(i)]\n \\item ${\\mathcal T}$ is V-minimal.\n \\item ${\\mathcal T}$ is $0$-h-minimal.\n \\item ${\\mathcal T}$ is $1$-h-minimal.\n\\end{enumerate}\n\\end{prop}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:vmin}]\n(iii) $\\Rightarrow$ (ii) is trivial.\n\n(i) $\\Rightarrow$ (iii): We use the criteria from Theorem~\\ref{thm:tame2vf}, so let $f\\colon K \\to K$ be $A$-definable,\nfor some $A \\subset K \\cup \\mathrm{RV}$.\n\nFirst note that by the Remark just above \\cite[Lemma~3.30]{HK}, adding parameters from $K \\cup \\mathrm{RV}$ to the language preserves V-minimality, so using compactness,\nthe results from \\cite{HK} hold uniformly in families parametrized by $K$ or $\\mathrm{RV}$.\n\nBy dimension theory (e.g.\\ \\cite[Lemma~3.55]{HK}),\n$f$ has only finitely many infinite fibers, i.e., Condition~(T2) from Theorem~\\ref{thm:tame2vf} holds.\nBy applying \\cite[Corollary~4.3]{HK} to all fibers $f^{-1}(b)$ of $f$ (where $b$ runs over $K$),\nwe find an $A$-definable map $\\rho\\colon K \\to \\mathrm{RV}^k$ (for some $k \\ge 0$) such that for each $\\xi\\in \\mathrm{RV}^k$, the restriction $f|_{\\rho^{-1}(\\xi)}$ is either constant or injective.\nApply \\cite[Corollary~5.9]{HK} to each injective restriction $f|_{\\rho^{-1}(\\xi)}$ and refine the map $\\rho$ accordingly, i.e.,\nsuch that afterwards, $f$ is ``nice'' on each open ball contained in one fiber of $\\rho$ in the sense of \\cite[Definition~5.8]{HK}.\nFinally, by applying \\cite[Corollary~4.3]{HK} to the graph of $\\rho$, we find a finite $A$-definable set $C$ $1$-preparing $\\rho$ (namely, the image of the map $c$ provided by the corollary).\nThen $f$ is nice on each ball $1$-next to $C$, and this implies Condition~(T1) from Theorem~\\ref{thm:tame2vf}.\n\n\n\n(ii) $\\Rightarrow$ (i): We prove the conditions from Definition~\\ref{defn:Vmin}:\n\n(0) Let $X \\subset K$ be definable, and let $C \\subset K$ be a finite set preparing $X$. Then $X$ can be written as a union of the form\n$\\bigcup_{c \\in C} X_c$, where $X_c = \\{c + x \\mid \\operatorname{rv}(x) \\in Z_c\\}$\nfor suitable definable sets $Z_c \\subset \\mathrm{RV}$. Using the assumption that definable subsets of $\\mathrm{RV}$ are already definable in the language ${\\mathcal L}_\\val$,\nwe obtain that each $X_c$ is a finite boolean combination of points, open balls, and closed balls. This then also follows for $X$.\n\n(1) holds by assumption.\n\n(2) is a special case of Lemma~\\ref{lem:intersection}.\n\n(3) By 0-h-minimality, there exists a finite $A$-definable set $C$ preparing $X$. This set $C$ cannot be disjoint from any $B_i$, since for any $c \\in C \\setminus B_i$, the ball $1$-next to $c$ containing $B_i$ is strictly bigger than $B_i$.\n\\end{proof}\n\n\n\\subsection{Some open questions}\nAs major future research leads we see the development of motivic integration and of applications to point counting  (\u00e0 la Pila-Wilkie) under Hensel minimality, as mentioned in the introduction. We finish the paper with some questions which are internal to this very paper.\nProbably the most \nimmediate question in this context is:\n\n\\begin{question}\nDoes $0$-h-minimality imply $1$-h-minimality, and\/or does $1$-h-minimality imply $\\omega$-h-minimality? (More generally: For which $\\ell < \\ell'$ does $\\ell$-h-minimality imply $\\ell'$-h-minimality?)\n\\end{question}\n\nIf $1$-h-minimality is not equivalent to $\\omega$-h-minimality, we have the following questions:\n\\begin{question}\nAre the ${\\mathcal T}_\\omin$-convex structures from Subsection~\\ref{sec:Tcon} (with ${\\mathcal T}_\\omin$ power-bounded) $\\omega$-h-minimal?\n\\end{question}\n\n\\begin{question}\nDoes $1$-h-minimality imply $\\omega$-h-minimality\nunder the assumptions from Proposition~\\ref{prop:vmin}?\n\\end{question}\n\nIn a somewhat opposite direction, one might wonder:\n\n\\begin{question}\nWhich results still hold if we only assume $0$-h-minimality, or even under Assumption~\\ref{ass:no-ctrl} (``Hensel minimality without control of parameters'')?\n\\end{question}\n\n\n\n\\medskip\n\nSuppose that ${\\mathcal L}'$ is an $\\mathrm{RV}$-expansion of ${\\mathcal L}$. Then $\\ell$-h-minimality of $\\operatorname{Th}_{{\\mathcal L}}(K)$ implies $\\ell$-h-minimality of $\\operatorname{Th}_{{\\mathcal L}'}(K)$\nfor $\\ell = 0, 1, \\omega$ (Theorem~\\ref{thm:resp:h}).\n\n\n\\begin{question}\\label{que:conv}\nDoes the converse also hold, i.e., does $\\ell$-h-minimality of $\\operatorname{Th}_{{\\mathcal L}'}(K)$ imply $\\ell$-h-minimality of $\\operatorname{Th}_{{\\mathcal L}}(K)$?\n\\end{question}\n\nNote that for values of $\\ell$ other than $0$, $1$, $\\omega$, we do not even know the direction from ${\\mathcal L}$ to ${\\mathcal L}'$.\n\n\n\\begin{remark}\nHensel minimality is not preserved by passing to reducts in general. Indeed, suppose that $\\operatorname{Th}_{{\\mathcal L}}(K)$ is $\\omega$-h-minimal and that $K$ is $\\aleph_0$-saturated.\nFix a ball $B = B_{<\\lambda}(a) \\subset K$ which is strictly contained in a ball disjoint from $\\acl_K(\\emptyset)$\n(so that $B$ cannot be prepared by a finite, $\\emptyset$-definable $C \\subset K$). Then $\\operatorname{Th}_{{\\mathcal L}(a,\\lambda)}(K)$ is $\\omega$-h-minimal but\nthe reduct $\\operatorname{Th}_{{\\mathcal L} \\cup \\{B\\}}(K)$ is not (where by ``$B$'', we mean a predicate for that ball).\n\\end{remark}\n\n\\medskip\n\nSuppose that $\\operatorname{Th}(K)$ is $\\omega$-h-minimal and that we have a definable coarsening $|\\cdot|_c$ of the valuation;\nwrite $K_c$ for $K$ considered as a valued field with the coarsened valuation, and write $k_c$ for the residue field of $K_c$,\nconsidered as a valued field with the valuation induced from $|\\cdot|$, and with the full induced structure by ${\\mathcal L}$.\n\n\\begin{question}\nBy Corollary~\\ref{cor:coarse},\n\\begin{enumerate}\n\\item if $\\operatorname{Th}(K)$ is $\\omega$-h-minimal, then so is $\\operatorname{Th}(K_c)$;\n\\end{enumerate}\nbut:\n\\begin{enumerate}\\stepcounter{enumi}\n \\item Does $\\omega$-h-minimality of $\\operatorname{Th}(K)$ imply $\\omega$-h-minimality of $\\operatorname{Th}(k_c)$?\n \\item Do $\\omega$-h-minimality of $\\operatorname{Th}(K_c)$ and $\\omega$-h-minimality of $\\operatorname{Th}(k_c)$ together imply $\\omega$-h-minimality of $K$?\n\\end{enumerate}\nAnd also:\n\\begin{enumerate}\\setcounter{enumi}{3}\n \\item Does (1) (and\/or (2), (3)) also hold for $0$- and\/or $1$-h-minimality?\n\\end{enumerate}\n\\end{question}\n\n\n\\medskip\n\n\nAny $C^1$-function $U \\subset {\\mathbb R}^n \\to {\\mathbb R}$ also has a strict derivative (see Definition~\\ref{defn:strict-der}).\nThis is not the case in valued fields:\n\n\\begin{example}\nDefine $f\\colon K^2 \\to K$ by $f(x,y)= x^2$ if $|x|^4 \\leq  |y|$ and $f(x,y) = x^3$ otherwise. This function is $C^1$ everywhere, but at $0$, the strict derivative does not exist, since $\\frac{f(x,x^4) - f(x,0)}{x^4} = x^{-2} - x^{-1}$, which diverges for $x \\to 0$.\n\\end{example}\n\nIn view of this example, and by our knowledge that strict $C^1$ is the better notion for rank one valued fields for several reasons (see e.g.~\\cite{BGlockN}, and where strict $C^1$ means that the strict derivative exists everywhere and is continuous), one may try to build a good working notion of definable strict $C^1$ submanifolds of $K^n$, assuming a suitable form of Hensel minimality.   The following is a first question in this direction.\n\n\\begin{question}\nDoes the Implicit Function Theorem hold for definable, strict $C^1$ functions, say, assuming $1$-h-minimality (with a well-chosen definition of ``strict $C^1$'')?\n\\end{question}\n\n\n\\medskip\n\n\n\nAddendum \\ref{add:cd:alg:range} of Cell Decomposition (Theorem~\\ref{thm:cd:alg:skol}) speaks about simultaneous preparation of the domain and range of functions,\nbut only in the one-variable case.\n\\begin{question}\nIs there a version of Addendum \\ref{add:cd:alg:range} for functions from $K^n$ to $K^m$ which works for $n$ and\/or $m$ bigger than $1$? What would even be the right formulation?\n\\end{question}\nThe following question might be related:\n\\begin{question}\nTheorem~\\ref{thm:tame2vf} provides a characterization of $1$-h-minimality in terms of functions from $K$ to $K$. Is there an analogous characterization of $\\omega$-h-minimality?\n\\end{question}\n\n\n\\medskip\n\n\nAnd finally: Theorem~\\ref{thm:T3\/2,mv} (order one Taylor approximations of functions in several variables) suggests\nthat we might have the following variant of Theorem~\\ref{thm:t-high-high} (higher order Taylor approximations of functions in several variables):\n\n\\begin{question}\\label{qu:t-high-high}\nGiven a definable function $f\\colon K^n \\to K$ and an integer $r \\ge 1$, does there exist a definable map $\\chi\\colon K^n \\to \\mathrm{RV}^k$ such that\n(\\ref{eq:t-high-high}) (or a similar kind of Taylor approximation) holds on each $n$-dimensional fiber of $\\chi$?\n\\end{question}\n\n\\begin{remark}\nSuch a result would be strictly stronger than Theorem~\\ref{thm:t-high-high}, which yields Taylor approximations\nonly on boxes disjoint from a lower-dimensional definable set $C$. Indeed, given $\\chi$, one can easily find a $C$\nsuch that every box disjoint from $C$ is contained in a fiber of $\\chi$\n(namely by $1$-preparing $\\chi$ fiberwise using Corollary~\\ref{cor:prep}). On the other hand,\nthe family of maximal boxes disjoint from $C$ cannot, in general, by parametrized by a tuple from $\\mathrm{RV}$.\n\\end{remark}\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\normalsize {In} quantum systems, measurement of product observables of two or more can contain information  about quantum correlations and quantum dynamics.   In the recent past, many theoretical and experimental works have been performed regarding the  measurements of product observables using weak measurements.   They have been used to resolve the Hardy's Paradox \\cite{1} with experimental verification \\cite{2}, EPR-Bohm experiment \\cite{3}, direct measurement of a density matrix \\cite{4}, reconstruction of  entangled quantum states \\cite{5}. It has also been reported that  a strange  quantum effect namely the ``Quantum Cheshire Cats\"  where the properties of a quantum particle can be disembodied (e.g., photon's position and polarization degrees of freedoms can be separated from each other) can be realized using product weak values \\cite{6}. See also the references \\cite{7} and \\cite{8} for the experimental test of  the existence  of Quantum Cheshire Cats and the exchange of  grins between two such Quantum Cheshire Cats, respectively.  Weak measurements of  product observables also play an important role in  understanding the quantum mechanics  such as Bell tests \\cite{9,10,11}, nonlocality via post-selection \\cite{12}.\\par \nHigher moment weak values are useful to obtain the weak-valued probability distribution \\cite{13} of an observable in the pre and post-selected systems (which will be shown in this work later).  Some applications  of  weak-valued probability distribution  are: (a) Ozawa's measurement- disturbance relation has experimentally been verified using weak-valued probability distribution \\cite{14}, (b) to obtain all of the values of the observables relevant to a Bell test experimentally, weak-valued probabilities have been used \\cite{15}, (c) the authors of \\cite{16} have shown that there is a connection between weak-valued joint probabilities and incompatibility. There are some other applications of weak-valued probabilities e.g., (d) experimental realization of the Quantum Box Problem \\cite{17}, (e) to resolve Hardy's paradox \\cite{2}, (f) justification of Scully \\emph{et al.}'s claim \\cite{18} that the momentum disturbance associated with which-way measurement in Young's double-slit experiment can be avoided has been shown by the negativity of the weak-valued probabilities corresponding to the momentum disturbance, which consequently have zero variance \\cite{19,20}, (g) to control the probe wave packet of the target system by pre and post-selections, one can use the higher moment weak values \\cite{21}, (h) to obtain the modular value of an observable in pre and post- selected systems, higher moment weak values can be used as there is an exact connection between them \\cite{22,23}.\n\\par\nWeak measurements in particular weak values are known to provide useful informations with simple experimental setups. The  ``weak value\" was first  introduced by  Aharonov, Albert, and Vaidman (AAV)  \\cite{24}. It  was inspired by the two-time formulation of the quantum-mechanical system \\cite{25}.  The mechanism of  AAV method is that they took the von Neumann measurement scheme  \\cite{26} one step forward  by considering the coupling coefficient very small i.e., weak followed by a strong measurement in succession. This formulation is characterized by the pre- and postselected states of the system. By preparing  a system initially  in the state $\\ket{\\psi}$ and  post-selecting  in the state $\\ket{\\phi}$, as a result we obtain the weak value of any observable ${A}$ which  is defined as \n\\begin{align}\n{\\braket{{A}_w}}^{\\phi}_{\\psi}=\\frac{\\braket{\\phi |{A}|\\psi}}{\\braket{\\phi |\\psi}}\\label{1}.\n\\end{align}\nThis is a complex number and the spatial and momentum displacements of the pointer state give the real and imaginary  parts of that weak value \\cite{27,28}, respectively. By this way, we get the full knowledge of the complex weak value. One of the exciting and interesting features of weak value is that it can lie outside the max-min range of the eigenvalues of the operator of interest. Simultaneously, measurement disturbance is quite small which gives one to perform further measurements or simultaneous measurement of multiple observables. \\par\nWeak measurements have been proven useful in understanding quantum systems such as  for the direct measurement of the wave function of a quantum system \\cite{29},  to calculate slow- and fast-light effects in birefringent photonic crystals \\cite{30}, the confirmation of the Heisenberg-Ozawa uncertainty relationship \\cite{31}, for detecting tiny spatial shifts \\cite{32,33}.  Weak value can also  be used to measure non-Hermitian operators \\cite{34,35}, in hot thermometry \\cite{36}, to detect entanglement universally in a two-qubit system \\cite{37}. \\par\n\\normalsize{\\textit{Product weak values:}} The observables used in the von Neumann measurement scheme  are of simple kinds. Direct measurement of product observables is extremely  difficult whether it is strong or weak. This difficulty arises from the fact that the measurement interaction in the von Neumann measurement scheme couples two different observables to a single pointer \\cite{27}. To overcome this, an approach using multi particle interaction Hamiltonian was proposed by Resch and Steinberg (RS) {\\cite{38}} applying AAV method. Namely,  they have used the Hamiltonian of the form ${H}=g_1{A}\\otimes{p}_x + g_2{B}\\otimes{p}_y$ with gaussian pointer states. Here $A$ and $B$ are two observables of the system. ${p}_x$ and ${p}_y$ are the pointer's momentum degrees of freedom in two different directions x and y, respectively. $g_1$ and $g_2$ are coupling coefficients between the system and the pointer for  two different directions, respectively.  By performing a second order expansion in the two-dimensional pointer displacement $\\braket{{X} {Y}}$, they showed that it is possible to extract the real part of the product weak value  for the case of commuting observables $[{A},{B}]=0$, namely $Re[\\braket{(AB)_w}_{\\psi}^{\\phi}]=2{\\braket{XY}}\/({g_1g_2t^2})-Re[(\\braket{A_w}_{\\psi}^{\\phi})^* \\braket{B_w}_{\\psi}^{\\phi}]$, where `t' is the interaction time.  For imaginary part of the product weak value, one needs to look into $\\braket{{X} {P}_y}$.  Note that, the weak value of a tensor product  observable  in a bipartite system (we will call it  product weak value in a bipartite system)  can also be calculated according to the RS method as the local observables are commuting. It's generalised version i.e., the product  weak values $\\braket{({A} {B})w}_{\\psi}^{\\phi}$ and it's higher orders i.e., $\\braket{({A} {B} {C}\\cdots)w}_{\\psi}^{\\phi}$ using N pointers' correlations  can be found in Ref. {\\cite{39}}.\\par\n\\normalsize{\\textit{Summary of the present work:}} In this work, we show that by introducing  an unique  orthogonal  state to the given post selection, it is possible to extract the higher moment weak values $\\braket{({A}^n)_w}_{\\psi}^{\\phi}$ in a single system  as well as the product weak values for the given  pure and mixed pre selected  states in a bipartite system separately.  We   show application of our results to  reconstruct quantum states of  single and bipartite systems. We have used higher moment weak values to reconstruct a  pure state of a single system and product weak values of  a bipartite system to reconstruct unknown pure and mixed states of that  bipartite system. Recently, Pan \\emph{et al.} \\cite{5} have used product weak values of projection operators  to reconstruct an unknown bipartite pure state. They have considered entangled pointer states as well as the modular values of  local projection operators  and the modular values of  sum of the local  projection operators \\cite{23} to evaluate product weak values. We, for the first time show that such product weak values can be realized locally for the case of both pure and mixed states. Also, to reconstruct the states of single and bipartite systems,  we have generalized the  measurement of projection operators to  arbitrary  observables. Our method can be generalized to the multipartite systems.  Further, we give a necessary separability inequality for finite dimensional systems using  product weak values. This inequality is violated by  certain class of entangled states  by cleverly choosing the product observables and the post selections. By such choices, The PPT criteria can be achieved for these class of entangled states. In particular, we give several  examples namely $(i)$ two-qubit Werner state (noisy singlet), $(ii)$ mixture of two-Bell states, $(iii)$ mixture of arbitrary pure entangled and maximally mixed states, $(iv)$ mixture of two arbitrary entangled states, $(v)$ mixture of four-Bell states, $(vi)$ two qudit  Werner states, $(vii)$ higher dimensional isotropic states. The criteria can potentially detect more classes of entangled states with suitably choosing product observables and post-selections. Finally we show that our methods of  \"extraction of product  and higher moment weak values\"  are robust against the errors which occur  due to the inappropriate choices of system observables and unsharp post-selections.\\par\nThis paper is organized as follows. In sec. \\ref{e II} we provide the formulation of our method.  In sec. \\ref{e III} we apply our method to reconstruct quantum states of single as well as bipartite systems separately. Entanglement detection criteria is shown in sec. \\ref{e IV}. We show the robustness of our method in sec. \\ref{e V} and finally conclude in sec. \\ref{e VI}.\n\\section{FORMULATION}\\label{e II}\nThe following identity which is sometimes referred as Vaidman's formula \\cite{40}  will be used to derive the main results of this paper\n\\begin{align}\n{A}\\ket{\\phi}=\\braket{{A}}_{\\phi}\\ket{\\phi} + \\braket{\\Delta A}_{\\phi} \\ket{\\phi^{\\perp}_{A}},\\label{2}\n\\end{align}\nwhere $A$ is an Hermitian operator and $\\ket{\\phi}$ is any quantum state vector in the Hilbert space $\\mathcal{H}$. The state vector  $\\ket{\\phi_{A}^{\\perp}}$ is orthogonal to  $\\ket{\\phi}$, $\\braket{{A}}_{\\phi}=\\braket{\\phi |{A}|\\phi}$ and $\\braket{\\Delta A}_{\\phi}=\\braket{\\phi^{\\perp}_{A} |{A}|\\phi}$. For the derivation see Appendix \\ref{A}\n\\subsection{Higher moment weak values}\\label{II A}\nIf $\\ket{\\psi}$, ${A}$ and $\\ket{\\phi}$ are the pre-selected state, the observable and the post-selected state, respectively, then the weak value of the observable $A$ is given by\n\\begin{align}\n{\\braket{{A}_w}}^{\\phi}_{\\psi}&=\\left(\\frac{\\braket{\\psi|{A}|\\phi}}{\\braket{\\psi|\\phi}}\\right)^{\\ast}\\!=\\braket{A}_{\\phi} +\\braket{\\Delta A}_{\\phi}\\frac{\\braket{\\phi_{A}^\\perp|\\psi}}{\\braket{\\phi|\\psi}},\\label{3}\n\\end{align}\nwhere we have used Eq. (\\ref{2}).  A  similar expression was considered in Ref. \\cite{41} to explain the origin of the complex  and anomalous nature of a weak value.  The Eq. (\\ref{3})  for the expression of the weak value will be useful for deriving the following result.\\par\n\\emph{Result 1:-}\nThe weak value of the operator $A^2$ which we call  the ``second moment weak value\" has the following expression\n \\begin{align}\n{\\braket{({A}^2)_w}}^{\\phi}_{\\psi}=\\braket{A}_\\phi \\Big({\\braket{{A}_w}}^{\\phi}_{\\psi} - {\\braket{{A}_w}}^{\\phi^{\\perp}_{A}}_{\\psi}\\Big) + {\\braket{{A}_w}}^{\\phi}_{\\psi}{\\braket{{A}_w}}^{\\phi^{\\perp}_{A}}_{\\psi},\\label{4}\n\\end{align}\nwhere ${\\braket{{A}_w}}^{\\phi}_{\\psi}$ and ${\\braket{{A}_w}}^{\\phi^{\\perp}_{A}}_{\\psi}$ are the weak values of the operator $A$ for the given pre selection $\\ket{\\psi}$ with two post-selections $\\ket{\\phi}$ and $\\ket{{\\phi^{\\perp}_{A}}}$, respectively.\n\\begin{proof}\n\\begin{align}\n{\\braket{({A}^2)_w}}^{\\phi}_{\\psi}&=\\left(\\frac{\\braket{\\psi |{A}^2|\\phi}}{\\braket{\\psi|\\phi}}\\right)^{\\ast}\\nonumber\\\\\n&=\\left(\\frac{1}{\\braket{\\psi|\\phi}}\\bra{\\psi}{A}[\\braket{A}_{\\phi}\\ket{\\phi} +\\braket{\\Delta A}_{\\phi}\\ket{\\phi_{A}^\\perp}]\\right)^\\ast \\nonumber\\\\\n&=\\left(\\braket{A}_{\\phi}\\frac{\\braket{\\psi |{A}|\\phi}}{\\braket{\\psi|\\phi}} +\\braket{\\Delta A}_{\\phi}\\frac{\\braket{\\psi |{A}|\\phi_{A}^\\perp}}{\\braket{\\psi|\\phi}}\\right)^\\ast \\nonumber\\\\\n&=\\braket{A}_{\\phi}\\frac{\\braket{\\phi |{A}|\\psi}}{\\braket{\\phi|\\psi}} +\\braket{\\Delta A}_{\\phi}\\frac{\\braket{\\phi_{A}^\\perp |{A}|\\psi}}{\\braket{\\phi_{A}^\\perp |\\psi}}\\frac{\\braket{\\phi_{A}^\\perp |\\psi}}{\\braket{\\phi|\\psi}}.\\label{5}\n\\end{align}\nFrom Eq. (\\ref{3}), using $\\braket{\\Delta A}_{\\phi}\\frac{\\braket{\\phi_{A}^\\perp|\\psi}}{\\braket{\\phi|\\psi}}=\\braket{A_w}^{\\phi}_{\\psi}-\\braket{A}_{\\phi} $ in Eq. (\\ref{5}), we will obtain Eq. (\\ref{4}).\n\\end{proof}\nIn the similar way we obtain all the higher moment weak values which take the general form as\n\\begin{align}\n{\\braket{{A}^n_w}}^{\\phi}_{\\psi}=\\braket{A}_\\phi \\hspace{-2pt}\\Big(\\hspace{-3pt}{\\braket{{A}^{n-1}_w}}^{\\phi}_{\\psi}\\! - \\braket{{A}^{n-1}_w}^{\\phi^{\\perp}_{A}}_{\\psi}\\hspace{-2pt}\\Big) \\!+ {\\braket{{A}^{n-1}_w}}^{\\phi^{\\perp}_{A}}_{\\psi}{\\braket{{A}_w}}^{\\phi}_{\\psi},\\label{6}\n\\end{align}\nfor $n=1,2,\\cdots$. \\par\nNow consider the second moment weak value i.e., Eq. (\\ref{4}), where ${\\braket{{A}_w}}^{\\phi}_{\\psi}$ and ${\\braket{{A}_w}}^{\\phi^{\\perp}_{A}}_{\\psi}$ are extractable from the one and the same experimental set-up for the post-selection of $\\ket{\\phi}$ and $\\ket{\\phi^{\\perp}_{A}}$, respectively  as these two states are orthogonal to each other. Note that, although the post-selection can be realized here in one and the same measurement set-up, nevertheless, in order to actually find out the weak values ${\\braket{{A}_w}}^{\\phi}_{\\psi}$ and ${\\braket{{A}_w}}^{\\phi^{\\perp}_{A}}_{\\psi}$, measurements of phase-space displacements for the two post-selected states $\\ket{\\phi}$ and $\\ket{\\phi^{\\perp}_{A}}$  need to be performed. $\\braket{A}_\\phi$ is the average value of $A$ for the post selected state $\\ket{\\phi}$. Extraction of higher moment weak values becomes extremely simple in two dimensional Hilbert space  discussed in the following.\\par\n{\\it  Two dimensional case:} In two dimensional Hilbert space,  there are only two pairwise orthogonal post-selections which occur at the same time and hence both the weak values with orthogonal post-selections can be extracted simultaneously. Particularly in this dimension, we find that only by knowing the weak  values ${\\braket{{A}_w}}^{\\phi}_{\\psi}$ and ${\\braket{{A}_w}}^{\\phi^{\\perp}_{A}}_{\\psi}$,  we are able to obtain all the higher moment weak values without any further complications in comparison to Ref.  \\cite{42}. So the  number of measurements is being reduced considerably than the earlier proposal \\cite{42}. \\par\n{\\it  Higher dimensional case:} In higher dimensional Hilbert space, to extract second moment weak value of the observable $A$, one can  perform  the projective measurements $\\{\\ket{\\phi}\\bra{\\phi}, \\ket{\\phi^{\\perp}_{A}}\\bra{\\phi^{\\perp}_{A}}, I-\\ket{\\phi}\\bra{\\phi} -\\ket{\\phi^{\\perp}_{A}}\\bra{\\phi^{\\perp}_{A}} \\}$ for post-selections.\\par\nIt can be shown that  the $n$-th moment weak value i.e., ${\\braket{A^{n}_w}}^{\\phi}_{\\psi}$, consists  $n$ number of different weak values, namely  ${\\braket{{A}_w}}^{\\phi}_{\\psi}$, ${\\braket{{A}_w}}^{\\phi^{\\perp}_{A}}_{\\psi}$,$\\cdots$,${\\braket{{A}_w}}^{(\\phi^{\\perp}_{A})^{n-1}}_{\\psi}$.  Here  $\\ket{(\\phi^{\\perp}_{A})^{n-1}}=\\ket{((\\phi^{\\perp}_{A})^{\\perp}_{A})\\cdots (n-1) \\hspace{1mm} times}=\\frac{1}{\\braket{\\Delta A}_{(\\phi^{\\perp}_{A})^{n-2}}}\\left(A-\\braket{A}_{(\\phi^{\\perp}_{A})^{n-2}}\\right)\\ket{(\\phi^{\\perp}_{A})^{n-2}}$. Now, if $n$ is even, then there exist pairwise orthogonal  post-selected states i.e.,\n$\\left(\\ket{\\phi},\\ket{\\phi^{\\perp}_{A}}\\right)$, $\\left(\\ket{(\\phi^{\\perp}_{A})^{\\perp}_{A}},\\ket{((\\phi^{\\perp}_{A})^{\\perp}_{A})^{\\perp}_{A}}\\right)$,$\\cdots$,$\\left(\\ket{(\\phi^{\\perp}_{A})^{n-2}},\\ket{(\\phi^{\\perp}_{A})^{n-1}}\\right)$. So, it is possible to obtain weak values  ${\\braket{{A}_w}}^{\\phi}_{\\psi}$ and  ${\\braket{{A}_w}}^{\\phi^{\\perp}_{A}}_{\\psi}$ simultaneously for the first pair of post-selected states, so on and so forth. So, effectively the total  number of measurements to be performed according to the AAV method to extract the $n$-th moment weak value is ${n}\/{2}$. For odd $\\hspace{0.5mm}$ $n$, the number of measurements is $(n+1)\/2$. Note that, all the measurements here are to be done by only  changing the post-selections while keeping  the observable $A$  fixed in the AAV method.  \\textit{Once we extract the $n$-th moment weak value, then all the lower moment weak values can be calculated  from the data of $n$-th moment weak value.}\\par \nThe higher moment weak values of an observable are inaccessible with Gaussian pointer states. The reason is that, in the RS method \\cite{38}, the higher moment weak value terms will vanish due to the  properties of the Gaussian  pointer state. The whole expression can be found in Ref. \\cite{42} (equation 4).  More specifically, it can be seen in the above mentioned expression that when the orbital angular momentum (OAM) of the pointer state is zero (which corresponds to the two dimensional Gaussian pointer state, i.e., OAM state with zero orbital angular momentum), the higher moment weak value terms vanish. To retrieve higher moment weak values, we need to use OAM states with non-zero orbital angular momentums. The key factor for such cases is that the two dimensional OAM states are not factorizable in two different directions for non-zero orbital angular momentums. In doing that one needs to engineer OAM states with higher winding numbers, or superpositions of OAM states to obtain  the higher moment weak values. This procedure  can become difficult as the moments increase. One needs to prepare pointer states with different combination of orbital angular momentum {\\cite{42}}. Moreover, there are several disadvantages of the RS method from experimental perspective which we will discuss later in this section. See \\cite{43} for a comment regarding the extraction of higher moment weak values. \\par\nAs an application, we will use the higher moment weak values to reconstruct an unknown pure state of a single system. See  Appendix \\ref{B} for the derivation of the second or higher moment weak values  for the mixed pre selected state case.\n\\subsection{Product weak values}{}\\label{II B}\n\\emph{Product weak values for pure pre-selected state:}\nThe product  weak value of the  observable  ${A}\\otimes {B}$ in a bipartite system  is given by\n\\begin{align}\n  \\braket{({A}\\otimes {B})_w}^{\\phi_A\\phi_B}_{\\psi_{AB}}=\\frac{\\braket{\\phi_A\\phi_B|({A}\\otimes {B})|\\psi_{AB}}}{\\braket{\\phi_A\\phi_B|\\psi_{AB}}},\\label{7}\n\\end{align}\n where the pre selection  is the bipartite pure state $\\ket{\\psi_{AB}}$ and the post-selection is a product state $\\ket{\\phi_A\\phi_B}=\\ket{\\phi_A}\\otimes \\ket{\\phi_B}$.\\par\nConsider the ``local weak value\" of the operator $A$\n \\begin{align}\n  \\braket{A^{local}_w}^{\\phi_A\\phi_B}_{\\psi_{AB}}&=\\frac{\\braket{\\phi_A\\phi_B|({A}\\otimes {I_B})|\\psi_{AB}}}{\\braket{\\phi_A\\phi_B|\\psi_{AB}}}\\nonumber\\\\\n  &=\\braket{A}_{\\phi_A} + \\braket{\\Delta A}_{\\phi_A}\\frac{\\braket{\\phi^{\\perp}_A\\phi_B|\\psi_{AB}}}{\\braket{\\phi_A\\phi_B|\\psi_{AB}}},\\label{8}\n\\end{align}\nwhere we have used Eq. (\\ref{2}) in the subsystem A. Eq. (\\ref{8}) will be used to derive the following result.\n \\begin{widetext}\n \\emph{Result 2:-} The product weak value in Eq. (\\ref{7}) can be realized via local weak values as\n\\begin{align}\n\\braket{(A\\otimes B)_w}^{\\phi_A\\phi_B}_{\\psi_{AB}}=&\\braket{A}_{\\phi_A} \\left({\\braket{B^{local}_w}}^{\\phi_A\\phi_B}_{\\psi_{AB}} - {\\braket{{B^{local}_w}}^{\\phi^{\\perp}_{A}\\phi_B}_{\\psi_{AB}}}\\right) + {\\braket{A^{local}_w}}^{\\phi_A\\phi_B}_{\\psi_{AB}}{\\braket{B^{local}_w}}^{\\phi^{\\perp}_{A}\\phi_B}_{\\psi_{AB}}\\label{9}\n\\end{align}\n where ${\\braket{{A}^{local}_w}}^{\\phi_A\\phi_B}_{\\psi_{AB}}$, ${\\braket{{B}^{local}_w}}^{\\phi_A\\phi_B}_{\\psi_{AB}}$ and ${\\braket{{B}^{local}_w}}^{\\phi^{\\perp}_{A}\\phi_B}_{\\psi_{AB}}$ are the ``local'' weak values and  $\\ket{\\phi^{\\perp}_{A}}=\\frac{1}{\\braket{\\Delta A}_{\\phi_A}}\\left(A - \\braket{A}_{\\phi_A}\\right)\\ket{\\phi_A}$ is given by  (\\ref{2}) for the subsystem A.\n \\begin{proof}\n\\begin{align}\n\\braket{({A}\\otimes {B})_w}^{\\phi_A\\phi_B}_{\\psi_{AB}}&=\\left(\\frac{\\braket{\\psi_{AB}|({A}\\otimes {B})|\\phi_A\\phi_B}}{\\braket{\\psi_{AB}|\\phi_A\\phi_B}}\\right)^\\ast\\nonumber\\\\\n&=\\left(\\frac{\\braket{\\psi_{AB}|(\\braket{A}_{\\phi_A}\\ket{\\phi_A} +\\braket{\\Delta A}_{\\phi_A}\\ket{\\phi_{A}^\\perp})\\otimes {B}|\\phi_B}}{\\braket{\\psi_{AB}|\\phi_A\\phi_B}}\\right)^\\ast\\nonumber\\\\\n&=\\braket{A}_{\\phi_A}\\frac{\\braket{\\phi_A\\phi_B|({I_A}\\otimes {B})|\\phi_{AB}}}{\\braket{\\phi_A\\phi_B|\\psi_{AB}}} + \\braket{\\Delta A}_{\\phi_A}\\frac{\\braket{\\phi^{\\perp}_A\\phi_B|({I_A}\\otimes {B})|\\phi_{AB}}}{\\braket{\\phi^{\\perp}_A\\phi_B|\\psi_{AB}}}\\frac{\\braket{\\phi^{\\perp}_A\\phi_B|\\psi_{AB}}}{\\braket{\\phi_A\\phi_B|\\psi_{AB}}}\\nonumber\\\\\n&=\\braket{A}_{\\phi_A}{\\braket{{B}^{local}_w}}^{\\phi_A\\phi_B}_{\\psi_{AB}} + {\\braket{{ B}^{local}_w}}^{\\phi^{\\perp}_A\\phi_B}_{\\psi_{AB}}\\left({\\braket{{A}^{local}_w}}^{\\phi_A\\phi_B}_{\\psi_{AB}} - \\braket{A}_{\\phi_A}\\right),\\nonumber\n\\end{align}\nwhere we have  used Eq. (\\ref{2}) in the second line for the subsystem A and Eq. (\\ref{8}) in the fourth line. After the manipulation, we have Eq. (\\ref{9}).\n\\end{proof}\n \\end{widetext}\\par\n\\emph{Product weak values in terms of local weak values:}  We have obtained a product weak value using only local weak values. Note that the local weak values ${\\braket{{B}^{local}_w}}^{\\phi_A\\phi_B}_{\\psi_{AB}}$ and ${\\braket{{ B}^{local}_w}}^{\\phi^{\\perp}_{A}\\phi_B}_{\\psi_{AB}}$ can be measured in the same experimental setup as the post-selected states $\\ket{\\phi_A}$ and $\\ket{\\phi^{\\perp}_{A}}$ are orthogonal to each other. We need another measurement setup for ${\\braket{{A}^{local}_w}}^{\\phi_A\\phi_B}_{\\psi_{AB}}$. So, effectively the total number of  measurements  to be performed according to the AAV method to extract the product weak value $\\braket{({A\\otimes B})_w}^{\\phi_A\\phi_B}_{\\psi_{AB}}$ is only two.  In experiment, local weak values like ${\\braket{{B}^{local}_w}}^{\\phi_A\\phi_B}_{\\psi_{AB}}$ can be realized as \n\\begin{align}\n  \\braket{B^{local}_w}^{\\phi_A\\phi_B}_{\\psi_{AB}}&=\\frac{\\braket{\\phi_B\\phi_A|(I_{A}\\otimes {B})|\\psi_{AB}}}{\\braket{\\phi_B\\phi_A|\\psi_{AB}}}\\label{10}\\\\\n  &=\\frac{\\braket{\\phi_B|{B}|\\psi^{\\phi_A}_B}}{\\braket{\\phi_B|\\psi_B^{\\phi_A}}},\\label{11}\n\\end{align}\nwhere $\\ket{\\psi_B^{\\phi_A}}=\\braket{\\phi_A|\\psi_{AB}}$ is an unnormalized state for the  subsystem B. That is, we first measure the projection operator $\\Pi_{\\phi_A}=\\ket{\\phi_A}\\bra{\\phi_A}$ on the subsystem A, then the  state of the subsystem B becomes $\\braket{\\phi_A|\\psi_{AB}}\/\\sqrt{\\braket{\\psi_B^{\\phi_A}|\\psi_B^{\\phi_A}}}$ which we consider to be pre-selected state for the subsystem B. This pre-selected state is unknown as $\\ket{\\psi_{AB}}$ is unkown. So, for each given projector related to  the subsystem A, there exists an unknown pre-selected state  of the subsystem B. The observable is $B$ and the post-selection is $\\ket{\\phi_B}$. So from Eq. (\\ref{11}), we  see that the local weak value $ \\braket{B^{local}_w}^{\\phi_A\\phi_B}_{\\psi_{AB}}$ can be realized  according to the AAV method related to the subsystem B. \\par\n\\emph{Product weak values for mixed pre-selected state:} If the pre selection of the bipartite system is a mixed state $\\rho_{AB}$ then, the product weak value of the observable ${A}\\otimes {B}$ is given by \n \\begin{align}\n  \\braket{({A}\\otimes {B})_w}^{\\phi_A\\phi_B}_{\\rho_{AB}}=\\frac{\\braket{\\phi_A\\phi_B|\\left({A}\\otimes {B}\\right)\\rho_{AB}|\\phi_A\\phi_B}}{\\braket{\\phi_A\\phi_B|\\rho_{AB}|\\phi_A\\phi_B}}.\\label{12}\n\\end{align}\nNote that generalization for mixed state of  Eq. (\\ref{9}) is not straightforward.\n  \\begin{widetext}\n \\emph{Result 3:-} The product weak value in Eq. (\\ref{12}) can be realized via local weak values as\n \\begin{align}\n\\braket{(A\\otimes B)_w}^{\\phi_A\\phi_B}_{\\rho_{AB}}=\\braket{A}_{\\phi_A}\\braket{B^{local}_w}^{\\phi_A\\phi_B}_{\\rho_{AB}} + \\frac{\\braket{\\Delta A}_{\\phi_A}}{2p(\\rho_{AB},\\phi_A\\phi_B)}\\sum_{i=1}^{m}\\bigg( &\\lambda^i_A \\braket{ B^{local}_w}^{i_A\\phi_B}_{\\rho_{AB}}p(\\rho_{AB},i_A\\phi_B) \\nonumber\\\\\n&+ {\\lambda^{\\prime}_A}^i \\braket{ B^{local}_w}^{{i^{\\prime}_A}\\phi_B}_{\\rho_{AB}}p(\\rho_{AB},{i^{\\prime}_A}\\phi_B) \\bigg),\\label{13}\n\\end{align}\nwhere \\{$\\lambda_A^i,\\ket{i_A}$\\} and \\{${\\lambda^{\\prime}_A}^i,\\ket{{i^{\\prime}_A}}$\\} satisfy the spectral decomposition for the  normal operators $\\ket{\\phi_A}\\bra{\\phi^{\\perp}_{A}} + \\ket{\\phi^{\\perp}_A}\\bra{\\phi_A}$ and $\\ket{\\phi_A}\\bra{\\phi^{\\perp}_{A}} - \\ket{\\phi^{\\perp}_A}\\bra{\\phi_A}$, respectively. $p(\\rho_{AB},\\phi_A\\phi_B)=\\braket{\\phi_A\\phi_B|\\rho_{AB}|\\phi_A\\phi_B}$  is the probability of obtaining the post-selected state $\\ket{\\phi_A\\phi_B}$ for  the given  pre selected state  $\\rho_{AB}$ and `$m$' is the  dimension of the  subsystem A.\\par\n \\begin{proof}\n \\begin{align}\n  \\braket{({A}\\otimes {B})_w}^{\\phi_A\\phi_B}_{\\rho_{AB}}&=\\frac{\\braket{\\phi_A\\phi_B|\\left({A}\\otimes {B}\\right)\\rho_{AB}|\\phi_A\\phi_B}}{\\braket{\\phi_A\\phi_B|\\rho_{AB}|\\phi_A\\phi_B}}\\nonumber\\\\\n  &=\\braket{A}_{\\phi_A}\\frac{\\braket{\\phi_A\\phi_B|\\left({I_A}\\otimes {B}\\right)\\rho_{AB}|\\phi_A\\phi_B}}{\\braket{\\phi_A\\phi_B|\\rho_{AB}|\\phi_A\\phi_B}} + \\braket{\\Delta A}_{\\phi_A}\\frac{\\braket{\\phi^{\\perp}_A\\phi_B|\\left({I_A}\\otimes {B}\\right)\\rho_{AB}|\\phi_A\\phi_B}}{\\braket{\\phi_A\\phi_B|\\rho_{AB}|\\phi_A\\phi_B}},\\label{14}\n\\end{align}\nwhere we have used Eq. (\\ref{2}). Now $\\braket{\\phi^{\\perp}_A\\phi_B|\\left({I_A}\\otimes {B}\\right)\\rho_{AB}|\\phi_A\\phi_B}=Tr\\left[\\left(\\ket{\\phi_A}\\bra{\\phi^{\\perp}_{A}}\\otimes \\ket{\\phi_B}\\bra{\\phi_{B}}\\right)(I_A\\otimes B)\\rho_{AB}\\right]$ can be calculated as \n\\begin{align}\n\\braket{\\phi^{\\perp}_A\\phi_B|\\left({I_A}\\!\\otimes \\!{B}\\right)\\rho_{AB}|\\phi_A\\phi_B} + \\braket{\\phi_A\\phi_B|\\hspace{-2pt}\\left({I_A}\\!\\otimes\\! {B}\\right)\\rho_{AB}|\\phi^{\\perp}_A\\phi_B}\\hspace{-2pt}=\\hspace{-2pt}Tr\\hspace{-2pt}\\left[\\left(\\{\\ket{\\phi_A}\\bra{\\phi^{\\perp}_{A}} + \\ket{\\phi^{\\perp}_A}\\bra{\\phi_{A}}\\}\\otimes \\ket{\\phi_B}\\bra{\\phi_{B}}\\right)\\hspace{-2pt}\\left(I_A\\otimes B\\right)\\hspace{-2pt}\\rho_{AB}\\right],\\label{15}\n\\end{align}\nwhere $\\ket{\\phi_A}\\bra{\\phi^{\\perp}_{A}} + \\ket{\\phi^{\\perp}_A}\\bra{\\phi_{A}}$ is a normal operator satisfying $XX^{\\dagger}=X^{\\dagger}X$ (where X is any operator)  and hence can be  written in spectral decomposition \n\\begin{align}\n\\ket{\\phi_A}\\bra{\\phi^{\\perp}_{A}} + \\ket{\\phi^{\\perp}_A}\\bra{\\phi_A}=\\sum_i^{m}{\\lambda_A^i \\ket{i_A}\\bra{i_A}},\\label{16}\n\\end{align}\nwhere \\{$\\ket{i_A}$\\} is the set of  eigenvectors with eigenvalues \\{$\\lambda_A^i$\\}. Using Eq. (\\ref{16}) in (\\ref{15}), we have\n\\begin{align}\n\\braket{\\phi^{\\perp}_A\\phi_B|\\left({I_A}\\otimes {B}\\right)\\rho_{AB}|\\phi_A\\phi_B} + \\braket{\\phi_A\\phi_B|\\left({I_A}\\otimes {B}\\right)\\rho_{AB}|\\phi^{\\perp}_A\\phi_B}&=\\sum_i^{m}{\\lambda_A^i\\braket{i_A\\phi_B|(I_B\\otimes B)\\rho_{AB}|i_A\\phi_B}}\\nonumber\\\\\n&=\\sum_i^{m}{\\lambda_A^i\\braket{B^{local}_w}^{i_A\\phi_B}_{\\rho_{AB}}p(\\rho_{AB},i_A\\phi_B)},\\label{17}\n\\end{align}\nwhere we have used the definition of  the weak value   $\\braket{ B_w}^{i_A\\phi_B}_{\\rho_{AB}}$ of the observable $B$  for the given pre and post-selections $\\rho_{AB}$ and $\\ket{i_A\\phi_B}$, respectively.  $p(\\rho_{AB},i_A\\phi_B)=\\braket{i_A\\phi_B|\\rho_{AB}|i_A\\phi_B}$ is the probability of obtaining the product state $\\ket{i_A\\phi_B}$. Now similarly,\n\\begin{align}\n\\braket{\\phi^{\\perp}_A\\phi_B|\\left({I_A}\\otimes {B}\\right)\\rho_{AB}|\\phi_A\\phi_B} - \\braket{\\phi_A\\phi_B|\\left({I_A}\\otimes {B}\\right)\\rho_{AB}|\\phi^{\\perp}_A\\phi_B}=\\sum_i^{m}{{\\lambda^{\\prime}_A}^i\\braket{B^{local}_w}^{i^{\\prime}_A}_{\\rho_{AB}}p(\\rho_{AB},{i^{\\prime}_A}\\phi_B)},\\label{18}\n\\end{align}\nwhere we have used the fact that $\\ket{\\phi_A}\\bra{\\phi^{\\perp}_{A}} - \\ket{\\phi^{\\perp}_A}\\bra{\\phi_A}$ is also a normal operator with the spectral decomposition \n\\begin{align}\n\\ket{\\phi_A}\\bra{\\phi^{\\perp}_{A}} - \\ket{\\phi^{\\perp}_A}\\bra{\\phi_A}=\\sum_i^{m}{{\\lambda^{\\prime}_A}^i \\ket{i^{\\prime}_A}\\bra{{i^{\\prime}_A}}}.\\label{19}\n\\end{align}\nNow adding two equations (\\ref{17}) and (\\ref{18}), we have \n\\begin{align}\n\\braket{\\phi^{\\perp}_A\\phi_B|\\left({I_A}\\otimes {B}\\right)\\rho_{AB}|\\phi_A\\phi_B}=\\frac{1}{2}\\sum_i^{m}{\\left(\\lambda_A^i\\braket{ B^{local}_w}^{i_A\\phi_B}_{\\rho_{AB}}p(\\rho_{AB},i_A\\phi_B) + {\\lambda^{\\prime}_A}^i \\braket{B^{local}_w}^{i^{\\prime}_A\\phi_B}_{\\rho_{AB}}p(\\rho_{AB},{i^{\\prime}_A}\\phi_B)\\right)}.\\label{20}\n\\end{align}\nUsing Eq. (\\ref{20}) in (\\ref{14}), we obtain Eq. (\\ref{13}).\n\\end{proof}\n\\end{widetext}\\par\n\\emph{Product weak values in terms of local weak values:}  The product weak value $\\braket{(A\\otimes B)_w}^{\\phi_A\\phi_B}_{\\rho_{AB}}$ consists the local weak values $\\braket{ B^{local}_w}^{\\phi_A\\phi_B}_{\\rho_{AB}}$,  $\\{\\braket{B^{local}_w}^{i_A\\phi_B}_{\\rho_{AB}}\\}^{m}_{i=1}$ and $\\{\\braket{B^{local}_w}^{i^{\\prime}_A\\phi_B}_{\\rho_{AB}}\\}^{m}_{i=1}$. Here `$m$' is the dimension of the subsystem A. Note that, $\\{\\braket{B^{local}_w}^{i_A\\phi_B}_{\\rho_{AB}}\\}^{m}_{i=1}$ can be measured in the same experimental setup according to the AAV method as the post-selections $\\{\\ket{i_A}\\}$ form complete set of orthogonal basis. Similarly  $\\{\\braket{B^{local}_w}^{{i^{\\prime}_A}\\phi_B}_{\\rho_{AB}}\\}^{m}_{i=1}$ can also be measured within the same experiment with the complete set of post-selections $\\{\\ket{i^{\\prime}_A}\\}$ according to the AAV method. \\emph{So, the total number of measurements to be performed according to the AAV method to extract  $\\braket{(A\\otimes B)_w}^{\\phi_A\\phi_B}_{\\rho_{AB}}$ is  only three}. Local weak values can be realized in the same way as discussed above in Eq. (\\ref{11}).\\par\n\\emph{Comparison with previous works:} In the Ref. {\\cite{27,38}}, authors have shown that it is also possible to extract the product weak values  by obtaining  local weak values of the observables separately as well as by looking into different  pointers' position and momentum correlations i.e., statistical averages of these different degrees of freedom. There are some cases where polarization degrees of freedom (polarization correlations) was considered instead of position and momentum variables \\cite{2}. For such cases statistical averages of those polarization degrees of freedom can be difficult to obtain or number of measurements will be large. Instead, our approach shows that  one needs to obtain only local weak values. Our approach is not necessarily  restricted with Gaussian pointer states. But in most of the previous works \\cite{27,38,46}, different type  of pointer states have been used with certain constraints unlike in our case. \\par \nOur methods are  easy to perform in experiments due to their local realizations. While in the previous works  \\cite{27,38,42,46} i.e., extraction of higher moment weak values and product weak values, there are N pointers' correlations to be measured. So the scalability of their methods face significant challenges. Moreover, from an experimental perspective, their schemes are hard to implement as the resources required to observe the correlations are of second order in terms  of the interaction coefficient.\\par\nIn the next section, we discuss about the applications of our results to reconstruct unknown quantum states in a single and bipartite systems.\n\\section{Quantum State Tomography} \\label{e III}\nIn the following,  we discuss some methods of  quantum state tomography of a single and bipartite system using  higher moments weak values and product weak values.\n{\\subsection{ State reconstruction of a single system}} \n\\subsubsection{Pure state}\nAs an application of higher moment weak values we reconstruct a  pure state. The method of  quantum state reconstruction using weak values was introduced by Lundeen \\emph{et al}. \\cite{lundeennature} as follows. Any  state can be written in computational basis $\\{\\ket{i}\\}$ as \n\\begin{align}\n\\ket{\\psi}=\\sum_{i=0}{\\alpha_i\\ket{i}},\\label{21}\n\\end{align}\nwhere $\\alpha_i=\\braket{i|\\psi}$. Now the weak value of a projection operator $\\Pi_{i}=\\ket{i}\\bra{i}$ is given by \n\\begin{align}\n\\braket{(\\Pi_{i})_w}_{\\psi}^{b}=\\frac{\\braket{b|{i}}\\braket{i|\\psi}}{\\braket{b|\\psi}},\\label{22}\n\\end{align}\nwith $\\braket{b|i}\\neq0$. where $\\ket{b}$ is a post-selection. So using Eq. (\\ref{22}), we finally construct the pure state Eq. (\\ref{21})\n\\begin{align}\n\\ket{\\psi}=\\sum_i{\\frac{\\braket{(\\Pi_{i})_w}_{\\psi}^{b}}{\\braket{b|{i}}}\\ket{i}}.\\label{23}\n\\end{align}\nThe complex number $\\braket{b|\\psi}$ is not taken into account as it corresponds to the global phase factor after normalization. So to measure a  pure state, we need to obtain weak values of the projection operators $\\Pi_i$ with pre selection $\\ket{\\psi}$ and post-selection $\\ket{b}$, respectively.\\par\nInstead of measuring weak values of the  projection operators individually, we want to use the higher moment weak values of the observable which satisfies spectral decomposition with those projection operators and using those higher moment weak values we will obtain weak values of the projection operators. Let the observable  be  \n\\begin{align}\nA=\\sum_i{a_i\\Pi_{i}},\\label{24}\n\\end{align}\nwhere $a_i$ are the eigenvalues of the observable $A$. Now the weak value and higher moment weak values of the observable are \n \\begin{align}\n\\braket{A_w}_{\\psi}^{b}=\\sum_i{a_i\\braket{(\\Pi_{i})_w}_{\\psi}^{b}},\\label{25}\\\\\n\\braket{A^n_w}_{\\psi}^{b}=\\sum_i{a^n_i\\braket{(\\Pi_{i})_w}_{\\psi}^{b}},\\label{26}\n\\end{align}\nwhere `$n$' is any positive integer. Eqs. (\\ref{25}) and (\\ref{26}) can be solved to obtain the weak values of projection operators for different `$n$'. For example in three dimensional Hilbert Space, we require only up to second moment weak values because one can use the completeness relation for the projection operators with pre selection $\\ket{\\psi}$ and post-selection $\\ket{b}$ \n \\begin{align}\n1=\\sum_i{\\braket{(\\Pi_{i})_w}_{\\psi}^{b}}.\\label{27}\n\\end{align}\nIn Appendix \\ref{C}, we explicitly show  how to solve the above equations to obtain the weak values of the projection operators.\\par\nFrom (\\ref{C1}), the highest moment weak value is $\\braket{A^{d-1}_w}_{\\psi}^{b}$ and as we have discussed in (\\ref{II A}) that, extracting the highest moment weak value is enough to calculate all the lower moments weak values. \\emph{Hence the total number of measurements needed to reconstruct a pure state is ${d}\/{2}$ if the dimension $d$ is even and $({d-1})\/{2}$ if the dimension $d$ is odd} (see \\ref{II A} for detail discussion). \\par\nTo compare with Lundeen \\emph{et al.} \\cite{29} and Wu \\cite{47}, the number of measurement operators which is the complete set of projection operators  with a fixed post-selection is $(d-1)$. Moreover, we measure only one  system operator $A$, but post-selections are to be changed, while, in their method, there are $(d-1)$  system operators (projection operators) to be measured according to the AAV method.\\par\nNote that the weak values of projection operators in Eq. (\\ref{22}) are exactly the weak-valued probabilities which were mentioned in the introduction section. \\par\n\\normalsize{\\textit{Alternative:\u2013}}\nThe weak value of an observable $C$ with pre and post-selections $\\ket{\\psi}$ and $\\ket{0}$, respectively  is\n\\begin{align}\n\\braket{C_w}_{\\psi}^{0}=\\frac{\\braket{0|C|\\psi}}{\\braket{0|\\psi}}.\\label{28}\n\\end{align}\nInserting the identity operator $I=\\sum_{i}{\\ket{i}\\bra{i}}$ in numerator of the right hand side of Eq. (\\ref{28}), we have\n\\begin{align}\n\\braket{C_w}_{\\psi}^{0} - C_{00}=\\sum_{i=1}{C_{0i}\\frac{\\alpha_i}{\\alpha_0}},\\label{29}\n\\end{align}\nwhere $C_{0i}=\\braket{0|C|i}$. Like Eq. (\\ref{29}), we have to measure a set of observables   to obtain the values of $\\alpha_1\/\\alpha_0$,  $\\alpha_2\/\\alpha_0$,$\\cdots$,$\\alpha_{(d-1)}\/\\alpha_0$  (see Appendix \\ref{C}). The value of $\\alpha_0$ can be obtained from the normalization condition. Here, the number of measurement operators is $(d-1)$. This method will be used to reconstruct an unknown quantum pure state of a bipartite system.\n\\subsubsection{Mixed state}\nMeasurement of a mixed state of a quantum system was also  introduced by  Lundeen \\emph{et al.} \\cite{4} using product weak values of two non commuting projection operators. It was later simplified by Shengjun Wu \\cite{47} where weak values of complete set of  projection operators with complete set of post-selected states have been used.  Here, we develop another method by using the weak values of arbitrary observables which can be thought as the generalization of the reference  \\cite{47}. This part is added here because we will use the same procedure while dealing with bipartite mixed state reconstruction using product weak values.\\par\nLet the unknown mixed state be the pre-selected state  then the weak value of the observable $C$ with post-selection $\\ket{j}$ is \n\\begin{align}\n\\braket{C_w}_{\\rho}^{j}=\\frac{\\braket{j|C\\rho|j}}{\\braket{j|\\rho|j}}.\\label{30}\n\\end{align}\nInserting the identity operator $I=\\sum_{i}{\\ket{i}\\bra{i}}$, we have\n\\begin{align}\np(\\rho,j)\\braket{C_w}_{\\rho}^{j}=\\sum_i{C_{ji}\\rho_{ij}},\\label{31}\n\\end{align}\nwhere $p(\\rho,j)=\\braket{j|\\rho|j}$ is the probability of obtaining the basis state $\\ket{j}$ as a post-selection and $C_{ji}=\\braket{j|C|i}$, $\\rho_{ij}=\\braket{i|\\rho|j}$. To obtain the $j$-th column  of the density matrix $\\rho$ from Eq. (\\ref{31}), we need to measure a set of  arbitrary observables according to the AAV method to get a set of equations like  (\\ref{31}) (see Appendix \\ref{D}). \\par\nTo compare with the work by Lundeen \\emph{et al.} \\cite{4} where each matrix element is directly obtainable via sequential measurements of two non-commuting projection operators in the AAV method. Different combinations of  position and momentum correlations are to be measured  where the correlations are of second order in terms  of the interaction coefficient. Our method is more efficient as we  only need to measure $(d-1)$ arbitrary  single observables  according to AAV method. We do not discard any post-measurements data and thus reduces the number of experimental runs (see Appendix \\ref{D}). \\par\nWeak measurement methods have  several key advantages for state reconstruction of a quantum system  over the  standard schemes \\cite{4}. For example, the state disturbance is minimum and thus it is possible for characterization of the state of a system during a physical  process in an experiment. Unlike standard schemes, global reconstruction is not required by our methods as states can be determined locally i.e., each matrix element can directly be accessed. Standard scheme typically requires $\\mathcal{O}(d^2)$ measurements, while our method require  $\\mathcal{O}(d-1)$ measurements for mixed state reconstruction.\\par\nRecently Vallone \\emph{et al.} \\cite{48} have shown: ``Strong Measurements Give a Better Direct Measurement of the Quantum Wave Function\".  Namely, they have considered the von Neumannn Hamiltonian with basis projection operator (of the system) and Pauli operators (of the two dimensional pointer) and coupling coefficient (without approximation).  After the evolution of the system and the pointer,  the system is projected in the uniform superposition of the basis states. After that, each wavefunctions or basis coefficients of the concerned system state are calculated using the experimental probabilities obtained from the measurement of the  two dimensional pointer's different observables.    To compare,\\\\\n{(i) In our method, the state disturbance is minimum. While in  the work of Vallone \\emph{et al.} \\cite{48}, the system will be disturbed strongly.} \\\\\n(ii)  In the  method of Vallone \\emph{et al.} \\cite{48},   the post-selection of the system has to be of particular forms  to make the scheme successful otherwise (a)  the systematic error (trace distance) will be independent of the interaction coefficient which is one of their main concerns in the scheme (b)  wavefunctions for each computational basis will diverge and hence the scheme will fail.  Dimension of the pointer's Hilbert space is considered to  be two dimensional (finite dimensional). By such restrictions, the method  can only be used for limited number of quantum systems  (e.g., optical systems). While in our methods, there are no such restrictions on pointer states. The most suitable ways can be applied to obtain single weak values. \\\\\n(iii) (a) In the  method of Vallone  \\emph{et al.} \\cite{48}, for pure state reconstruction of a single system, there are effectively `$d-1$' number of projection operators and for each projection operator, three different measurement observables are needed. So, the total number of measurement settings is `$3(d-1)$'. While in our method, the measurement settings are nearly `$d\/2$' using higher moment weak values. (b) For the mixed state reconstruction, they need two independent pointers, three different joint pointer operators  and `$d-1$' number of projection operators and one post-selection in the system \\cite{49}. Using such combinations, one need to calculate  mean values of  different combination of tripartite observables. In our method, `$d-1$' number of weak values  are required and there are no  such joint operations. \\\\\n(iv)  Vallone  \\emph{et al.} \\cite{48}  have shown  that strong measurements outperform weak measurements  in both the ``precision and accuracy\" for arbitrary quantum states in most cases. In our case, by performing the experiment many times on identically prepared systems, it is possible to reduce the uncertainty in the mean pointer displacement to any arbitrary precision \\cite{50} (in order to obtain the  real and imaginary part of the weak values)\\\\\n(v) For the given finite ensemble  size, our scheme can't give better performance than the methods of  \\cite{48,49}  in terms of precision and accuracy \\cite{51,52}. \n\\subsection{State reconstruction of a bipartite system}\n\\subsubsection{Pure state}\nIn the above,  we introduced reconstruction of an unknown pure state of a given  system using single observable (or projection) weak values. For the  reconstruction of a bipartite  state, one needs to measure product  weak values namely the weak values of the tensor product observables.  In standard scheme i.e., von Neumann measurement scheme, measurement of product observables can not be realized directly  as it requires the interaction Hamiltonian of the two distant subsystems to be of the form $H\\propto (A\\otimes B)$  which implies an instantaneous interaction between the two distant subsystems (a relativistic constraint). In this section, we use our version of  product weak values (\\ref{9}) in a bipartite system to  reconstruct  a    pure  state  following the same method of Eq. (\\ref{29}) as we saw for the   pure state case in a single quantum system.\\par\nThe pure state of a bipartite system can be written in computational basis as\n\\begin{align}\n\\ket{\\psi_{AB}}=\\sum_{ij}{\\alpha_{ij}\\ket{i_A}\\otimes\\ket{j_B}},\\label{32}\n\\end{align}\nwhere $\\alpha_{ij}=\\braket{ij|\\psi_{AB}}$ and $\\ket{ij}=\\ket{i_A}\\otimes\\ket{j_B}$. \n The product  weak value of the  observable  $C_{A}\\otimes C_{B}$ in a bipartite system  is given by \n\\begin{align}\n  \\braket{(C_{A}\\otimes C_{B})_w}^{00}_{\\psi_{AB}}=\\frac{\\braket{00|(C_{A}\\otimes C_{B})|\\psi_{AB}}}{\\braket{00|\\psi_{AB}}},\\label{33}\n\\end{align}\nwhere $\\ket{\\psi_{AB}}$ is the bipartite pre selected state which is to be reconstructed. $\\ket{00}=\\ket{0_A}\\otimes\\ket{0_B}$ is the post-selected state. Now inserting identity operator $I=\\sum_{ij}{\\ket{ij}\\bra{ij}}$ of the joint Hilbert space in Eq. (\\ref{33}),  we have\n\\begin{align}\n\\braket{( \\hspace{-1pt}C_{A}\\!\\otimes \\!C_{B} \\hspace{-1pt})_w}^{00}_{\\psi_{AB}} \\hspace{-4pt}-  \\hspace{-2pt}\\big[C_{A}\\big]_{00} \\hspace{-1pt}\\big[C_{B}\\big]_{00} \\hspace{-2pt}= \\hspace{-8pt}\\sum_{i,j\\neq (0,0)} \\hspace{-10pt}\\big[C_{A}\\big]_{0i} \\hspace{-1pt}\\big[C_{B}\\big]_{0j}\\frac{\\alpha_{ij}}{\\alpha_{00}}, \\label{34}\n\\end{align}\n where  $\\left[C_{A}\\right]_{0i}\\left[C_{B}\\right]_{0j}=\\braket{00|C_{A}\\otimes C_{B}|ij}$ and $\\alpha_{00}\\neq 0$. Again we have to solve a matrix equation using  Eq. (\\ref{34}) for a set of product operators  to obtain the values $\\frac{\\alpha_{ij}}{\\alpha_{00}}$ (see Appendix \\ref{E}) .\\par\nWe have found  using the matrix equation (\\ref{E1}) that we do not require to measure all the product weak values. For example,  consider a two-qubit system where \n\\begin{equation}\n\\begin{split}\nC^{(1)}_{A}\\otimes C^{(1)}_{B}&=I^A\\otimes \\sigma_x^B ,\\hspace{2mm}  C^{(2)}_{A}\\otimes C^{(2)}_{B}=\\sigma_x^A\\otimes I^B\\label{35}\\\\[10pt]\n&C^{(3)}_{A}\\otimes C^{(3)}_{B}=\\sigma_x^A\\otimes \\sigma_x^B,\n\\end{split}\n\\end{equation}\nthen the square matrix in  (\\ref{E1}) becomes an identity matrix having nonzero determinant and hence all the coefficients can be determined. \\emph{This way, the number of measurements of product weak values can be reduced}. In this particular case, we need only one product weak value i.e., $\\braket{(\\sigma_x^A\\otimes \\sigma_x^B)_w}^{00}_{\\psi_{AB}}$ to be measured and according to Eq. (\\ref{9}), it can be calculated using local weak values $\\braket{ (\\sigma_x^B)_w}^{00}_{\\psi_{AB}}$, $\\braket{(\\sigma_x^B)_w}^{10}_{\\psi_{AB}}$ and $\\braket{(\\sigma_x^A)_w}^{00}_{\\psi_{AB}}$.   So the total number of local weak values  is only three    to reconstruct two-qubit  pure state.  Note that, the local weak value  $\\braket{(\\sigma_x^B)_w}^{10}_{\\psi_{AB}}$ can be calculated by using  $\\braket{(\\sigma_x^B)_w}^{00}_{\\psi_{AB}}$ with the completeness relation for $\\ket{0}$ and $\\ket{1}$.\\par\nTo compare with the method of Pan \\emph{et al.} \\cite{5}, our method  is experimentally simple because it  does not depend on the nature of  the pointer's state (entangled or product) and  locally measurable (using local weak values only) while in their method, the use of  entangled pointer's states are necessary (which might not be an easy task to perform) and local modular values as well as  modular values of the sum of the local operators are required. For certain cases, some of the probability amplitudes with the entangled pointer's states are considered to be sufficiently small.  Complicated situations may arise for higher dimensions and multi-partite systems because of the entangled pointer states. Our method can be generalized  both in higher dimensions and multi-partite systems with local weak value measurements only. In the method of  Pan \\emph{et al.} \\cite{5},  The number of product weak values is $(m-1)(n-1)$ and each product weak value  consists one  modular value of the sum of the two local projectors as well as two local projector modular values. Here `$m$' and `$n$' are the number of dimensions of the subsystems A and B, respectively. \\emph{In our  method, there are  $(d-1)$ (where $d=mn$) numbers of product weak values and each product weak value can be extracted with only  two numbers of AAV type weak measurements}. But as we have seen for the case of two-qubit system (\\ref{35}), we do not need to calculate  $(d-1)$ numbers of product weak values all the time. For example, effectively we need only two local weak values to reconstruct the pure state of the two-qubit system. Note that, in Ref. \\cite{5} for two-qubit system, the total number of measurements is three in which one is the  modular value of the sum of two local projectors and two local projector modular values. So, in most of the cases, it is  possible to reduce  the  number of  product weak values  considerably  in our method of  state reconstruction.\n\\subsubsection{Mixed state}{}\nTo reconstruct a mixed state of a bipartite system, we will use the method of Eq. (\\ref{31}).  Now, let the pre-selection of the system be $\\rho_{AB}$ which is unknown and post-selection be any computational  basis state $\\ket{kl}$. Then the product weak value of the   operator $C_A\\otimes C_B$ is given by\n\\begin{align}\n  \\braket{(C_{A}\\otimes C_{B})_w}^{kl}_{\\rho_{AB}}=\\frac{\\braket{kl|\\left(C_{A}\\otimes C_{B}\\right)\\rho_{AB}|kl}}{\\braket{kl|\\rho_{AB}|kl}}.\\label{36}\n\\end{align}\nNow inserting the identity operator $I=\\sum_{i,j}{\\ket{ij}\\bra{ij}}$ in Eq. (\\ref{36}), we have\n\\begin{align}\n  p(\\rho_{AB},kl) \\hspace{-2pt}\\braket{( \\hspace{-1pt}C_{A}\\!\\otimes\\! C_{B} \\hspace{-1pt})_w}^{kl}_{\\rho_{AB}} \\hspace{-2pt}= \\hspace{-2pt}\\sum_{i,j}\\left[C_{A}\\right]_{ki} \\hspace{-1pt}\\left[C_{B}\\right]_{lj} \\hspace{-1pt}[\\rho_{AB}]_{ij,kl},\\label{37}\n\\end{align}\nwhere $\\left[C_{A}\\right]_{ki}\\left[C_{B}\\right]_{lj}=\\braket{kl|C_{A}\\otimes C_{B}|ij}$, $[\\rho_{AB}]_{ij,kl}=\\braket{ij|\\rho_{AB}|kl}$ and $p(\\rho_{AB},kl)=\\braket{kl|\\rho_{AB}|kl}$ is the probability of the successful post-selection $\\ket{kl}$.\nSo to obtain the \\emph{kl}-th column of the density matrix $\\rho_{AB}$, we have to form a matrix equation using  Eq. (\\ref{37}) for a set of product operators (see Appendix \\ref{F}).\\par\nClearly, mixed state reconstruction is more resource intensive than the pure state case in a bipartite system.  The number of product weak values to be calculated here is ($d-1$) and each product weak value can be extracted with only  three numbers of AAV type weak  measurements (see \\ref{II B}). Here $d=m n$, where `m' and `n' are dimensions of the subsystems A and B, respectively.  \\emph{We will  get advantage of using  matrix equation} (\\ref{ F1}) \\emph{where for some cases we do not require to calculate all the product weak values as we have seen  for the case of bipartite pure state reconstruction}. \\par\nIt is  important to note  that, our method of calculating product weak values for pure and mixed states in a bipartite system can also be applied for  projection operators and hence one can reconstruct  pure  state using the state reconstruction method of Ref. \\cite{5}. For mixed state reconstruction, one should look to the  Ref. \\cite{47} by considering   bipartite system conditions.\\par\nFull knowledge of the state of a quantum system is always crucial  to understand a system better and for controlling quantum technologies. In particular, the measurement of bipartite (multi-partite) states are useful for information transfer, cryptography protocols, etc. They are also used to study nonlocality, quantum discord, entanglement entropy, etc.  We have shown the application of product and higher moment weak values as quantum state reconstruction of a single and bipartite systems only. The calculations of product weak values  of a bipartite system  are  even more  fascinating because of their local realizations. We can have  applications  of  product weak values  to extract informations about multi-partite systems for future technologies. Product weak values with local realization can find it's applications in quantum steering, to perform some nonlocal tasks, etc.\n\\section{Entanglement detection}\\label{e IV}\nDue to an immense application of  entangled systems \\cite{53,54}, it is, by default,   an  important task in the field of quantum information to detect whether the shared states are entangled or not.  Here we show that product weak values (introduced in sec. \\ref{e II}) can be used to detect entanglement of a bipartite system's state. Product weak values are experimentally accessible quantities and we have discussed in sec. \\ref{e II} how one can do  that.    We have found  a necessary separability criteria for finite-dimensional systems.  By  clever choices of product observable and post-selections, it is possible to achieve the PPT criteria for entanglement detection of several important class of entangled bipartite states. Our method of entanglement detection can definitely be used for more class of entangled states.\\par\n There are some existing  necessary separability criteria \\cite{53,54} for detection of entanglement for finite-dimensional systems  based on local uncertainty relation (standard deviation based) \\cite{55}, entropic uncertainty relations \\cite{56}, separability inequalities on Bell correlations \\cite{57} (which are exponentially stronger than the corresponding  local reality inequalities), etc. It is worth mentioning here that  Uffink and Seevink  provided  a single separability inequality \\cite{58}, (although the choice of the observables being state-dependent)  quadratic in nature  is used to detect separability \/ entanglement  of all two-qubit states. \\par\nThe separable states are considered to be of the following form \n\\begin{align}\n\\rho=\\sum_i{p_i\\rho_A^i\\otimes \\rho_B^i}\\label{38},\n\\end{align}\nwhere $\\rho_A^i=\\ket{\\psi_A^i}\\bra{\\psi_A^i}$, $\\rho_B^i=\\ket{\\psi_B^i}\\bra{\\psi_B^i}$ and $\\sum_i p_i=1$.  We will consider the following quantity which is directly connected to the product weak value (\\ref{12}) for mixed states\n\\begin{widetext}\n\\begin{align}\n&\\left| \\braket{\\phi_A\\phi_B|(A\\otimes B)\\rho|\\phi_A\\phi_B} \\right|^2\\nonumber\\\\\n&=\\left| \\sum_i p_i\\braket{\\phi_A|A\\rho_A^i|\\phi_A} \\braket{\\phi_B|B\\rho_B^i|\\phi_B} \\right|^2\\nonumber\\\\\n&=\\left| \\sum_i \\left\\{\\sqrt{p_i}\\frac{\\braket{\\phi_A|A\\rho_A^i|\\phi_A}}{\\sqrt{\\braket{\\phi_A|\\rho_A^i|\\phi_A}}}\\sqrt{\\braket{\\phi_B|\\rho_B^i|\\phi_B}}\\right\\} \\left\\{\\sqrt{p_i}{\\sqrt{\\braket{\\phi_A|\\rho_A^i|\\phi_A}}} \\frac{\\braket{\\phi_B|B\\rho_B^i|\\phi_B}}{\\sqrt{\\braket{\\phi_B|\\rho_B^i|\\phi_B}}}\\right\\}\\right|^2\\nonumber\\\\\n& \\leq \\left( \\sum_i p_i\\frac{\\left|\\braket{\\phi_A|A\\rho_A^i|\\phi_A}\\right|^2}{\\braket{\\phi_A|\\rho_A^i|\\phi_A}}\\braket{\\phi_B|\\rho_B^i|\\phi_B} \\right)\\left( \\sum_i p_i\\braket{\\phi_A|\\rho_A^i|\\phi_A}\\frac{\\left|\\braket{\\phi_B|B\\rho_B^i|\\phi_B}\\right|^2}{\\braket{\\phi_B|\\rho_B^i|\\phi_B}} \\right)\\nonumber\\\\\n&=\\left( \\sum_i p_i\\braket{\\phi_A|A|\\psi_A^i}{\\braket{\\psi_A^i|A|\\phi_A}}\\braket{\\phi_B|\\psi_B^i}\\braket{\\psi_B^i|\\phi_B}\\right)\\left( \\sum_i p_i\\braket{\\phi_A|\\psi_A^i}{\\braket{\\psi_A^i|\\phi_A}}\\braket{\\phi_B|B|\\psi_B^i}\\braket{\\psi_B^i|B|\\phi_B}\\right)\\nonumber\\\\\n&=\\braket{\\phi_A\\phi_B|(A\\otimes I)\\rho(A\\otimes I)|\\phi_A\\phi_B}\\braket{\\phi_A\\phi_B|(I\\otimes B)\\rho(I\\otimes B)|\\phi_A\\phi_B}\\nonumber\\\\\n&=\\braket{\\phi_A|A\\rho_A^{\\phi_B}A|\\phi_A}\\braket{\\phi_B|B\\rho_B^{\\phi_A}B|\\phi_B}\\nonumber\\\\\n&=\\braket{\\phi_A|A^2|\\phi_A}\\braket{\\phi_B|B^2|\\phi_B}\\braket{\\phi_A^{\\prime}|\\rho_A^{\\phi_B}|\\phi_A^{\\prime}}\\braket{\\phi_B^{\\prime}|\\rho_B^{\\phi_A}|\\phi_B^{\\prime}}\\label{39},\n\\end{align}\nwhere we have applied the Cauchy-schwarz inequality,  $\\rho_X^{\\phi_Y}=\\braket{\\phi_Y|\\rho|\\phi_Y}$ and $\\ket{\\phi_X^{\\prime}}=X\\ket{\\phi_X}\/\\sqrt{\\braket{\\phi_X|X^2|\\phi_X}}$ with $X\\neq Y$ and X,Y=\\{A,B\\}.  The quantity $\\braket{\\phi_X^{\\prime}|\\rho_X^{\\phi_Y}|\\phi_X^{\\prime}}$ can experimentally be obtained  in the following way.  At the first stage, measure the projection operator $\\Pi_{\\phi_Y}=\\ket{\\phi_Y}\\bra{\\phi_Y}$ in the subsystem `Y' on the shared bipartite state $\\rho$. The collapsed state in the subsystem `X' is now $\\rho_X^{\\phi_Y}$, which is also the prepared state in this subsystem. At the second stage, measure the projection operator $\\Pi_{\\phi_X^{\\prime}}=\\ket{\\phi_X^{\\prime}}\\bra{\\phi_X^{\\prime}}$  in the subsystem `X'. Note that, $\\braket{\\phi_X|X^2|\\phi_X}$ can be obtained by just knowing the matrix form of the operator $X$ and $\\ket{\\phi_X}$.\n\\end{widetext}\\par\nThe violation of the above inequality will imply entanglement of the given bipartite state.  Now, the following  examples will show the potential of the above inequality to detect entanglement of certain class of entangled states. \\\\\n\\textit{(i)} \\textit{Two-qubit Werner state} (noisy singlet):\n\\begin{align}\n\\rho=p\\ket{\\psi_{AB}^-}\\bra{\\psi_{AB}^-}+(1-p)\\frac{I_A\\otimes I_B}{4},\\nonumber\n\\end{align}\nwhere $\\ket{\\psi_{AB}^-}=\\frac{1}{\\sqrt{2}}(\\ket{01}-\\ket{10})$. By choosing $A=\\sigma^x_A$, $B=\\sigma^x_B$, $\\ket{\\phi_A}=\\ket{1}$ and $\\ket{\\phi_B}=\\ket{0}$, it can be shown that the inequality is violated for $p>1\/3$ (PPT criterion).\\\\\n\\textit{(ii)} \\textit{Mixture of two Bell states}:\n\\begin{align}\n\\rho=p\\ket{\\phi_{AB}^+}\\bra{\\phi_{AB}^+}+(1-p)\\ket{\\phi_{AB}^-}\\bra{\\phi_{AB}^-},\\nonumber\n\\end{align}\nwhere $\\ket{\\phi_{AB}^+}=\\frac{1}{\\sqrt{2}}(\\ket{00}+\\ket{11})$ and $\\ket{\\phi_{AB}^-}=\\frac{1}{\\sqrt{2}}(\\ket{00}-\\ket{11})$.\\par\nConsider  $A=\\sigma^x_A$, $B=\\sigma^x_B$, $\\ket{\\phi_A}=\\ket{1}$ and $\\ket{\\phi_B}=\\ket{1}$. The inequality is violated for $p\\neq 1\/2$ (PPT criterion).\\\\\n\\emph{(iii)} The following density operator \n\\begin{align}\n\\rho=p\\ket{\\psi_{AB}}\\bra{\\psi_{AB}} +(1-p)\\frac{I_A\\otimes I_B}{4}\\nonumber\n\\end{align}  \nwhere $\\ket{\\psi_{AB}}=a\\ket{00}+b\\ket{11}$ and $|a|^2+|b|^2=1$, is entangled if and only if $p > 1\/(1+4|ab|)$ (PPT criterion).\\par\nUsing the above separability criterion with the choices $A=\\sigma_A^x$, $B=\\sigma_B^x$,  $\\ket{\\phi_A}=\\ket{1}$ and $\\ket{\\phi_B}=\\ket{1}$. The inequality is violated for $p > 1\/(1+4|ab|)$.\\\\\n\\textit{(iv)} The density operator \n\\begin{align}\n\\rho=p\\ket{\\psi_{AB}^{(1)}}\\bra{\\psi_{AB}^{(1)}} +(1-p)\\ket{\\psi_{AB}^{(2)}}\\bra{\\psi_{AB}^{(2)}}\\nonumber\n\\end{align}  \nwhere $\\ket{\\psi_{AB}^{(1)}}=b_1\\ket{01}+c_1\\ket{10}$, $\\ket{\\psi_{AB}^{(2)}}=b_2\\ket{01}+c_2\\ket{10}$,  $|b_1|^2+|c_1|^2=1$ and $|b_2|^2+|c_2|^2=1$, is entangled if and only if (PPT criterion)  $|pb_1^*c_1+(1-p)b_2^*c_2| > 0$.\\par \nUsing the  separability criterion with the choices $A=\\sigma_A^x$, $B=\\sigma_B^x$,  $\\ket{\\phi_A}=\\ket{0}$ and $\\ket{\\phi_B}=\\ket{1}$. The inequality is violated for $|pb_1^*c_1+(1-p)b_2^*c_2| > 0$.\\\\\n\\textit{(v) Mixture of 4-Bell states:} \n\\begin{align}\n\\rho=&p_1\\ket{\\psi_{AB}^+}\\bra{\\psi_{AB}^+}+p_2\\ket{\\psi_{AB}^-}\\bra{\\psi_{AB}^-}\\nonumber\\\\&\n+p_3\\ket{\\phi_{AB}^+}\\bra{\\phi_{AB}^+}+p_4\\ket{\\phi_{AB}^-}\\bra{\\phi_{AB}^-},\\nonumber\n\\end{align}\nwhere $\\ket{\\psi_{AB}^+}=\\frac{1}{\\sqrt{2}}(\\ket{00}+\\ket{11})$,  $\\ket{\\psi_{AB}^-}=\\frac{1}{\\sqrt{2}}(\\ket{00}-\\ket{11})$ and $p_1+p_2+p_3+p_4=1$. This density matrix is entangled if and only if $p_i >1\/2$, $p_j < 1\/2$, $i\\neq j$ and $i,j=1,2,3,4$ (PPT criterion).\\par\nConsider (a) $A=\\sigma^x_A$, $B=\\sigma^x_B$, $\\ket{\\phi_A}=\\ket{0}$ and $\\ket{\\phi_B}=\\ket{0}$. The inequality is violated for $p_1> 1\/2$ or $p_2>1\/2$, (b) $A=\\sigma^x_A$, $B=\\sigma^x_B$, $\\ket{\\phi_A}=\\ket{0}$ and $\\ket{\\phi_B}=\\ket{1}$. The  inequality is violated for $p_3> 1\/2$ or $p_4>1\/2$.\\\\\n\\textit{(vii) Two qudit Werner  states \\cite{59}:}\n\\begin{align}\n\\rho=(1-p)\\frac{2}{d^2+d}P^{(+)} + p\\frac{2}{d^2-d}P^{(-)}, \\hspace{3mm} 0\\leq p\\leq 1,\\nonumber\n\\end{align}\nwhere the projectors $P^{(+)}=(I+V)\/2$, $P^{(-)}=(I-V)\/2$ with identity $I$ and flip operation $V=\\sum_{i,j=0}^{d-1}{\\ket{i}\\bra{j}\\otimes \\ket{j}\\bra{i}}$, and $\\{\\ket{i}\\}$ is the basis states. The state $\\rho$  is entangled if and only if  $p>1\/2$  (PPT criterion).\\par\nTo see, which values of `$p$' are achievable via the separability inequality Eq. (\\ref{39}), we first calculate the entanglement condition and then will see some physically implementable systems. Consider $\\bra{i^{\\prime}_A i^{\\prime}_B}C_A\\otimes C_B=\\bra{j^{\\prime}_A j^{\\prime}_B}$, where  $\\braket{i^{\\prime}_A|j^{\\prime}_A}=0$ or $\\braket{i^{\\prime}_B|j^{\\prime}_B}=0$ or both. Then from Eq. (\\ref{39}), the LHS - RHS   becomes\n\\begin{align}\n|\\Lambda^{(-)}\\braket{j^{\\prime}_A|i^{\\prime}_B}\\braket{j^{\\prime}_B|i^{\\prime}_A}\\hspace{-1mm}|^2 - &\n\\left[\\Lambda^{(+)}+\\Lambda^{(-)}|\\hspace{-1mm}\\braket{j^{\\prime}_A|i^{\\prime}_B}\\hspace{-1mm}|^2\\right]\\nonumber\\\\\n& \\times\\left[\\Lambda^{(+)}+\\Lambda^{(-)}|\\hspace{-1mm}\\braket{j^{\\prime}_B|i^{\\prime}_A}\\hspace{-1mm}|^2\\right]\\label{40}\n\\end{align}\nwhere $\\Lambda^{(+)}=\\frac{1-p}{d^2+d} + \\frac{p}{d^2-d}$ and $\\Lambda^{(-)}=\\frac{1-p}{d^2+d} - \\frac{p}{d^2-d}$. Now by making the choices $|\\hspace{-1mm}\\braket{j^{\\prime}_B|i^{\\prime}_A}\\hspace{-1mm}|=|\\hspace{-1mm}\\braket{j^{\\prime}_A|i^{\\prime}_B}\\hspace{-1mm}|=1$, it is easy to show that LHS-RHS (\\ref{40}) is alway positive for $p> \\frac{3(d-1)}{2(2d-1)}$ which is the entanglement condition for the Werner state in $d\\otimes d$. It known that for $\\frac{1}{2}<p\\leq \\frac{3(d-1)}{2(2d-1)}$ and $p> \\frac{3(d-1)}{2(2d-1)}$, the Werner state is bound entangled (conjectured) and distillable respectively.  That is, our separability criterion (\\ref{39}) is able to detect the distillability   of the Werner state but not the bound entanglement (if any). In Appendix \\ref{G}, we give the examples of how to fulfil  the choices we made here in the physical systems.\\\\\n\\textit{(vi) Higher dimensional isotropic states \\cite{60}:}\n\\begin{align}\n\\rho=p\\ket{\\psi^+_{AB}}\\bra{\\psi^+_{AB}}+(1-p)\\frac{I_A\\otimes I_B}{d^2},\\nonumber\n\\end{align}\nwhere $\\ket{\\psi_{AB}^+}=\\frac{1}{\\sqrt{d}}(\\sum_{i=1}^d\\ket{i_Ai_B}$ and `$d$' is the dimension of the subsytems. \\par \nBy choosing the spin flip operators  $A=\\sigma^x_A$, $B=\\sigma^x_B$ such that $(\\sigma^x_A\\otimes \\sigma_B^x)\\ket{i_Ai_B}=\\ket{j_Aj_B}$, $j_A\\neq i_A$, $j_B\\neq i_B$,  and $\\ket{\\phi_A}=\\ket{i_A}$,  $\\ket{\\phi_B}=\\ket{i_B}$, it can be shown that the inequality is violated for $p>1\/(d+1)$ (PPT criterion).\\par\nIn comparison with most of the existing works, the above separability inequality is easier to implement in experiments due to the simple realization of weak measurement and less number of measurement settings. In particular, compared to the case of universal (i.e., state-independent) detection of two-qubit entanglement using two copies of the state at a time and using the notion of weak values \\cite{37}, the aforesaid inequality (\\ref{39}) (involving product weak values) uses only a single copy of the bipartite state ${\\rho}$ at a time. Moreover, the criterion is resource-wise better than tomography, based on local realization and dependent on one type of measurement set-up. \\par\nWe do believe that the separability inequality (\\ref{39}) is of universal nature  at least for the set of all two-qubit states. Needless to say that the choice of the local observables $A$, $B$ as well as the post-selected state $\\ket{{\\phi}_A{\\phi}_B}$ do depend upon the choice of the input bi-partite state ${\\rho}$.\n\\section{Robustness}\\label{e V}\nIn AAV method, the coupling between  the system and the pointer is extremely small and hence the state collapse  is avoided. During the process, any resolution is insufficient to distinguish the different eigenvalues of the observable. Nevertheless, by performing the experiment many times on identically prepared systems, it is possible to  reduce the uncertainty in the mean pointer displacement to any arbitrary precision \\cite{50}. \\par \nThere  are other type of errors which  are inevitable due to   the inappropriate  choices of system observables and unsharp post-selections. Here, we show that our methods of  ``extraction of product and higher moment weak values\" are robust against them.\\\\ \n\\textsl{(i)} \\normalsize{\\textit{Error in choice of observable:}}  In experiment, let's say, we want to measure a spin-1\/2 observable according to the AAV method but due to some  technical difficulties, we are  unable to  measure the actual spin-1\/2 observable (slightly changed $\\theta$ and $\\phi$, where $\\theta$ and $\\phi$ define a point on the bloch sphere). Now let `$A$' be the correct observable while `$A^e$' be the erroneous one such that  $\\left|A-A^e\\right| \\leq \\delta$, where $\\left|X\\right| = Tr\\sqrt{X^{\\dagger}X}$   is the trace norm of a square matrix X. So the error occurring  in the weak value is given by \n\\begin{align}\n\\Delta(\\rho,A,\\phi)&=\\left|{\\braket{A_w}^{\\phi}_{\\rho} - \\braket{A^e_w}^{\\phi}_{\\rho}}\\right|=\\frac{\\left|\\braket{\\phi|(A-A^e)\\rho|\\phi}\\right|}{\\braket{\\phi|\\rho|\\phi}}\\nonumber\\\\\n&\\leq \\frac{\\left|\\braket{\\phi|(A-A^e)\\rho|\\phi}\\right|}{m}, \\label{41}\n\\end{align}\nwhere `m' is the minimum of the probabilities for all the possible choices  of rank-one post-selections with a given pre-selection. Now consider the spectral decomposition $A-A^e=\\sum_i{\\lambda_i\\ket{i}\\bra{i}}$ where $\\{\\ket{i}\\}$ is the complete set of orthogonal basis. Then\n\\begin{align}\n\\Delta(\\rho,A,\\phi)&\\leq\\frac{1}{m}\\left|\\sum \\lambda_i \\braket{\\phi|i}\\braket{i|\\rho|\\phi} \\right|\\nonumber\\\\\n&\\leq\\frac{1}{m}\\sum_i \\left|\\lambda_i\\right| \\left| \\braket{\\phi|i}\\braket{i|\\rho|\\phi}\\right|. \\label{42}\n\\end{align}\nNote that $\\left|A-A^e\\right|=\\sum_i \\left|\\lambda_i\\right|$ and since $0\\leq \\rho \\leq \\mathbbm{1} $, it can easily be shown that $\\left| \\braket{\\phi|i}\\braket{i|\\rho|\\phi}\\right|\\leq 1$. Hence\n\\begin{align}\n\\Delta(\\rho,A,\\phi)&\\leq\\frac{1}{m}\\sum_i \\left|\\lambda_i\\right|=\\frac{\\left|A-A^e\\right|}{m}\\leq \\frac{\\delta}{m}. \\label{43}\n\\end{align}\n\\textsl{(ii)} \\normalsize{\\textit{Noisy post-selection:}} Now we consider another type of error which is common  in experiments is due to the  unsharp post-selections. Let us assume that the unsharp post-selection is a mixture of the true post-selection $\\ket{\\phi}$ with probability $(1-\\epsilon)$ and noise state $\\sigma$ with probability $\\epsilon$ \n\\begin{align}\n \\Phi^{\\epsilon}=(1-\\epsilon)\\ket{\\phi}\\bra{\\phi} + \\epsilon\\sigma,\\label{44}\n\\end{align}\nwhere $\\epsilon$ is a sufficiently small positive quantity. Then the difference between the perturbed and true weak values is \n\\begin{align}\n\\braket{A_w}^{ \\Phi^{\\epsilon}}_{\\rho} - \\braket{A_w}^{\\phi}_{\\rho}\\approx\\epsilon\\left[ \\frac{Tr(\\sigma A \\rho) - \\braket{A_w}^{\\phi}_{\\rho} Tr(\\sigma\\rho)}{\\braket{\\phi|\\rho|\\phi}} \\right].\\label{45}\n\\end{align}\\par\nSo in both the cases (Eqs. (\\ref{43}) and (\\ref{45})), the weak values are robust. Now it is not hard to realize that product and higher moment weak values are also robust. The only thing we need to do is to replace the observable $A$ by  $A^2$ for a single system and  $C_A\\otimes C_B$ for a bipartite system in  Eqs. (\\ref{43}) and (\\ref{45}). Hence the weak values which we have used to reconstruct the state of a single and bipartite systems are also robust.\\\\\n\\section{Conclusion}\\label{e VI}\nWe have derived the methods  of extracting  higher moment weak values and product weak values using Vaidman's formula.  Such higher moment weak values are  calculated using only the weak values of that observable with pairwise orthogonal post-selections.  Two dimensional Hilbert space becomes the simplest case for extracting the  higher moment weak values. Our methods turn out to be  simple from experimental perspective as we don't need to measure the N pointer's correlations as required in the previous works. Previously, it was thought that with  Gaussian pointers' states, it is not possible to obtain the higher moment weak values but we have shown that  instead of looking for different pointer states (e.g., OAM states)   to obtain the higher moment weak values, we  can use  Vaidman's formula. To extract the product weak values in a bipartite system, we have again used  Vaidman's formula in one of the subsystems. The product weak values can be calculated using only local weak values. The key factor for such local realization is that the action of the local operator on the local post-selected state is equivalent to the superposition of that post-selected state and a unique orthogonal state to that given post-selected state. Our method can be used to verify  Hardy's Paradox, to confirm the existence of quantum Cheshire cats, to perform EPR-Bohm experiment, to realize non-locality via post-selections, etc.\\par\nAs an application, we have shown how to  reconstruct quantum states of a single and bipartite systems separately. We have used higher moment weak values to reconstruct  an unknown pure  state of a single system. The number of measurements are nearly half of the measurements  required in previous works. Mixed state reconstruction has been shown using arbitrary observbles.  We have used product weak values for  reconstruction  of   pure and mixed  states in a bipartite system. Such reconstructions become simply feasible in experiment using only the measurements of local weak values.  In the previous works, projection measurement operators were the central for direct quantum state tomography. But we have generalized it to any arbitrary observables for both single and bipartite systems. Comparisons between the previous works and our work have been  considered from various perspective (e.g., number of measurements according to the  AAV method and experimental feasibility).  { A necessary separability criteria (in terms of an inequality) for finite dimensional bi-partite systems using  product weak values has been derived. This inequality is turned out to be strong as the PPT criteria can be achieved for  certain class of entangled states  by cleverly choosing the product observables and the post selections.   The criteria can, in principle detect more classes of entangled states with suitably choosing product observables and post-selections}.  Finally, we have shown that our methods are robust against   the errors which  are inevitable due to   the inappropriate  choices of system observables and unsharp  post-selections. Our method can easily be extended to multi-partite systems.\\par\n{\\bf{Acknowledgment}}: Sahil is thankful to the QIC group at HRI, Allahabad, for making an arrangement for visiting the group during which, part of the work was done. SM acknowledges financial support from the Visiting Postdoctoral Programme of IMSc, Chennai. We would like to thank  AK Pan and PK Panigrahi for bringing an useful comment \\cite{43} on higher moment weak values calculation to our attention. \n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Section-introduction}\n\nGiven a compact domain $T\\subset\\mathbb{R}^2$ with piecewise smooth boundary,\nconsider the straight line motion of a particle inside $T$, with specular reflections\nin $\\partial T$. Let $f:M\\to M$ be the {\\em billiard map}, where\n$M=\\partial T\\times[-\\tfrac{\\pi}{2},\\tfrac{\\pi}{2}]$ with the convention that $(r,\\theta)\\in M$\nrepresents $r=$ collision position at $\\partial T$ and $\\theta=$ angle of collision.\nThe map $f$ has a natural invariant Liouville measure $d\\mu=\\cos\\theta drd\\theta$.\nSina{\\u\\i} proved that dispersing billiards are uniformly hyperbolic systems with\ndiscontinuities \\cite{Sinai-billiards}, hence the Liouville measure is ergodic.\n\n\\medskip\nFor a while uniform hyperbolicity was the only mechanism to generate chaotic billiards,\nuntil Bunimovich constructed examples of ergodic nowhere dispersing\nbilliards \\cite{Bunimovich-close-to-scattering,Bunimovich-ergodic-properties,Bunimovich-Nowhere-dispersing}.\nThese billiards, known as {\\em Bunimovich billiards}, are non-uniformly hyperbolic:\n$\\mu$--almost every point has one positive Lyapunov exponent and one negative Lyapunov exponent,\nsee \\cite[Chapter 8]{Chernov-Markarian}.\nIn this paper we construct symbolic models for non-uniformly hyperbolic billiard maps\nsuch as Bunimovich billiards. Assume that the billiard table $T$ satisfies\nthe conditions of \\cite[Part V]{Katok-Strelcyn}, and\nlet $h$ be the Kolmogorov-Sina{\\u\\i} entropy of $\\mu$.\n\n\\begin{theorem}\\label{Thm-billiard}\nIf $\\mu$ is ergodic and $h>0$ then there exists a topological Markov shift\n$(\\Sigma,\\sigma)$ and a H\\\"older continuous map $\\pi:\\Sigma\\to M$ s.t.:\n\\begin{enumerate}[$(1)$]\n\\item $\\pi\\circ \\sigma=f\\circ\\pi$.\n\\item $\\pi$ is surjective and finite-to-one on a set of full $\\mu$--measure.\n\\end{enumerate}\n\\end{theorem}\n\nOther examples of non-uniformly hyperbolic billiard maps are\n\\cite{Wojtkowski-principles,Bunimovich-lemons}. See section \\ref{Section-preliminaries} for\nthe definition of topological Markov shifts.\n\n\\begin{corollary}\\label{corollary-periodic}\nUnder the above assumptions, $\\exists C>0$ and $p\\geq 1$ s.t. $f$ has at least\n$Ce^{hnp}$ periodic points of period $np$ for all $n\\geq 1$. \n\\end{corollary}\n\nCorollary \\ref{corollary-periodic} is consequence of Theorem \\ref{Thm-billiard}\nand the work of Gurevi{\\v{c}} \\cite{Gurevich-Topological-Entropy,Gurevich-Measures-Of-Maximal-Entropy},\nas in \\cite[Thm. 1.1]{Sarig-JAMS}. It is related to an estimate of Chernov \\cite{Chernov-91}.\nThe integer $p$ is the period of $(\\Sigma,\\sigma)$, hence $p=1$ iff $(\\Sigma,\\sigma)$\nis topologically mixing. Since $\\mu$ is mixing, we expect that the symbolic\ncoding of Theorem \\ref{Thm-billiard}\ncan be improved to give a topologically mixing $(\\Sigma,\\sigma)$.\nTheorem \\ref{Thm-billiard} is consequence of the main result\nof this paper, Theorem \\ref{Thm-main}, and of an argument of Katok and Strelcyn\n\\cite[Section I.3]{Katok-Strelcyn}.\nThe statement of Theorem \\ref{Thm-main} is technical, so we first introduce some notation. \n\n\n\\medskip\nLet $M$ be a smooth Riemannian surface with finite diameter, possibly with boundary.\nWe assume that the diameter of $M$ is smaller than one\\footnote{Just multiply the\nmetric by a sufficiently small constant.}.\nLet $\\mathfs D^+,\\mathfs D^-$ be closed\nsubsets of $M$.\nFix $f:M\\backslash\\mathfs D^+\\to M$ a diffeomorphism\nonto its image, s.t. $f$ has an inverse $f^{-1}:M\\backslash\\mathfs D^-\\to M$ that\nis a diffeomorphism onto its image.\n\n\\medskip\n\\noindent\n{\\sc Set of discontinuities $\\mathfs D$:} The {\\em set of discontinuities of $f$} is\n$\\mathfs D:=\\mathfs D^+\\cup \\mathfs D^-$. \n\n\\medskip\nIf $x\\not\\in\\bigcup_{n\\in\\mathbb{Z}}f^n(\\mathfs D)$ then $f^n(x)$ is well-defined for all $n\\in\\mathbb{Z}$,\nand for every $y=f^n(x)$ there is a neighborhood $U\\ni y$ s.t. $f\\restriction_U,f^{-1}\\restriction_{U}$\nare diffeomorphisms onto their images. We require some regularity conditions on $M,f$.\nThe first four assumptions are on the geometry of $M$.\nGiven $x\\in M\\backslash\\mathfs D$, let ${\\rm inj}(x)$ denote the {injectivity radius} of $M$ at $x$,\nand let $\\exp{x}$ be the {\\em exponential map} at $x$, wherever it can be defined.\nGiven $r>0$, let $B_x[r]\\subset T_xM$ be the ball with center 0 and radius $r$.\nThe Riemannian metric on $M$ induces a Riemannian metric on $TM$, called the {\\em Sasaki\nmetric}, see e.g. \\cite[\\S2]{Burns-Masur-Wilkinson}.\nDenote the Sasaki metric by $d_{\\rm Sas}(\\cdot,\\cdot)$.\nSimilarly, we denote the Sasaki metric on $TB_x[r]$ by the same notation, and the context\nwill be clear in which space we are. For nearby small vectors, the Sasaki metric is\nalmost a product metric in the following sense. Given a geodesic $\\gamma$ joining $y$ to $x$,\nlet $P_\\gamma:T_yM\\to T_xM$ be the parallel transport along $\\gamma$.\nIf $v\\in T_xM$, $w\\in T_yM$ then\n$d_{\\rm Sas}(v,w)\\asymp d(x,y)+\\|v-P_\\gamma w\\|$ as $d_{\\rm Sas}(v,w)\\to 0$, see e.g.\n\\cite[Appendix A]{Burns-Masur-Wilkinson}. The rate of convergence depends on the\ncurvature tensor of the metric on $M$. Here are the first two assumptions on $M$.\n\n\\medskip\n\\noindent\n{\\sc Regularity of $\\exp{x}$:} $\\exists a>1$ s.t. for all\n$x\\in M\\backslash\\mathfs D$ there is $d(x,\\mathfs D)^a<\\mathfrak r(x)<1$ \ns.t. for $D_x:=B(x,2\\mathfrak r(x))$ the following holds:\n\\begin{enumerate}[ii]\n\\item[(A1)] If $y\\in D_x$ then ${\\rm inj}(y)\\geq 2\\mathfrak r(x)$, $\\exp{y}^{-1}:D_x\\to T_yM$\nis a diffeomorphism onto its image, and\n$\\tfrac{1}{2}(d(x,y)+\\|v-P_{y,x}w\\|)\\leq d_{\\rm Sas}(v,w)\\leq 2(d(x,y)+\\|v-P_{y,x} w\\|)$ for all $y\\in D_x$ and\n$v\\in T_xM,w\\in T_yM$ s.t. $\\|v\\|,\\|w\\|\\leq 2\\mathfrak r(x)$, where \t\n$P_{y,x}:=P_\\gamma$ is the radial geodesic $\\gamma$ joining $y$ to $x$.\n\\item[(A2)] If $y_1,y_2\\in D_x$ then\n$d(\\exp{y_1}v_1,\\exp{y_2}v_2)\\leq 2d_{\\rm Sas}(v_1,v_2)$ for $\\|v_1\\|$, $\\|v_2\\|\\leq 2\\mathfrak r(x)$,\nand $d_{\\rm Sas}(\\exp{y_1}^{-1}z_1,\\exp{y_2}^{-1}z_2)\\leq 2[d(y_1,y_2)+d(z_1,z_2)]$\nfor $z_1,z_2\\in D_x$ whenever the expression makes sense.\nIn particular $\\|d(\\exp{x})_v\\|\\leq 2$ for $\\|v\\|\\leq 2\\mathfrak r(x)$,\nand $\\|d(\\exp{x}^{-1})_y\\|\\leq 2$ for $y\\in D_x$.\n\\end{enumerate}\n\n\\medskip\nThe next two assumptions are on the regularity of $d\\exp{x}$.\nFor $x,x'\\in\\ M\\backslash\\mathfs D$, let $\\mathfs L _{x,x'}:=\\{A:T_xM\\to T_{x'}M:A\\text{ is linear}\\}$\nand $\\mathfs L _x:=\\mathfs L_{x,x}$. \nThen the parallel transport $P_{y,x}$ considered in (A1) is in $\\mathfs L_{y,x}$.\nGiven $y\\in D_x,z\\in D_{x'}$ and $A\\in \\mathfs L_{y,z}$,\nlet $\\widetilde{A}\\in\\mathfs L_{x,x'}$, $\\widetilde{A}:=P_{z,x'} \\circ A\\circ P_{x,y}$.\nBy definition, $\\widetilde{A}$ depends on $x,x'$ but different basepoints define\na map that differs from $\\widetilde{A}$ by pre and post composition with isometries.\nIn particular, $\\|\\widetilde{A}\\|$ does not depend on the choice of $x,x'$.\nSimilarly, if $A_i\\in\\mathfs L_{y_i,z_i}$ then $\\|\\widetilde{A_1}-\\widetilde{A_2}\\|$ does\nnot depend on the choice of $x,x'$.\nDefine the map $\\tau=\\tau_x:D_x\\times D_x\\to \\mathfs L_x$\nby $\\tau(y,z)=\\widetilde{d(\\exp{y}^{-1})_z}$, where we use the identification\n$T_v(T_{y}M)\\cong T_{y}M$ for all $v\\in T_yM$.\n\n\n\\medskip\n\\noindent\n{\\sc Regularity of $d\\exp{x}$:}\n\\begin{enumerate}[ii]\n\\item[(A3)] If $y_1,y_2\\in D_x$ then\n$\n\\|\\widetilde{d(\\exp{y_1})_{v_1}}-\\widetilde{d(\\exp{y_2})_{v_2}}\\|\n\\leq d(x,\\mathfs D)^{-a}d_{\\rm Sas}(v_1,v_2)\n$\nfor all $\\|v_1\\|,\\|v_2\\|\\leq 2\\mathfrak r(x)$, and \n$\\|\\tau(y_1,z_1)-\\tau(y_2,z_2)\\|\\leq d(x,\\mathfs D)^{-a}[d(y_1,y_2)+d(z_1,z_2)]$\nfor all $z_1,z_2\\in D_x$.\n\\item[(A4)] If $y_1,y_2\\in D_x$ then the map $\\tau(y_1,\\cdot)-\\tau(y_2,\\cdot):D_x\\to \\mathfs L_x$\nhas Lipschitz constant $\\leq d(x,\\mathfs D)^{-a}d(y_1,y_2)$.\n\\end{enumerate}\n\n\\medskip\nConditions (A1)--(A2) guarantee that the exponential maps and their inverses\nare well-defined and have uniformly bounded Lipschitz constants in balls\nof radii $d(x,\\mathfs D)^a$.\nCondition (A3) controls the Lipschitz constants of the derivatives of these maps,\nand condition (A4) controls the Lipschitz constants of their second derivatives.\nHere are some case when (A1)--(A4) are satisfied, in increasing order of generality:\n\\begin{enumerate}[$\\circ$]\n\\item The curvature tensor $R$ of $M$ is globally bounded, e.g. when $M$ is the\nphase space of a billiard map.\n\\item $R,\\nabla R,\\nabla^2 R,\\nabla^3R$ grow at most polynomially\nfast with respect to the distance to $\\mathfs D$, e.g. when $M$ is a moduli space\nof curves equipped with the Weil-Petersson metric \\cite{Burns-Masur-Wilkinson}.\n\\end{enumerate}\nNow we discuss the assumptions on $f$.\n\n\\medskip\n\\noindent\n{\\sc Regularity of $f$:} There are constants $0<\\beta<1<b$ s.t. for all  $x\\in M\\backslash\\mathfs D$:\n\\begin{enumerate}[ii]\n\\item[(A5)] If $y\\in D_x$ then $\\|df_y^{\\pm 1}\\|\\leq d(x,\\mathfs D)^{-b}$.\n\\item[(A6)] If $y_1,y_2\\in D_x$ and $f(y_1),f(y_2)\\in D_{x'}$ then\n$\\|\\widetilde{df_{y_1}}-\\widetilde{df_{y_2}}\\|\\leq \\mathfrak Kd(y_1,y_2)^\\beta$,\nand if $y_1,y_2\\in D_x$ and $f^{-1}(y_1),f^{-1}(y_2)\\in D_{x''}$ then\n$\\|\\widetilde{df_{y_1}^{-1}}-\\widetilde{df_{y_2}^{-1}}\\|\\leq \\mathfrak Kd(y_1,y_2)^\\beta$.\n\\end{enumerate}\n\n\\medskip\nAlthough technical, conditions (A5)--(A6) hold in most cases of interest, e.g.\nif $\\|df^{\\pm 1}\\|,\\|d^2f^{\\pm 1}\\|$ grow at most polynomially fast with respect to\nthe distance to $\\mathfs D$.\nWe finally define the measures we code. Fix $\\chi>0$.\n\n\\medskip\n\\noindent\n{\\sc $\\chi$--hyperbolic measure:} An $f$--invariant probability measure on $M$ is called\n{\\em $\\chi$--hyperbolic} if $\\mu$--a.e. $x\\in M$ has one Lyapunov exponent $>\\chi$\nand another $<-\\chi$.\n\n\\medskip\n\\noindent\n{\\sc $f$--adapted measure:} An $f$--invariant measure on $M$ is called {\\em $f$--adapted}\nif\n$$\\int_M \\log d(x,\\mathfs D)d\\mu(x)>-\\infty.$$\nA fortiori $\\mu(\\mathfs D)=0$.\n\n\n\n\n\\begin{theorem}\\label{Thm-main}\nLet $M,f$ satisfy conditions {\\rm (A1)--(A6)}. For all $\\chi>0$,\nthere exists a topological Markov shift $(\\Sigma,\\sigma)$\nand a H\\\"older continuous map $\\pi:\\Sigma\\to M$ s.t.:\n\\begin{enumerate}[$(1)$]\n\\item $\\pi\\circ \\sigma=f\\circ\\pi$.\n\\item  $\\pi[\\Sigma^\\#]$ has full $\\mu$--measure for every $f$--adapted $\\chi$--hyperbolic measure $\\mu$.\n\\item For all $x\\in \\pi[\\Sigma^\\#]$, $\\#\\{\\underline v\\in\\Sigma^\\#:\\pi(\\underline v)=x\\}<\\infty$.\n\\end{enumerate}\n\\end{theorem}\n\nAbove, $\\Sigma^\\#$ is the {\\em recurrent set} of $\\Sigma$, see section \\ref{Section-preliminaries}.\nEvery $\\sigma$--invariant measure $\\widehat{\\mu}$ is carried by $\\Sigma^\\#$,\nhence its projection $\\mu=\\widehat{\\mu}\\circ\\pi^{-1}$ has the same entropy\nas $\\widehat{\\mu}$ (this follows from the Abramov-Rokhlin formula \\cite{Abramov-Rokhlin}).\nIn particular, the topological entropy of $(\\Sigma,\\sigma)$ is at most that of $(M,f)$.\nOn the other direction, every $f$--adapted $\\chi$--hyperbolic measure $\\mu$ has\na lift $\\widehat{\\mu}$ with the same entropy. If we know that $\\chi$--hyperbolic measures are\n$f$--adapted then the topological entropies of $(\\Sigma,\\sigma)$ and $(M,f)$ coincide,\nand their measures of maximal entropy are related. In this case, Corollary \\ref{corollary-periodic}\nhas a potentially stronger statement: for every $\\varepsilon>0$, $\\exists C>0$ and $p\\geq 1$ s.t. $f$ has at\nleast $Ce^{(H-\\varepsilon)np}$ periodic points of period $np$ for all $n\\geq 1$,\nwhere $H$ is the topological entropy of $\\Sigma$.\nAt the moment, we are not aware of general results assuring that $\\chi$--hyperbolic measures\nare $f$--adapted, except when the measure is Liouville \\cite[Section I.3]{Katok-Strelcyn}.\n\n\\medskip\nWe now discuss the applicability of Theorem \\ref{Thm-billiard}. Let us restrict ourselves to\nbilliard tables with finitely many boundary components, otherwise many degeneracies\ncan occur (see e.g. \\cite[Part V]{Katok-Strelcyn}). Assumptions (A1)--(A6) are satisfied\nif all boundary components are $C^3$. The precise conditions that guarantee \nnon-uniform hyperbolicity are unknown, so we mention two classes of billiard tables $T$\nwhose billiard maps are non-uniformly hyperbolic:\n\\begin{enumerate}[$\\circ$]\n\\item Sina{\\u\\i} billiard: every component of $\\partial T$ is dispersing.\nIn this case, the billiard map exhibits uniform hyperbolicity.\n\\item Bunimovich billiard: $\\partial T$ is the union of finitely many segments\nand arcs of circles s.t. each of these arcs belongs to a disc contained in $T$.\nWhen this happens, non-uniform hyperbolicity is ensured via a focusing-defocusing mechanism,\nsee \\cite[Chapter 8]{Chernov-Markarian}. See Figure \\ref{figure-billiards} for some examples.\n\\end{enumerate}\n\n\\begin{figure}[hbt!]\n\\centering\n\\def12cm{12cm}\n\\input{billiards.pdf_tex}\\label{figure-billiards}\\caption{Examples of Bunimovich billiards:\n(a) pool table with pockets, (b) stadium, (c) flower.}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Related literature}\n\nThe construction of Markov partitions and symbolic dynamics for uniformly hyperbolic\ndiffeomorphisms and flows in compact manifolds laid its foundation during the late sixties and\nearly seventies through the works of Adler \\& Weiss\n\\cite{Adler-Weiss-PNAS,Adler-Weiss-Similarity-Toral-Automorphisms},\nSina{\\u\\i} \\cite{Sinai-Construction-of-MP,Sinai-MP-U-diffeomorphisms},\nBowen \\cite{Bowen-MP-Axiom-A,Bowen-Symbolic-Flows}, and\nRatner \\cite{Ratner-MP-three-dimensions,Ratner-MP-n-dimensions}.\nBelow we discuss other contexts.\n\n\n\\medskip\n\\noindent\n{\\sc Billiards:} These are the main examples of maps with discontinuities.\nKatok and Strelcyn constructed invariant manifolds for non-uniformly hyperbolic\nbilliard maps which include Bunimovich billiards \\cite{Katok-Strelcyn}.\nBunimovich, Chernov and Sina{\\u\\i} constructed countable Markov partitions for two-dimensional\ndispersing billiard maps \\cite{Bunimovich-Chernov-Sinai}.\nAll these results are for Liouville measures. Up to our knowledge,\nour result is the first symbolic coding of uniformly and non-uniformly hyperbolic\nbilliard maps for general measures.\n\n\n\\medskip\n\\noindent\n{\\sc Tower extensions of billiard maps:} Young constructed tower extensions\nfor certain two-dimensional dispersing billiard maps \\cite{Young-towers}.\nContrary to our case, Young's tower extensions provide codings which are usually infinite-to-one,\nhence it is unclear that $\\chi$--hyperbolic measures can be lifted to the\nsymbolic space without increasing its entropy. Nevertheless, such tower extensions\nguarantee exponential decay of correlations for certain two-dimensional dispersing billiard maps.\t  \n\n\\medskip\n\\noindent\n{\\sc Non-uniformly hyperbolic three-dimensional flows:} The first author and Sarig\nconstructed symbolic models for non-uniformly hyperbolic three-dimensional flows with positive\nspeed \\cite{Lima-Sarig}. The idea is to take a Poincar\\'e section and analyze the Poincar\\'e return map $f$.\nThe Poincar\\'e map $f$ has discontinuities, but its derivative is uniformly bounded inside the set\nof continuities. Hence the methods of \\cite{Sarig-JAMS} apply more easily.\n\n\n\n\\medskip\n\\noindent\n{\\sc Weil-Petersson flow:} Moduli spaces of curves possess natural negatively\ncurved incomplete K\\\"ahler metrics, called {\\em Weil-Petersson metrics}. The geodesic\nflow of one such metric is called the {\\em Weil-Petersson flow}, and it preserves a canonical\nLiouville measure.\nThe properties of the Weil-Petersson metric are intimately related to the hyperbolic\ngeometry of surfaces, and this partly explains the recent interest in the dynamics\nof the Weil-Petersson flow. Burns, Masur and Wilkinson proved that the Liouville\nmeasure is hyperbolic \\cite{Burns-Masur-Wilkinson}.\nFor that, they combined results of Wolpert and McMullen to show that the\nWeil-Petersson metric explodes at most polynomially fast while approaching\nthe boundary of the Deligne-Mumford compactification of the moduli space of curves,\nhence the Weil-Petersson flow satisfies the assumptions of Katok and Strelcyn \\cite{Katok-Strelcyn}.\nThe construction of symbolic dynamics for the Weil-Petersson flow is still open.\n\n\\medskip\nAs pointed out by Sarig \\cite[pp. 346]{Sarig-JAMS}, our main result (Theorem \\ref{Thm-main})\ncan be regarded as a step towards the construction of Markov partitions capturing\nmeasures of maximal entropy for surface maps with discontinuities with positive topological\nentropy, such as Bunimovich billiards.\nMotivated by this, we ask the following question.\n\n\\smallskip\n\n\\noindent\n{\\sc Question:} Let $f$ be a billiard map with topological entropy $H>0$. Does $f$ have a measure\nof maximal entropy? If it does, is it $f$--adapted? Is it Bernoulli? \n\n\\medskip\nA positive answer to this question would imply that $\\exists C>0$ s.t. $f$ has at least \n$Ce^{Hn}$ periodic points of period $n$, for all $n\\geq 1$.\n\n\\medskip\nIn \\cite[pp. 858]{Burns-Masur-Wilkinson} it was suggested\nthat one of the assumptions (in their notation, the compactness of $\\overline{N}$)\ncan be relaxed to the assumption that $N$ has finite diameter.\nThe main reason not to claim this is that they use \\cite{Katok-Strelcyn}, whose framework\nassumes $\\overline N$ to be compact. We only assume finite diameter, hence\nour work is a step towards the relaxation of the assumptions of \\cite{Katok-Strelcyn}\nto the context mentioned in \\cite{Burns-Masur-Wilkinson}.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Methodology}\n\nThe proof of Theorem \\ref{Thm-main} is based on \\cite{Sarig-JAMS} and \\cite{Lima-Sarig},\nand it follows the steps below:\n\\begin{enumerate}[(1)]\n\\item If $\\mu$ is $f$--adapted and $\\chi$--hyperbolic, then $\\mu$--a.e. $x\\in M$\nhas a Pesin chart $\\Psi_x:[-Q_\\varepsilon(x),Q_\\varepsilon(x)]^2\\to M$ s.t.\n$\\lim_{n\\to\\infty}\\tfrac{1}{n}\\log Q_\\varepsilon(f^n(x))=0$.\n\\item Define $\\varepsilon$--double charts $\\Psi_x^{p^s,p^u}$, the two-sided versions of Pesin charts\nthat control separately the local forward and local backward hyperbolicity at $x$.\n\\item Construct a countable collection $\\mathfs A$ of $\\varepsilon$--double charts that are dense\nin the space of all $\\varepsilon$--double charts. The notion of denseness is defined in terms of\nfinitely many parameters of $x$.\n\\item Define the transition between $\\varepsilon$--double charts s.t. $p^s,p^u$\nare as maximal as possible. This is important to establish the inverse theorem (Theorem \\ref{Thm-inverse}). \n\\item Apply a Bowen-Sina{\\u\\i} refinement (following \\cite{Bowen-LNM}).\nThe resulting partition defines a topological Markov shift $(\\Sigma,\\sigma)$ and a map\n$\\pi:\\Sigma\\to M$ satisfying Theorem \\ref{Thm-main}.\n\\end{enumerate}\n\n\\medskip\nContrary to \\cite{Sarig-JAMS,Lima-Sarig}, we do not require $M$ to be compact (not even to have bounded curvature)\nneither $f$ to have uniformly bounded $C^{1+\\beta}$ norm. As a consequence,\nwe have to control the parameters appearing in the construction more carefully.\nIn the methodology of proof above,\nthis is reflected in steps (1), (3), (4). Steps (2) and (5) work almost verbatim as in \\cite{Sarig-JAMS}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Preliminaries}\\label{Section-preliminaries}\n\nLet $\\mathfs G=(V,E)$ be an oriented graph, where $V=$ vertex set and $E=$ edge set.\nWe denote edges by $v\\to w$, and we assume that $V$ is countable.\n\n\\medskip\n\\noindent\n{\\sc Topological Markov shift (TMS):} A {\\em topological Markov shift} (TMS) is a pair $(\\Sigma,\\sigma)$\nwhere\n$$\n\\Sigma:=\\{\\text{$\\mathbb{Z}$--indexed paths on $\\mathfs G$}\\}=\n\\left\\{\\underline{v}=\\{v_n\\}_{n\\in\\mathbb{Z}}\\in V^{\\mathbb{Z}}:v_n\\to v_{n+1}, \\forall n\\in\\mathbb{Z}\\right\\}\n$$\nand $\\sigma:\\Sigma\\to\\Sigma$ is the left shift, $[\\sigma(\\underline v)]_n=v_{n+1}$. \nThe {\\em recurrent set} of $\\Sigma$ is\n$$\n\\Sigma^\\#:=\\left\\{\\underline v\\in\\Sigma:\\exists v,w\\in V\\text{ s.t. }\\begin{array}{l}v_n=v\\text{ for infinitely many }n>0\\\\\nv_n=w\\text{ for infinitely many }n<0\n\\end{array}\\right\\}.\n$$\nWe endow $\\Sigma$ with the distance $d(\\underline v,\\underline w):={\\rm exp}[-\\min\\{|n|\\in\\mathbb{Z}:v_n\\neq w_n\\}]$.\n\n\\medskip\nWrite $a=e^{\\pm\\varepsilon}b$ when $e^{-\\varepsilon}\\leq \\frac{a}{b}\\leq e^\\varepsilon$,\nand $a=\\pm b$ when $-|b|\\leq a\\leq |b|$. Given an open set $U\\subset \\mathbb{R}^n$ and $h:U\\to \\mathbb{R}^m$,\nlet $\\|h\\|_0:=\\sup_{x\\in U}\\|h(x)\\|$ denote the $C^0$ norm of $h$. For $0<\\beta<1$,\nlet $\\Hol{\\beta}(h):=\\sup\\frac{\\|h(x)-h(y)\\|}{\\|x-y\\|^\\beta}$ \nwhere the supremum ranges over distinct elements $x,y\\in U$.\nIf $h$ is differentiable, let\n$\\|h\\|_1:=\\|h\\|_0+\\|dh\\|_0$ denote its $C^1$ norm, and\n$\\|h\\|_{1+\\beta}:=\\|h\\|_{C^1}+\\Hol{\\beta}(dh)$ its $C^{1+\\beta}$ norm.\nGiven $x\\in M$, remember that $B_x[r]\\subset T_xM$ is the ball with center $0\\in T_xM$ and radius $r$.\nAlso define $R[r]:=[-r,r]^2\\subset\\mathbb{R}^2$.\n\n\\medskip\nThe diameter of $M$ is less than one, hence we can assume that $a=b$: just change\n$a,b$ to $\\max\\{a,b\\}$. For symmetry and simplification purposes,\nwe will sometimes use (A3)--(A5) in the weaker forms below.\nDefine $\\rho(x):=d(\\{f^{-1}(x),x,f(x)\\},\\mathfs D)$, then (A3)--(A5) imply\nthat for all $x\\in M\\backslash\\mathfs D$:\n\\begin{enumerate}[ii]\n\\item[(A3)'] If $y_1,y_2\\in D_x$ then\n$\n\\|\\widetilde{d(\\exp{y_1})_{v_1}}-\\widetilde{d(\\exp{y_2})_{v_2}}\\|\n\\leq \\rho(x)^{-a}d_{\\rm Sas}(v_1,v_2)\n$\nfor all $\\|v_1\\|,\\|v_2\\|\\leq 2\\mathfrak r(x)$,\nand \n$\\|\\tau(y_1,z_1)-\\tau(y_2,z_2)\\|\\leq d(x,\\mathfs D)^{-a}[d(y_1,y_2)+d(z_1,z_2)]$\nfor all $z_1,z_2\\in D_x$.\n\\item[(A4)'] If $y_1,y_2\\in D_x$ then the map $\\tau(y_1,\\cdot)-\\tau(y_2,\\cdot):D_x\\to \\mathfs L_x$\nhas Lipschitz constant $\\leq \\rho(x)^{-a}d(y_1,y_2)$.\n\\item[(A5)'] If $y\\in D_x$ then $\\|df_y^{\\pm 1}\\|\\leq \\rho(x)^{-a}$.\n\\end{enumerate}\nHere is a consequence of (A5) and the inverse theorem, written in symmetric form:\n\\begin{enumerate}[(A7)]\n\\item[(A7)] $\\|df^{\\pm 1}_x\\|\\geq m(df^{\\pm 1}_x)\\geq \\rho(x)^a$.\n\\end{enumerate}\nAbove, $m(A):=\\|A^{-1}\\|^{-1}$. For the ease of reference, we collect (A1)--(A7) in Appendix A\nin the format we will use in the text.\n\n\\medskip\nWe note that $\\mu$ is $f$--adapted iff $\\int \\log\\rho(x)d\\mu>-\\infty$.\nIf $\\mu$ is $f$--adapted then by $\\mu$--invariance the functions\n$-\\log d(f^{-1}(x),\\mathfs D),-\\log d(x,\\mathfs D),-\\log d(f(x),\\mathfs D)$ are in $L^1(\\mu)$,\nhence is also their maximum $-\\log\\rho(x)$. The reverse implication is proved similarly.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Linear Pesin theory}\n\nIn this section we construct changes of coordinates that make $df$ a hyperbolic matrix.\nSince we are dealing with the action of the derivative only, the closeness of $x$ to $\\mathfs D$\nis irrelevant.\n\n\\medskip\nFix $\\chi>0$, and let ${\\rm NUH}_\\chi$ be the set\nof $x\\in M\\backslash \\bigcup_{n\\in\\mathbb{Z}}f^n(\\mathfs D)$ for which\nthere are vectors $\\{e^s_{f^n(x)}\\}_{n\\in\\mathbb{Z}}$, $\\{e^u_{f^n(x)}\\}_{n\\in\\mathbb{Z}}$ s.t. for every\n$y=f^n(x)$, $n\\in\\mathbb{Z}$, it holds:\n\\begin{enumerate}[(1)]\n\\item $e^{s\/u}_y\\in T_yM$, $\\|e^{s\/u}_y\\|=1$.\n\\item ${\\rm span}(df^m_ye^{s\/u}_y)={\\rm span}(e^{s\/u}_{f^m(y)})$ for all $m\\in\\mathbb{Z}$.\n\\item $\\lim_{m\\to\\pm\\infty}\\tfrac{1}{m}\\log\\|df^m_y e^s_y\\|<-\\chi$ and\n$\\lim_{m\\to\\pm\\infty}\\tfrac{1}{m}\\log\\|df^m_y e^u_y\\|>\\chi$.\n\\item $\\lim_{m\\to\\pm\\infty}\\tfrac{1}{m}\\log|\\sin\\alpha(f^m(y))|=0$, where\n$\\alpha(f^m(y))=\\angle(e^s_{f^m(y)},e^u_{f^m(y)})$.\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Oseledets-Pesin reduction}\n\nWe represent $df_x$ as a hyperbolic matrix.\n\n\n\n\n\\medskip\n\\noindent\n{\\sc Parameters $s(x),u(x)$:} For $x\\in{\\rm NUH}_\\chi$, define\n$$\ns(x):=\\sqrt{2}\\left(\\sum_{n\\geq 0}e^{2n\\chi}\\|df^n_xe^s_x\\|^2\\right)^{1\/2} \\text{ and }\nu(x):=\\sqrt{2}\\left(\\sum_{n\\geq 0}e^{2n\\chi}\\|df^{-n}_xe^u_x\\|^2\\right)^{1\/2}.\n$$\n\n\\medskip\nThese numbers are well-defined because $x\\in{\\rm NUH}_\\chi$, and $s(x),u(x)\\geq \\sqrt{2}$.\nLet $e_1=(1,0),e_2=(0,1)$ be the canonical basis of $\\mathbb{R}^2$.\n\n\\medskip\n\\noindent\n{\\sc Linear map $C_\\chi(x):$} For $x\\in{\\rm NUH}_\\chi$, let\n$C_\\chi(x):\\mathbb{R}^2\\to T_xM$ be the linear map s.t.\n$$\nC_\\chi(x):e_1\\mapsto \\frac{e^s_x}{s(x)}\\ ,\\ C_\\chi(x): e_2\\mapsto \\frac{e^u_x}{u(x)}\\cdot\n$$\n\n\\medskip\nGiven a linear transformation, let $\\|\\cdot\\|$ denote its sup norm and $\\|\\cdot\\|_{\\rm Frob}$ its\nFrobenius norm\\footnote{The Frobenius norm\nof a $2\\times 2$ matrix $A=\\left[\\begin{array}{cc}a & b\\\\ c & d\\end{array}\\right]$ is\n$\\|A\\|_{\\rm Frob}=\\sqrt{a^2+b^2+c^2+d^2}$.}. The Frobenius norm is equivalent to the usual sup norm,\nwith $\\|\\cdot\\|\\leq \\|\\cdot\\|_{\\rm Frob}\\leq \\sqrt{2}\\|\\cdot\\|$.\n  \n\\begin{lemma}\\label{Lemma-linear-reduction}\nFor all $x\\in{\\rm NUH}_\\chi$, the following holds:\n\\begin{enumerate}[{\\rm (1)}]\n\\item $\\|C_\\chi(x)\\|\\leq \\|C_\\chi(x)\\|_{\\rm Frob}\\leq 1$\nand $\\|C_\\chi(x)^{-1}\\|_{\\rm Frob}=\\tfrac{\\sqrt{s(x)^2+u(x)^2}}{|\\sin\\alpha(x)|}$.\n\\item $C_\\chi(f(x))^{-1}\\circ df_x\\circ C_\\chi(x)$ is a diagonal matrix with diagonal entries $A,B\\in\\mathbb{R}$\ns.t. $|A|<e^{-\\chi}$ and $|B|>e^\\chi$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(a) In the basis $\\{e_1,e_2\\}$ of $\\mathbb{R}^2$ and the basis $\\{e^s_x,(e^s_x)^\\perp\\}$ of $T_xM$, $C_\\chi(x)$ takes the\nform $\\left[\\begin{array}{cc}\\tfrac{1}{s(x)}& \\tfrac{\\cos\\alpha(x)}{u(x)}\\\\ 0&\\tfrac{\\sin\\alpha(x)}{u(x)}\\end{array}\\right]$,\nhence $\\|C_\\chi(x)\\|_{\\rm Frob}^2=\\tfrac{1}{s(x)^2}+\\tfrac{1}{u(x)^2}\\leq 1$. The inverse of\n$C_\\chi(x)$ is\n$\\left[\\begin{array}{cc}s(x)& -\\tfrac{s(x)\\cos\\alpha(x)}{\\sin\\alpha(x)}\\\\ 0&\\tfrac{u(x)}{\\sin\\alpha(x)}\\end{array}\\right]$,\ntherefore $\\|C_\\chi(x)^{-1}\\|_{\\rm Frob}=\\tfrac{\\sqrt{s(x)^2+u(x)^2}}{|\\sin\\alpha(x)|}$.\n\n\\medskip\n\\noindent\n(b) It is clear that $e_1,e_2$ are eigenvectors of $C_\\chi(f(x))^{-1}\\circ df_x\\circ C_\\chi(x)$.\nWe calculate the eigenvalue of $e_1$ (the calculation of the eigenvalue of $e_2$ is similar).\nSince $df_xe^s_x=\\pm\\|df_xe^s_x\\|e^s_{f(x)}$,\n$[df_x\\circ C_\\chi(x)](e_1)=\\pm df_x\\left[\\tfrac{e^s_x}{s(x)}\\right]=\\pm\\tfrac{\\|df_xe^s_x\\|}{s(x)}e^s_{f(x)}$, hence\n$[C_\\chi(f(x))^{-1}\\circ df_x\\circ C_\\chi(x)](e_1)=\\pm\\|df_xe^s_x\\|\\tfrac{s(f(x))}{s(x)}e_1$.\nThus $A:=\\pm\\|dfe^s_x\\|\\tfrac{s(f(x))}{s(x)}$ is the eigenvalue of $e_1$. Note that\n\\begin{align*}\ns(f(x))^2&=\\tfrac{2}{e^{2\\chi}\\|df_xe^s_x\\|^2}\\sum_{n\\geq 1}e^{2n\\chi}\\|df^n_xe^s_x\\|^2\n=\\tfrac{s(x)^2-2}{e^{2\\chi}\\|df_xe^s_x\\|^2}<\\tfrac{s(x)^2}{e^{2\\chi}\\|df_xe^s_x\\|^2},\n\\end{align*}\ntherefore $|A|<e^{-\\chi}$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The set ${\\rm NUH}_\\chi^*$}\n\nWe need to control the exponential rate decay of the distance of\ntrajectories to the set of discontinuities $\\mathfs D$.\n\n\\medskip\n\\noindent\n{\\sc Regular set:} We define the {\\em regular set of $f$} by \n$$\n{\\rm Reg}:=\\left\\{x\\in M\\backslash\\mathfs D:\\lim_{n\\to\\pm\\infty}\n\\tfrac{1}{|n|}\\log\\rho(f^n(x))=0\\right\\}.\n$$\n\n\n\n\n\\medskip\n\\noindent\n{\\sc The set ${\\rm NUH}_\\chi^*$:} It is the set of $x\\in{\\rm NUH}_\\chi$ with the following properties:\n\\begin{enumerate}[(1)]\n\\item $x\\in{\\rm Reg}$.\n\\item There exist sequences $n_k,m_k\\to\\infty$ s.t.\n$C_\\chi(f^{n_k}(x)),C_\\chi(f^{-m_k}(x))\\to C_\\chi(x)$. \n\\item $\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|C_\\chi(f^n(x))\\|=0$.\n\\item $\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|C_\\chi(f^n(x))^{-1}\\|=0$.\n\\end{enumerate}\n\n\\medskip\nThe next lemma shows that relevant measures are carried by ${\\rm NUH}_\\chi^*$.\n\n\n\n\\begin{lemma}\\label{Lemma-adaptedness}\nIf $\\mu$ is $f$--adapted and $\\chi$--hyperbolic, then $\\mu[{\\rm NUH}_\\chi^*]=1$.\n\\end{lemma}\n\n\n\\begin{proof}\nBy (A5) and the $f$--adaptedness of $\\mu$, $\\int \\log^+\\|df^{\\pm 1}\\|d\\mu<\\infty$\nhence the Oseledets theorem applies to the cocycle $df^n$ and\nmeasure $\\mu$.\nSince $\\mu$ is $\\chi$--hyperbolic, $\\mu[{\\rm NUH}_\\chi]=1$.\nBy $f$--adaptedness and the Birkhoff ergodic theorem\\footnote{Here we are using that if $\\varphi:M\\to\\mathbb{R}$ satisfies\n$\\int |\\varphi|d\\mu<\\infty$ then $\\liminf_{n\\to\\pm\\infty}\\tfrac{1}{n}\\varphi(f^n(x))=0$ $\\mu$--a.e.\nIndeed, by the Birkhoff theorem\n$\\widetilde\\varphi(x)=\\lim_{n\\to\\infty}\\tfrac{1}{n}\\sum_{i=0}^{n-1}\\varphi(f^i(x))$ exists $\\mu$--a.e., hence\n$\\lim_{n\\to\\infty}\\tfrac{1}{n}\\varphi(f^n(x))=\\lim_{n\\to\\infty}\\left[\\tfrac{1}{n}\\sum_{i=0}^{n}\\varphi(f^i(x))\n-\\tfrac{1}{n}\\sum_{i=0}^{n-1}\\varphi(f^i(x))\\right]=0$ $\\mu$--a.e. The same argument works\nfor $n\\to-\\infty$.},\n$\\mu({\\rm Reg})=1$.\nBy the Poincar\\'e recurrence theorem, (2) holds $\\mu$--a.e. It remains to check (3)--(4).\n\n\\medskip\nFor $x\\in{\\rm NUH}_\\chi$, let $D_\\chi(x):=C_\\chi(f(x))^{-1}\\circ df_x\\circ C_\\chi(x)$.\nThis defines a cocycle $D_\\chi^{(n)}$ on ${\\rm NUH}_\\chi$.\nWe first show that we can apply the Oseledets theorem for $D_\\chi^{(n)}$ and $\\mu$.\nBy lemma \\ref{Lemma-linear-reduction} and its proof,\n$D_\\chi(x)=\\left[\\begin{array}{cc}A(x) & 0 \\\\ 0 & B(x) \\end{array}\\right]$\nwhere $A(x)^2=e^{-2\\chi}\\tfrac{s(x)^2-2}{s(x)^2}$ and $B(x)^2=e^{2\\chi}\\tfrac{u(f(x))^2}{u(f(x))^2-2}$.\nWe have $\\|D_\\chi(x)\\|=|B(x)|$ and $\\|D_\\chi(x)^{-1}\\|=|A(x)|^{-1}$,\ntherefore we wish to show that\n\\begin{align*}\n\\int \\log |A(x)|d\\mu(x)>-\\infty\\ \\text{ and }\\ \\int \\log |B(x)|d\\mu(x)<\\infty.\n\\end{align*}\nWe prove the first inequality (the second inequality is proved similarly). By (A6),\n$s(x)^2\\geq 2(1+e^{2\\chi}\\|df_xe^s_x\\|^2)\\geq 2(1+e^{2\\chi}\\rho(x)^{2a})$\nhence\n$$\nA(x)^2=e^{-2\\chi}\\tfrac{s(x)^2-2}{s(x)^2}=e^{-2\\chi}\\left(1-\\tfrac{2}{s(x)^2}\\right)\n\\geq\\tfrac{\\rho(x)^{2a}}{1+e^{2\\chi}\\rho(x)^{2a}}\\geq\\tfrac{\\rho(x)^{2a}}{1+e^{2\\chi}}\\,\\cdot\n$$\nTherefore\n$$\n\\int \\log|A(x)|d\\mu(x)\\geq a\\int\\log \\rho(x)d\\mu(x)-\\tfrac{1}{2}\\log(1+e^{2\\chi})>-\\infty.\n$$\nBy a similar reasoning, $\\int \\log |B(x)|d\\mu(x)<\\infty$.\nTherefore we can apply the Oseledets theorem for $D_\\chi^{(n)}$ and $\\mu$:\nthere is an $f$--invariant set $X\\subset{\\rm NUH}_\\chi$ with $\\mu(X)=1$ s.t. every\n$x\\in X$ satisfies (2) and $\\lim_{n\\to\\infty}\\tfrac{1}{n}\\log\\|D_\\chi^{(n)}(x)\\|$ exists.\nWe claim that (3)--(4) hold in $X$.\n\n\\medskip\nWe first show that the Lyapunov exponents of $D_\\chi^{(n)}$ and $df^n$ coincide in $X$.\nFix $x\\in X$, and take $n_k\\to\\infty$ s.t. $C_\\chi(f^{n_k}(x))\\to C_\\chi(x)$.\nSince\n$\\|D_\\chi^{(n)}(x)\\|\\leq \\|C_\\chi(f^{n}(x))^{-1}\\|\\|df^n_x\\|\\|C_\\chi(x)\\|\\leq \\|C_\\chi(f^{n}(x))^{-1}\\|\\|df^n_x\\|$,\n\\begin{align*}\n&\\lim_{n\\to\\infty}\\tfrac{1}{n}\\log\\|D_\\chi^{(n)}(x)\\|=\\limsup_{k\\to\\infty}\\tfrac{1}{n_k}\\log\\|D_\\chi^{(n_k)}(x)\\|\\\\\n&\\leq \\limsup_{k\\to\\infty}\\tfrac{1}{n_k}\\log\\|C_\\chi(f^{n_k}(x))^{-1}\\|+\\limsup_{k\\to\\infty}\\tfrac{1}{n_k}\\log\\|df^{n_k}_x\\|\n=\\lim_{n\\to\\infty}\\tfrac{1}{n}\\log\\|df^n_x\\|.\n\\end{align*}\nSimilarly, $\\|df^n_x\\|\\leq \\|C_\\chi(f^n(x))\\|\\|D_\\chi^{(n)}(x)\\|\\|C_\\chi(x)^{-1}\\|\\leq\\|D_\\chi^{(n)}(x)\\|\\|C_\\chi(x)^{-1}\\|$,\nthus\n\\begin{align*}\n&\\lim_{n\\to\\infty}\\tfrac{1}{n}\\log\\|df^n_x\\|=\\limsup_{k\\to\\infty}\\tfrac{1}{n_k}\\log\\|df^{n_k}_x\\|\n\\leq \\limsup_{k\\to\\infty}\\tfrac{1}{n_k}\\log\\|D_\\chi^{(n_k)}(x)\\|\\\\\n&=\\lim_{n\\to\\infty}\\tfrac{1}{n}\\log\\|D_\\chi^{(n)}(x)\\|.\n\\end{align*}\nHence $\\lim_{n\\to\\infty}\\tfrac{1}{n}\\log\\|D_\\chi^{(n)}(x)\\|=\\lim_{n\\to\\infty}\\tfrac{1}{n}\\log\\|df^n_x\\|$.\nApplying the same argument along the sequence $m_k\\to\\infty$ for which\n$C_\\chi(f^{-m_k}(x))\\to C_\\chi(x)$, we obtain\n\\begin{equation}\\label{equality-spectra}\n\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|D_\\chi^{(n)}(x)\\|=\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|df^n_x\\|.\n\\end{equation}\n\n\\medskip\nSince $\\|C_\\chi(\\cdot)\\|\\leq 1$, $\\limsup_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|C_\\chi(f^n(x))\\|\\leq 0$. Reversely,\nthe inequality $\\|df^n_x\\|\\leq \\|C_\\chi(f^n(x))\\|\\|D_\\chi^{(n)}(x)\\|\\|C_\\chi(x)^{-1}\\|$ implies\n$$\n\\liminf_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|C_\\chi(f^n(x))\\|\\geq \\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|df^n_x\\|-\n\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|D_\\chi^{(n)}(x)\\|=0.\n$$\nThis proves (3). A similar argument to the proof of (3) does {\\em not} give (4). For that, we introduce some normalizing matrices. \nLet $\\lambda_1(x),\\lambda_2(x)$ be the Lyapunov exponents of $df^n$ at $x$. By (\\ref{equality-spectra}),\n$D_\\chi^{(n)}$ has the same Lyapunov exponents at $x$. Taking\n$\\Lambda_\\chi(x):=\\left[\\begin{array}{cc}\\lambda_1(x) & 0 \\\\ 0 & \\lambda_2(x)\\end{array}\\right]$,\nwe have $\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log \\|(D_\\chi^{(n)}(x)\\Lambda_\\chi(x)^{-n})^{\\pm 1}\\|=0$.\n\n\\medskip\nSimilarly, we can define $\\Lambda(x):T_xM\\to T_xM$ by $\\Lambda(x)e^s_x=\\lambda_1(x)e^s_x$\nand $\\Lambda(x)e^u_x=\\lambda_2(x)e^u_x$ and observe that\n$\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|(df^n_x\\Lambda(x)^{-n})^{\\pm 1}\\|=0$. Since\n$\\Lambda_\\chi(x)=C_\\chi(x)^{-1} \\Lambda(x) C_\\chi(x)$, it follows that\n\\begin{align*}\n&C_\\chi(f^n(x))^{-1}=D_\\chi^{(n)}(x)C_\\chi(x)^{-1}(df^n_x)^{-1}\\\\\n&=[D_\\chi^{(n)}(x)\\Lambda_\\chi(x)^{-n}][\\Lambda_\\chi(x)^n C_\\chi(x)^{-1} \\Lambda(x)^{-n}]\n[df^n_x\\Lambda(x)^{-n}]^{-1}\\\\\n&=[D_\\chi^{(n)}(x)\\Lambda_\\chi(x)^{-n}]C_\\chi(x)^{-1} [df^n_x\\Lambda(x)^{-n}]^{-1}\n\\end{align*}\nand hence\n\\begin{align*}\n&\\limsup_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|C_\\chi(f^n(x))^{-1}\\|\\\\\n&\\leq\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log \\|D_\\chi^{(n)}(x)\\Lambda_\\chi(x)^{-n}\\|+\n\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|(df^n_x\\Lambda(x)^{-n})^{-1}\\|=0.\n\\end{align*}\nSince $\\liminf_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|C_\\chi(f^n(x))^{-1}\\|\\geq 0$, property (4) holds.\nHence $X$ satisfies (2)--(4) and $\\mu[X]=1$. Therefore\n$X\\cap{\\rm Reg}\\subset {\\rm NUH}_\\chi^*$ has full $\\mu$--measure.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Non-linear Pesin theory}\n\nWe now define charts that make $f$ itself look like a hyperbolic matrix.\n\n\\medskip\n\\noindent\n{\\sc Pesin chart $\\Psi_x$:} For $x\\in{\\rm NUH}_\\chi$, let\n$\\Psi_x:R[\\mathfrak r(x)]\\to M$, $\\Psi_x:=\\exp{x}\\circ C_\\chi(x)$.\n$\\Psi_x$ is called the {\\em Pesin chart at $x$}.\n\n\\medskip\nGiven $x\\in M\\backslash\\mathfs D$, let $\\iota_x:T_xM\\to \\mathbb{R}^2$ be an isometry.\nIf $y\\in D_x$ and $A:\\mathbb{R}^2\\to T_yM$ is a linear map,\nwe can define $\\widetilde{A}:\\mathbb{R}^2\\to \\mathbb{R}^2$, $\\widetilde{A}:=\\iota_x\\circ P_{y,x}\\circ A$.\nAgain, $\\widetilde A$ depends on $x$ but $\\|\\widetilde{A}\\|$ does not.\n\n\n\\begin{lemma}\\label{Lemma-Pesin-chart}\nThe Pesin chart $\\Psi_x$ is a diffeomorphism onto its image. Moreover:\n\\begin{enumerate}[{\\rm (1)}]\n\\item $\\Psi_x$ is $2$--Lipschitz and $\\Psi_x^{-1}$ is $2\\|C_\\chi(x)^{-1}\\|$--Lipschitz.\n\\item $\\|\\widetilde{d(\\Psi_x)_{v_1}}-\\widetilde{d(\\Psi_x)_{v_2}}\\|\\leq d(x,\\mathfs D)^{-a}\\|v_1-v_2\\|$\nfor all $v_1,v_2\\in R[\\mathfrak r(x)]$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSince $C_\\chi(x)$ is a contraction,\n$C_\\chi(x)R[\\mathfrak r(x)]\\subset B_x[2\\mathfrak r(x)]$\nand so $\\Psi_x$ is well-defined with inverse $C_\\chi(x)^{-1}\\circ \\exp{x}^{-1}$.\nIt is a diffeomorphism because $C_\\chi(x)$ and $\\exp{x}$ are.\n\n\\medskip\n\\noindent\n(1) By (A2), $\\Psi_x$ is $2$--Lipschitz and\n$\\Psi_x^{-1}$ is $2\\|C_\\chi(x)^{-1}\\|$--Lipschitz.\n\n\\medskip\n\\noindent\n(2) Since $C_\\chi(x)v_i\\in B_x[2\\mathfrak r(x)]$, (A3) implies that\n\\begin{align*}\n&\\|\\widetilde{d(\\Psi_x)_{v_1}}-\\widetilde{d(\\Psi_x)_{v_2}}\\|=\n\\|\\widetilde{d(\\exp{x})_{C_\\chi(x)v_1}}\\circ C_\\chi(x)-\n\\widetilde{d(\\exp{x})_{C_\\chi(x)v_2}}\\circ C_\\chi(x)\\|\\\\\n&\\leq d(x,\\mathfs D)^{-a}\\|C_\\chi(x)v_1-C_\\chi(x)v_2\\|\\leq d(x,\\mathfs D)^{-a}\\|v_1-v_2\\|.\n\\end{align*}\n\\end{proof}\n\n\\medskip\nGiven $\\varepsilon>0$, let $I_\\varepsilon:=\\{e^{-\\frac{1}{3}\\varepsilon n}:n\\geq 0\\}$.\n\n\\medskip\n\\noindent\n{\\sc Parameter $Q_\\varepsilon(x)$:} For $x\\in{\\rm NUH}_\\chi$, let\n$Q_\\varepsilon(x):=\\max\\{q\\in I_\\varepsilon:q\\leq \\widetilde Q_\\varepsilon(x)\\}$, where\n$$\n\\widetilde Q_\\varepsilon(x)=\\varepsilon^{3\/\\beta}\n\\min\\left\\{\\|C_\\chi(x)^{-1}\\|_{\\rm Frob}^{-24\/\\beta},\\|C_\\chi(f(x))^{-1}\\|^{-12\/\\beta}_{\\rm Frob}\\rho(x)^{72a\/\\beta}\\right\\}.\n$$\n\n\\medskip\nThe term $\\varepsilon^{3\/\\beta}$ will allow to absorb multiplicative constants.\nThe choice of $Q_\\varepsilon(x)$ guarantees that\nthe composition $\\Psi_{f(x)}^{-1}\\circ f\\circ \\Psi_x$ is well-defined in $R[10Q_\\varepsilon(x)]$\nand it is close to a linear hyperbolic map (Theorem \\ref{Thm-non-linear-Pesin}),\nand it allows to compare nearby Pesin charts (Proposition \\ref{Lemma-overlap}).\nWe have the following bounds:\n\\begin{align*}\n&Q_\\varepsilon(x)\\leq \\varepsilon^{3\/\\beta}, \\|C_\\chi(x)^{-1}\\|Q_\\varepsilon(x)^{\\beta\/24}\\leq \\varepsilon^{1\/8},\n\\|C_\\chi(f(x))^{-1}\\|Q_\\varepsilon(x)^{\\beta\/12}\\leq \\varepsilon^{1\/4},\\\\\n&\\rho(x)^{-a}Q_\\varepsilon(x)^{\\beta\/72}<\\varepsilon^{1\/24}.\n\\end{align*}\n\n\n\\begin{lemma}[Temperedness lemma]\\label{Lemma-temperedness}\nIf $x\\in{\\rm NUH}_\\chi^*$, then\n$$\n\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log Q_\\varepsilon(f^n(x))=0.\n$$\n\\end{lemma}\n\n\n\\begin{proof}\nClearly $\\limsup_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log Q_\\varepsilon(f^n(x))\\leq 0$.\nReversely, $x\\in{\\rm Reg}$ implies that $\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\rho(f^n(x))=0$.\nBy property (4) in the definition of ${\\rm NUH}_\\chi^*$,\n$\\lim_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log\\|C_\\chi(f^n(x))^{-1}\\|=0$ hence $\\liminf_{n\\to\\pm\\infty}\\tfrac{1}{|n|}\\log Q_\\varepsilon(f^n(x))\\geq 0$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The map $f$ in Pesin charts}\n\n\n\\begin{theorem}\\label{Thm-non-linear-Pesin}\nThe following holds for all $\\varepsilon>0$ small enough: If $x\\in{\\rm NUH}_\\chi$\nthen $f_x:=\\Psi_{f(x)}^{-1}\\circ f\\circ\\Psi_x$ is well-defined on\n$R[10Q_\\varepsilon(x)]$ and satisfies:\n\\begin{enumerate}[{\\rm (1)}]\n\\item $d(f_x)_0=C_\\chi(f(x))^{-1}\\circ df_x\\circ C_\\chi(x)$.\n\\item $f_x(v_1,v_2)=(Av_1+h_1(v_1,v_2),Bv_2+h_2(v_1,v_2))$ for $(v_1,v_2)\\in R[10Q_\\varepsilon(x)]$ where:\n\\begin{enumerate}[{\\rm (a)}]\n\\item $|A|<e^{-\\chi}$ and $|B|>e^\\chi$, cf. Lemma \\ref{Lemma-linear-reduction}.\n\\item $h_1(0,0)=h_2(0,0)=0$ and $\\nabla h_1(0,0)=\\nabla h_2(0,0)=0$.\n\\item $\\|h_1\\|_{1+\\beta\/2}<\\varepsilon$ and $\\|h_2\\|_{1+\\beta\/2}<\\varepsilon$.\n\\end{enumerate}\n\\item $\\|df_x\\|_0<\\tfrac{2(1+e^{2\\chi})}{\\rho(x)^a}$.\n\\end{enumerate}\nThe norms above are taken in $R[10Q_\\varepsilon(x)]$.\nA similar statement holds for $f_x^{-1}:=\\Psi_x^{-1}\\circ f^{-1}\\circ \\Psi_{f(x)}$.\n\\end{theorem}\n\n\n\\begin{proof}\nThe first step is to show that $f_x:R[10Q_\\varepsilon(x)]\\to\\mathbb{R}^2$ is well-defined.\nUsing that $C_\\chi(x)$ is a contraction, $C_\\chi(x)R[10Q_\\varepsilon(x)]\\subset B_x[20Q_\\varepsilon(x)]$.\nSince $C_\\chi(f(x))^{-1}$ is globally defined, it is enough to show that\n$$\n(f\\circ\\exp{x})(B_x[20Q_\\varepsilon(x)])\\subset \\exp{f(x)}(B_{f(x)}[2\\mathfrak r(f(x))]).\n$$\nFor small $\\varepsilon>0$ we have:\n\\begin{enumerate}[$\\circ$]\n\\item $20Q_\\varepsilon(x)<2\\mathfrak r(x)\\Rightarrow\\exp{x}$ is well-defined on $B_x[20Q_\\varepsilon(x)]$. By (A2),\n$\\exp{x}$ maps $B_x[20Q_\\varepsilon(x)]$ diffeomorphically into $B(x,40Q_\\varepsilon(x))$.\n\\item $40Q_\\varepsilon(x)<2\\mathfrak r(x)\\Rightarrow B(x,40Q_\\varepsilon(x))\\subset B(x,2\\mathfrak r(x))$. \nBy (A5), $f$ maps $B(x,40Q_\\varepsilon(x))$ diffeomorphically into $B(f(x),40 \\rho(x)^{-a}Q_\\varepsilon(x))$.\n\\item $40\\rho(x)^{-a}Q_\\varepsilon(x)<\\tfrac{\\mathfrak r(f(x))}{2}\\Rightarrow B(f(x),40 \\rho(x)^{-a}Q_\\varepsilon(x))\\subset\nB\\left(f(x),\\frac{\\mathfrak r(f(x))}{2}\\right)$. By (A2),\n$\\exp{f(x)}^{-1}$ maps $B\\left(f(x),\\frac{\\mathfrak r(f(x))}{2}\\right)$ diffeomorphically into\n$B_{f(x)}[\\mathfrak r(f(x))]$.\n\\end{enumerate}\nTherefore $f_x:R[10Q_\\varepsilon(x)]\\to\\mathbb{R}^2$ is a diffeomorphism onto its image.\n\n\\medskip\nWe check (1)--(2). Property (1) is clear since\n$d(\\Psi_x)_0=C_\\chi(x)$ and $d(\\Psi_{f(x)})_0=C_\\chi(f(x))$. By Lemma \\ref{Lemma-linear-reduction},\n$d(f_x)_0=\\left[\\begin{array}{cc}A & 0 \\\\ 0 & B\\end{array}\\right]$ with $|A|<e^{-\\chi}$ and\n$|B|>e^\\chi$. Define $h_1,h_2:R[10Q_\\varepsilon(x)]\\to\\mathbb{R} $ by \n$f_x(v_1,v_2)=(Av_1+h_1(v_1,v_2),Bv_2+h_2(v_1,v_2))$. Then (a)--(b) are automatically\nsatisfied. It remains to prove (c).\n\n\\medskip\n\\noindent\n{\\sc Claim:} $\\|d(f_x)_{w_1}-d(f_x)_{w_2}\\|\\leq \\tfrac{\\varepsilon}{3}\\|w_1-w_2\\|^{\\beta\/2}$\nfor all $w_1,w_2\\in R[10Q_\\varepsilon(x)]$.\n\n\\medskip\nBefore proving the claim, let us show how to conclude (c). Let $h=(h_1,h_2)$.\nIf $\\varepsilon>0$ is small enough then $R[10Q_\\varepsilon(x)]\\subset B_x[1]$. Applying the claim with $w_2=0$,\nwe get $\\|dh_w\\|\\leq \\frac{\\varepsilon}{3}\\|w\\|^{\\beta\/2}<\\tfrac{\\varepsilon}{3}$. By the mean value inequality,\n$\\|h(w)\\|\\leq \\tfrac{\\varepsilon}{3}\\|w\\|<\\tfrac{\\varepsilon}{3}$, hence $\\|h\\|_{1+\\beta\/2}<\\varepsilon$.\n\n\\begin{proof}[Proof of the claim.]\nFor $i=1,2$, define\n$$\nA_i= \\widetilde{d(\\exp{f(x)}^{-1})_{(f\\circ \\exp{x})(w_i)}}\\,,\\\nB_i=\\widetilde{df_{\\exp{x}(w_i)}}\\,,\\ C_i=\\widetilde{d(\\exp{x})_{w_i}}.\n$$\nWe first estimate $\\|A_1 B_1 C_1-A_2 B_2 C_2\\|$.\n\\begin{enumerate}[$\\circ$]\n\\item By (A2), $\\|A_i\\|\\leq 2$. By (A2), (A3), (A5):\n\\begin{align*}\n&\\|A_1-A_2\\|\\leq d(f(x),\\mathfs D)^{-a}d((f\\circ \\exp{x})(w_1),(f\\circ \\exp{x})(w_2))\\\\\n&\\leq 2d(x,\\mathfs D)^{-a}d(f(x),\\mathfs D)^{-a}\\|w_1-w_2\\|\\leq 2\\rho(x)^{-2a}\\|w_1-w_2\\|.\n\\end{align*}\n\\item By (A5), $\\|B_i\\|\\leq \\rho(x)^{-a}$. By (A2) and (A6):\n$$\n\\|B_1-B_2\\|\\leq \\mathfrak K d(\\exp{x}(w_1),\\exp{x}(w_2))^{\\beta}\\leq 2\\mathfrak K\\|w_1-w_2\\|^\\beta.\n$$\n\\item By (A2), $\\|C_i\\|\\leq 2$. By (A3):\n$$\n\\|C_1-C_2\\|\\leq d(x,\\mathfs D)^{-a}\\|w_1-w_2\\|\\leq \\rho(x)^{-a}\\|w_1-w_2\\|.\n$$\n\\end{enumerate}\nBy a crude approximation, we get $\\|A_1 B_1 C_1-A_2 B_2 C_2\\|\\leq 24\\mathfrak K\\rho(x)^{-3a}\\|w_1-w_2\\|^\\beta$.\nNow we estimate $\\|d(f_x)_{w_1}-d(f_x)_{w_2}\\|$:\n\\begin{align*}\n&\\|d(f_x)_{w_1}-d(f_x)_{w_2}\\|\\leq \\|C_\\chi(f(x))^{-1}\\| \\|A_1 B_1 C_1-A_2 B_2 C_2\\| \\|C_\\chi(x)\\|\\\\\n&\\leq 24\\mathfrak K\\rho(x)^{-3a}\\|C_\\chi(f(x))^{-1}\\|\\|w_1-w_2\\|^\\beta.\n\\end{align*}\nSince $\\|w_1-w_2\\|<40Q_\\varepsilon(x)$, if $\\varepsilon>0$ is small enough then\n\\begin{align*}\n&24\\mathfrak K\\rho(x)^{-3a}\\|C_\\chi(f(x))^{-1}\\|\\|w_1-w_2\\|^{\\beta\/2}\n\\leq 200\\mathfrak K\\rho(x)^{-3a}\\varepsilon^{3\/2}\\|C_\\chi(f(x))^{-1}\\|^{-5}\\rho(x)^{36a}\\\\\n&\\leq 200\\mathfrak K\\varepsilon^{3\/2}<\\varepsilon.\n\\end{align*}\nThis completes the proof of the claim.\n\\end{proof}\n\n\\medskip\n\\noindent\n(3) In the proof of Lemma \\ref{Lemma-adaptedness} we showed that\n$\\|d(f_x)_0\\|=|B(x)|\\leq \\tfrac{\\sqrt{1+e^{2\\chi}}}{\\rho(x)^a}<\\tfrac{1+e^{2\\chi}}{\\rho(x)^a}$.\nBy part (2) above, if $w\\in R[10Q_\\varepsilon(x)]$ then\n$\\|d(f_x)_w\\|\\leq \\varepsilon\\|w\\|^{\\beta\/2}+\\tfrac{1+e^{2\\chi}}{\\rho(x)^a}<\\tfrac{2(1+e^{2\\chi})}{\\rho(x)^a}$,\nsince $\\varepsilon\\|w\\|^{\\beta\/2}<1<\\tfrac{1+e^{2\\chi}}{\\rho(x)^a}$ for small $\\varepsilon>0$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The overlap condition}\\label{section-overlap}\n\nWe now want to change coordinates from $\\Psi_x$ to $\\Psi_y$ when $x,y$\nare ``sufficiently close''. Even when $x$ and $y$ are very close, the behavior of $C_\\chi(x)$ and $C_\\chi(y)$\nmight differ, so we need to compare them.\nWe will eventually consider Pesin charts with different domains.\n\n\\medskip\n\\noindent\n{\\sc Pesin chart $\\Psi_x^\\eta$:} It is restriction of $\\Psi_x$ to $R[\\eta]$, where $0<\\eta\\leq Q_\\varepsilon(x)$.\n\n\\medskip\n\\noindent\n{\\sc $\\varepsilon$--overlap:} Two Pesin charts $\\Psi_{x_1}^{\\eta_1},\\Psi_{x_2}^{\\eta_2}$ are said to\n{\\em $\\varepsilon$--overlap} if $\\tfrac{\\eta_1}{\\eta_2}=e^{\\pm\\varepsilon}$ and if there is $x\\in M$ s.t.\n$x_1,x_2\\in D_x$ and $d(x_1,x_2)+\\|\\widetilde{C_\\chi(x_1)}-\\widetilde{C_\\chi(x_2)}\\|<(\\eta_1\\eta_2)^4$.\n\n\\medskip\nWe write $\\Psi_{x_1}^{\\eta_1}\\overset{\\varepsilon}{\\approx}\\Psi_{x_2}^{\\eta_2}$.\nWe claim that if $\\varepsilon>0$ is small enough, then $\\Psi_{x_1}^{\\eta_1}\\overset{\\varepsilon}{\\approx}\\Psi_{x_2}^{\\eta_2}$\nimplies that $\\Psi_{x_i}(R[10Q_\\varepsilon(x_i)])\\subset D_{x_1}\\cap D_{x_2}$\n(and hence we can apply (A1)--(A3) without mentioning $x$). We prove the inclusion for $i=1$.\nStart noting that, since $d(x_1,x_2)<\\varepsilon d(x_2,\\mathfs D)$,\n$d(x_1,\\mathfs D)=d(x_2,\\mathfs D)\\pm d(x_1,x_2)=(1\\pm\\varepsilon)d(x_2,\\mathfs D)$.\nBy Lemma \\ref{Lemma-Pesin-chart}(1),\n$\\Psi_{x_1}(R[10Q_\\varepsilon(x_1)])\\subset B(x_1,40Q_\\varepsilon(x_1))$. This ball\nis contained in $D_{x_1}$ since $40 Q_\\varepsilon(x_1)\\ll40\\varepsilon^{3\/\\beta}\\rho(x_1)^a<\\mathfrak r(x_1)$.\nWe have\n$$\n\\Psi_{x_1}(R[10Q_\\varepsilon(x_1)])\\subset B(x_1,40 Q_\\varepsilon(x_1))\\subset\nB(x_2,40 Q_\\varepsilon(x_1)+d(x_1,x_2)).\n$$\nSince\n$40Q_\\varepsilon(x_1)+d(x_1,x_2)\\leq 40\\varepsilon^{3\/\\beta}(1+\\varepsilon)^ad(x_2,\\mathfs D)^a+\nd(x_2,\\mathfs D)^a<2\\mathfrak r(x_2)$ for small $\\varepsilon>0$, it follows that\n$\\Psi_{x_1}(R[10Q_\\varepsilon(x_1)])\\subset D_{x_2}$.\nThe next proposition shows that $\\varepsilon$--overlap is strong enough to guarantee\nthat the Pesin charts are close.\n\n\n\\begin{proposition}\\label{Lemma-overlap}\nThe following holds for $\\varepsilon>0$ small enough.\nIf $\\Psi_{x_1}^{\\eta_1}\\overset{\\varepsilon}{\\approx}\\Psi_{x_2}^{\\eta_2}$ then:\n\\begin{enumerate}[{\\rm (1)}]\n\\item {\\sc Control of $s,u$:}\n$\\frac{s(x_1)}{s(x_2)}=e^{\\pm(\\eta_1\\eta_2)^3}$ and $\\frac{u(x_1)}{u(x_2)}=e^{\\pm(\\eta_1\\eta_2)^3}$.\n\\item {\\sc Control of $\\alpha$:} $\\frac{|\\sin\\alpha(x_1)|}{|\\sin\\alpha(x_2)|}=e^{\\pm(\\eta_1\\eta_2)^3}$.\n\\item {\\sc Overlap:} $\\Psi_{x_i}(R[e^{-2\\varepsilon}\\eta_i])\\subset \\Psi_{x_j}(R[\\eta_j])$ for $i,j=1,2$.\n\\item {\\sc Change of coordinates:} For $i,j=1,2$, the map $\\Psi_{x_i}^{-1}\\circ\\Psi_{x_j}$\nis well-defined in $R[d(x_j,\\mathfs D)^a]$,\nand $\\|\\Psi_{x_i}^{-1}\\circ\\Psi_{x_j}-{\\rm Id}\\|_{1+\\beta\/2}<\\varepsilon(\\eta_1\\eta_2)^2$\nwhere the norm is taken in $R[d(x_j, \\mathfs{D})^{2a}]$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} Assume $x_1,x_2\\in D_x$, and let $C_i=\\widetilde{C_\\chi(x_i)}$.\nBy assumption, $d(x_1,x_2)+\\|C_1-C_2\\|<(\\eta_1\\eta_2)^4$. Note that $\\Psi_{x_i}=\\exp{x_i}\\circ P_{x,x_i}\\circ C_i$.\n\n\\medskip\n\\noindent\n(1) We prove the estimate for $s$ (the calculation for $u$ is similar).\nSince $\\varepsilon>0$ is small, it is enough to prove that $\\left|\\tfrac{s(x_1)}{s(x_2)}-1\\right|<\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3$.\nWe have $s(x_i)^{-1}=\\|C_\\chi(x_i)e_1\\|=\\|C_ie_1\\|$, hence\n$|s(x_1)^{-1}-s(x_2)^{-1}|=|\\|C_1e_1\\|-\\|C_2e_1\\||\\leq \\|C_1-C_2\\|<(\\eta_1\\eta_2)^4$.\nAlso\n$s(x_1)=\\|C_\\chi(x_1)e_1\\|^{-1}\\leq \\|C_\\chi(x_1)^{-1}\\|<\\tfrac{\\varepsilon^{3\/\\beta}}{Q_\\varepsilon(x_1)}\n<\\tfrac{\\varepsilon^{3\/\\beta}}{\\eta_1\\eta_2}$,\ntherefore\n$$\n\\left|\\tfrac{s(x_1)}{s(x_2)}-1\\right|=s(x_1)|s(x_1)^{-1}-s(x_2)^{-1}|<\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3.\n$$\n\n\\medskip\n\\noindent\n(2) We use the general inequality for an invertible linear transformation $L$:\n\\begin{equation}\\label{gen-ineq-angles}\n\\frac{1}{\\|L\\|\\|L^{-1}\\|}\\leq \\frac{|\\sin\\angle(Lv,Lw)|}{|\\sin\\angle(v,w)|}\\leq \\|L\\|\\|L^{-1}\\|.\n\\end{equation}\nApply this to $L=C_1C_2^{-1}$, $v=C_2e_1$, $w=C_2e_2$ to get that\n$$\n\\frac{1}{\\|C_1C_2^{-1}\\|\\|C_2C_1^{-1}\\|}\\leq \\frac{\\sin\\alpha(x_1)}{\\sin\\alpha(x_2)}\\leq \\|C_1C_2^{-1}\\|\\|C_2C_1^{-1}\\|.\n$$\nWe have $\\|C_1C_2^{-1}-{\\rm Id}\\|\\leq\\|C_1-C_2\\|\\|C_2^{-1}\\|<\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3$,\nand by symmetry $\\|C_2C_1^{-1}-{\\rm Id}\\|<\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3$, therefore\n$\\|C_1C_2^{-1}\\|\\|C_2C_1^{-1}\\|<[1+\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3]^2<e^{2\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3}<e^{(\\eta_1\\eta_2)^3}$.\nThe left hand side estimate is proved similarly.\n\n\\medskip\n\\noindent\n(3) We prove that $\\Psi_{x_1}(R[e^{-2\\varepsilon}\\eta_1])\\subset \\Psi_{x_2}(R[\\eta_2])$.\nIf $v\\in R[e^{-2\\varepsilon}\\eta_1]$ then\n$\\|C_\\chi(x_1)v\\|\\leq \\sqrt{2}e^{-2\\varepsilon}\\eta_1<2\\mathfrak r(x)$,\nhence by (A1):\n$$d_{\\rm Sas}(C_\\chi(x_1)v,C_\\chi(x_2)v)\\leq 2(d(x_1,x_2)+\\|C_1v-C_2v\\|)\\leq 2(\\eta_1\\eta_2)^4.$$\nBy (A2), \n$d(\\Psi_{x_1}(v),\\Psi_{x_2}(v))\\leq 4(\\eta_1\\eta_2)^4\\Rightarrow\\Psi_{x_1}(v)\\in B(\\Psi_{x_2}(v),4(\\eta_1\\eta_2)^4)$.\nBy Lemma \\ref{Lemma-Pesin-chart}(1),\n$B(\\Psi_{x_2}(v),4(\\eta_1\\eta_2)^4)\\subset \\Psi_{x_2}(B)$ where\n$B\\subset \\mathbb{R}^2$ is the ball with center $v$ and radius $8\\|C_2^{-1}\\|(\\eta_1\\eta_2)^4$,\nhence it is enough that $B\\subset R[\\eta_2]$. If $w\\in B$ then\n$\\|w\\|_\\infty\\leq \\|v\\|_\\infty+8\\|C_2^{-1}\\|(\\eta_1\\eta_2)^4\\leq (e^{-\\varepsilon}+8\\varepsilon^{3\/\\beta})\\eta_2<\\eta_2$\nfor $\\varepsilon>0$ small enough.\n\n\n\\medskip\n\\noindent\n(4) The proof that $\\Psi_{x_2}^{-1}\\circ \\Psi_{x_1}$ is well-defined in\n$R[d(x_1,\\mathfs D)^a]$ is similar to the proof of (3). The only difference is in the last estimate:\nif $\\varepsilon>0$ is small enough then for $w\\in B$ it holds\n\\begin{align*}\n&\\|w\\|\\leq \\|v\\|+8\\|C_2^{-1}\\|(\\eta_1\\eta_2)^4\\leq \\sqrt{2}d(x_1,\\mathfs D)^a+8(\\eta_1\\eta_2)^3\\\\\n&\\leq [\\sqrt{2}(1+\\varepsilon)^a+8\\varepsilon^{3\/\\beta}]d(x_2,\\mathfs D)^a<2\\mathfrak r(x_2).\n\\end{align*}\nNow:\n\\begin{align*}\n&\\Psi_{x_2}^{-1}\\circ \\Psi_{x_1}-{\\rm Id}=C_2^{-1}\\circ\\exp{x_2}^{-1}\\circ\\exp{x_1}\\circ C_1-{\\rm Id}\\\\\n&=[C_2^{-1}\\circ P_{x_2,x}]\\circ[\\exp{x_2}^{-1}\\circ\\exp{x_1}-P_{x_1,x_2}]\\circ [P_{x,x_1}\\circ C_1]+C_2^{-1}(C_1-C_2)\\\\\n&=[C_2^{-1}\\circ P_{x_2,x}]\\circ[\\exp{x_2}^{-1}-P_{x_1,x_2}\\circ\\exp{x_1}^{-1}]\\circ\\Psi_{x_1}+C_2^{-1}(C_1-C_2).\n\\end{align*}\nWe calculate the $C^{1+\\beta\/2}$ norm of $[\\exp{x_2}^{-1}-P_{x_1,x_2}\\circ\\exp{x_1}^{-1}]\\circ\\Psi_{x_1}$\nin the domain $R[d(x_1, \\mathfs{D})^{2a}]$.\nBy Lemma \\ref{Lemma-Pesin-chart}(1), $\\|d\\Psi_{x_1}\\|_0\\leq 2$\nand\n$$\n\\Hol{\\beta\/2}(d\\Psi_{x_1})\\leq d(x_1,\\mathfs D)^{-a}4d(x_1,\\mathfs{D})^{2a(1-\\beta\/2)}=4d(x_1,\\mathfs D)^{a(1-\\beta)}\n< 4.\n$$\nCall $\\Theta:=\\exp{x_2}^{-1}-P_{x_1,x_2}\\circ\\exp{x_1}^{-1}$. For $\\varepsilon>0$ small enough, inside $D_{x_1}$ we have:\n\\begin{enumerate}[$\\circ$]\n\\item By (A2),\n$\\|\\Theta(v)\\|\\leq d_{\\rm Sas}(\\exp{x_2}^{-1}(v),\\exp{x_1}^{-1}(v))\\leq 2d(x_1,x_2)\\leq 2\\varepsilon^{6\/\\beta}(\\eta_1\\eta_2)^3$\nthus $\\|\\Theta\\circ \\Psi_{x_1}\\|_0<\\varepsilon^{2\/\\beta}(\\eta_1\\eta_2)^3$.\n\\item By (A3), $\\|d\\Theta_v\\|=\\|\\tau(x_2,v)-\\tau(x_1,v)\\|\\leq d(x_1,\\mathfs D)^{-a}d(x_1,x_2)\n<\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3$.\nHence $\\|d\\Theta\\|_0<\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3$ and\n$\\|d(\\Theta\\circ\\Psi_{x_1})\\|_0\\leq 2\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3<\\varepsilon^{2\/\\beta}(\\eta_1\\eta_2)^3$.\n\\item By (A4),\n\\begin{align*}\n&\\|\\widetilde{d\\Theta_v}-\\widetilde{d\\Theta_w}\\|=\\|[\\tau(x_2,v)-\\tau(x_1,v)]-[\\tau(x_2,w)-\\tau(x_1,w)]\\|\\\\\n&\\leq d(x_1,\\mathfs D)^{-a}d(x_1,x_2)\\|v-w\\\n\\end{align*}\nhence ${\\rm Lip}(d\\Theta)\\leq d(x_1,\\mathfs D)^{-a}d(x_1,x_2)$.\n\\item Using that\n$$\n\\Hol{\\beta\/2}(d(\\Theta_1\\circ\\Theta_2))\\leq \\|d\\Theta_1\\|_0\\Hol{\\beta\/2}(d\\Theta_2)+\n{\\rm Lip}(d\\Theta_1)\\|d\\Theta_2\\|_0^{2}4d(x_1,\\mathfs{D})^{2a(1-\\beta\/2)}\n$$\nfor $\\Theta_2$ with domain $R[d(x_1,\\mathfs{D})^{2a}]$, we get that\n\\begin{align*}\n&\\Hol{\\beta\/2}[d(\\Theta\\circ\\Psi_{x_1})]\\leq \\|d\\Theta\\|_0\\Hol{\\beta\/2}(d\\Psi_{x_1})+\n{\\rm Lip}(d\\Theta)\\|d\\Psi_{x_1}\\|_0^2 4d(x_1,\\mathfs{D})^{2a(1-\\beta\/2)}\\\\\n&<4\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3+ \nd(x_1,\\mathfs D)^{-a}d(x_1,x_2)16d(x_1,\\mathfs{D})^{2a(1-\\beta\/2)}\\\\\n&<4\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3+16\\varepsilon^{6\/\\beta}(\\eta_1\\eta_2)^3<\\varepsilon^{2\/\\beta}(\\eta_1\\eta_2)^3.\n\\end{align*}\n\\end{enumerate}\nThis implies that $\\|\\Theta\\circ\\Psi_{x_1}\\|_{1+\\beta\/2}<3\\varepsilon^{2\/\\beta}(\\eta_1\\eta_2)^3$, hence\n$$\n\\|C_2^{-1}\\circ P_{x_2,x}\\circ\\Theta\\circ\\Psi_{x_1}\\|_{1+\\beta\/2}\\leq \\|C_2^{-1}\\|3\\varepsilon^{2\/\\beta}(\\eta_1\\eta_2)^3\n\\leq 3\\varepsilon^{2\/\\beta}(\\eta_1\\eta_2)^2.\n$$\nThus\n$\\|\\Psi_{x_2}^{-1}\\circ \\Psi_{x_1}-{\\rm Id}\\|_{1+\\beta\/2}\\leq\n3\\varepsilon^{2\/\\beta}(\\eta_1\\eta_2)^2+\\|C_2^{-1}\\|(\\eta_1\\eta_2)^4<\n3\\varepsilon^{2\/\\beta}(\\eta_1\\eta_2)^2+\\varepsilon^{3\/\\beta}(\\eta_1\\eta_2)^3<4\\varepsilon^{2\/\\beta}(\\eta_1\\eta_2)^2<\\varepsilon(\\eta_1\\eta_2)^2$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The map $f_{x,y}$}\n\nLet $x,y\\in{\\rm NUH}_\\chi$, and assume that $\\Psi_{f(x)}^{\\eta}\\overset{\\varepsilon}{\\approx}\\Psi_y^{\\eta'}$.\nWe want to change $\\Psi_{f(x)}$ by $\\Psi_y$ in $f_x$ and obtain a result\nsimilar to Theorem \\ref{Thm-non-linear-Pesin}.\n\n\\medskip\n\\noindent\n{\\sc The maps $f_{x,y}$ and $f_{x,y}^{-1}$:} If $\\Psi_{f(x)}^{\\eta}\\overset{\\varepsilon}{\\approx}\\Psi_y^{\\eta'}$,\ndefine the map $f_{x,y}:=\\Psi_y^{-1}\\circ f\\circ \\Psi_x$.\nIf $\\Psi_{x}^{\\eta}\\overset{\\varepsilon}{\\approx}\\Psi_{f^{-1}(y)}^{\\eta'}$, define\n$f_{x,y}^{-1}:=\\Psi_x^{-1}\\circ f^{-1}\\circ \\Psi_y$. \n\n\\medskip\nAny meaningful estimate of the regularity of $f_{x,y}$ in the $C^{1+\\beta\/2}$ norm cannot be better than\nthat of Theorem \\ref{Thm-non-linear-Pesin}. In order to keep estimates of size $\\varepsilon$, we\nconsider the $C^{1+\\beta\/3}$ norm.\n\n\\begin{theorem}\\label{Thm-non-linear-Pesin-2}\nThe following holds for all $\\varepsilon>0$ small enough:\nIf $x,y\\in{\\rm NUH}_\\chi$ and $\\Psi_{f(x)}^{\\eta}\\overset{\\varepsilon}{\\approx}\\Psi_{y}^{\\eta'}$, then\n$f_{x,y}$ is well-defined in $R[10Q_\\varepsilon(x)]$ and can be written as\n$f_{x,y}(v_1,v_2)=(Av_1+h_1(v_1,v_2),Bv_2+h_2(v_1,v_2))$ where:\n\\begin{enumerate}[{\\rm (a)}]\n\\item $|A|<e^{-\\chi}$, $|B|>e^{\\chi}$, cf. Lemma \\ref{Lemma-linear-reduction}.\n\\item $\\|h_i(0)\\|<\\varepsilon\\eta$, $\\|\\nabla h_i(0)\\|<\\varepsilon\\eta^{\\beta\/3}$, and\n$\\Hol{\\beta\/3}(\\nabla h_i)<\\varepsilon$ where the norm is taken in $R[10Q_\\varepsilon(x)]$.\n\\end{enumerate}\nIf $\\Psi_{x}^{\\eta}\\overset{\\varepsilon}{\\approx}\\Psi_{f^{-1}(y)}^{\\eta'}$\nthen a similar statement holds for $f_{x,y}^{-1}$.\n\\end{theorem} \n\n\n\\begin{proof}\nWe write $f_{x,y}=(\\Psi_y^{-1}\\circ\\Psi_{f(x)})\\circ f_x=:g\\circ f_x$ and see it as a\nsmall perturbation of $f_x$.\nBy Theorem \\ref{Thm-non-linear-Pesin}(2--3),\n$$\nf_x(0)=0,\\ \\|d(f_x)\\|_0< \\tfrac{2(1+e^{2\\chi})}{\\rho(x)^a},\\ \\|d(f_x)_v-d(f_x)_w\\|\\leq \\varepsilon\\|v-w\\|^{\\beta\/2}\n$$\nfor $v,w\\in R[10Q_\\varepsilon(x)]$, where the $C^0$ norm is taken in $R[10Q_\\varepsilon(x)]$,\nand by Proposition \\ref{Lemma-overlap}(4) we have\n$$\n\\|g-{\\rm Id}\\|<\\varepsilon(\\eta\\eta')^2,\\ \\|d(g-{\\rm Id})\\|_0<\\varepsilon(\\eta\\eta')^2,\\ \\|dg_v-dg_w\\|\\leq\\varepsilon(\\eta\\eta')^2\\|v-w\\|^{\\beta\/2}\n$$\nfor $v,w\\in R[d(f(x),\\mathfs{D})^{2a}]$, where the $C^0$ norm is taken in this same domain.\n\n\\medskip\nWe first prove that $f_{x,y}$ is well-defined in $R[10Q_\\varepsilon(x)]$.\nWe have\n$$\nf_x(R[10Q_\\varepsilon(x)])\\subset B(0,40(1+e^{2\\chi})\\rho(x)^{-a}Q_\\varepsilon(x))\\subset R[d(f(x),\\mathfs D)^{2a}]\n$$\nsince\n$40(1+e^{2\\chi})\\rho(x)^{-a}Q_\\varepsilon(x)<40(1+e^{2\\chi})\\varepsilon^{3\/\\beta}d(f(x),\\mathfs D)^{2a}<d(f(x),\\mathfs D)^{2a}$\nfor $\\varepsilon>0$ small enough. By Proposition \\ref{Lemma-overlap}(4), $f_{x,y}$ is well-defined.\n \n\\medskip\nNow we prove (b). Let $h:=(h_1,h_2)=g\\circ f_x-d(f_x)_0$.\nThen $\\|h(0)\\|=\\|g(0)\\|<\\varepsilon(\\eta\\eta')^2<\\varepsilon\\eta$\nand for $\\varepsilon>0$ small enough:\n\\begin{align*}\n&\\|\\nabla h(0)\\|\\leq \\|dg_0\\circ d(f_x)_0-d(f_x)_0\\|\\leq \\|d(g-{\\rm Id})_0\\|\\|d(f_x)_0\\|\\\\\n&<\\varepsilon(\\eta\\eta')^2 2(1+e^{2\\chi})\\rho(x)^{-a}<\\varepsilon\\eta\\eta' 2\\varepsilon^{3\/\\beta}(1+e^{2\\chi})<\\varepsilon\\eta^{\\beta\/3}.\n\\end{align*}\nFinally, since $f_x(R[10Q_\\varepsilon(x)])\\subset R[d(f(x),\\mathfs D)^{2a}]$, if $\\varepsilon>0$ is small enough then\nfor all $v,w\\in R[10Q_\\varepsilon(x)]$ it holds:\n\\begin{align*}\n&\\|dh_v-dh_w\\|=\\|dg_{f_x(v)}\\circ d(f_x)_v-dg_{f_x(w)}\\circ d(f_x)_w\\|\\\\\n&\\leq \\|dg_{f_x(v)}-dg_{f_x(w)}\\|\\|d(f_x)_v\\|+\\|dg_{f_x(w)}\\|\\|d(f_x)_v-d(f_x)_w\\|\\\\\n&\\leq \\varepsilon(\\eta\\eta')^2\\|f_x(v)-f_x(w)\\|^{\\beta\/2}\\|d(f_x)\\|_0+\\varepsilon\\|dg\\|_0\\|v-w\\|^{\\beta\/2}\\\\\n&\\leq (\\varepsilon(\\eta\\eta')^2\\|d(f_x)\\|_0^{1+\\beta\/2}+40\\varepsilon\\|dg\\|_0Q_\\varepsilon(x)^{\\beta\/6})\\|v-w\\|^{\\beta\/3}\\\\\n&\\leq (4\\eta^2(1+e^{2\\chi})^2\\rho(x)^{-2a}+ 80Q_\\varepsilon(x)^{\\beta\/6})\\varepsilon\\|v-w\\|^{\\beta\/3}\\\\\n&\\leq (4\\varepsilon^{6\/\\beta}(1+e^{2\\chi})^2+ 80\\varepsilon^{1\/2})\\varepsilon\\|v-w\\|^{\\beta\/3}<\\varepsilon\\|v-w\\|^{\\beta\/3}.\n\\end{align*}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Double charts and the graph transform method}\n\nWe now define $\\varepsilon$--double charts.\nFor $\\varepsilon>0$ small, define $\\delta_\\varepsilon:=e^{-\\varepsilon n}\\in I_\\varepsilon$ where $n$ is the unique positive integer s.t.\n$e^{-\\varepsilon n}<\\varepsilon\\leq e^{-\\varepsilon(n-1)}$. In particular, $\\delta_\\varepsilon<\\varepsilon$.\n\n\\medskip\n\\noindent\n{\\sc $\\varepsilon$--double chart:} An {\\em $\\varepsilon$--double chart} is a pair of Pesin charts\n$\\Psi_x^{p^s,p^u}=(\\Psi_x^{p^s},\\Psi_x^{p^u})$ where $p^s,p^u\\in I_\\varepsilon$\nwith $0<p^s,p^u\\leq \\delta_\\varepsilon Q_\\varepsilon(x)$.\n\n\\medskip\nThe parameters $p^s\/p^u$ control the local forward\/backward hyperbolicity at $x$.\nThey are a way of separating the future and past dynamics. This will be better explained\nbelow, when we introduce the parameters $q_\\varepsilon,q_\\varepsilon^s,q_\\varepsilon^u$.\n\n\\medskip\n\\noindent\n{\\sc Edge $v\\overset{\\varepsilon}{\\rightarrow}w$:} Given $\\varepsilon$--double charts $v=\\Psi_x^{p^s,p^u}$\nand $w=\\Psi_y^{q^s,q^u}$, we draw an edge from $v$ to $w$ if the following conditions are\nsatisfied:\n\\begin{enumerate}[iii\\,]\n\\item[(GPO1)] $\\Psi_{f(x)}^{q^s\\wedge q^u}\\overset{\\varepsilon}{\\approx}\\Psi_y^{q^s\\wedge q^u}$\nand $\\Psi_{f^{-1}(y)}^{p^s\\wedge p^u}\\overset{\\varepsilon}{\\approx}\\Psi_x^{p^s\\wedge p^u}$.\n\\item[(GPO2)] $p^s=\\min\\{e^\\varepsilon q^s,\\delta_\\varepsilon Q_\\varepsilon(x)\\}$ and $q^u=\\min\\{e^\\varepsilon p^u,\\delta_\\varepsilon Q_\\varepsilon(y)\\}$.\n\\end{enumerate}\n\n\\medskip\n(GPO1) allows to pass from an $\\varepsilon$--double chart at $x$\nto an $\\varepsilon$--double chart at $y$ and vice-versa. (GPO2) is a greedy recursion\nthat implies that the local hyperbolicity parameters are the largest as possible.\nIt implies that $\\tfrac{p^s\\wedge p^u}{q^s\\wedge q^u}=e^{\\pm\\varepsilon}$.\n(GPO2) will be crucial in the proof of the inverse theorem (Theorem \\ref{Thm-inverse}).\n\n\\medskip\n\\noindent\n{\\sc $\\varepsilon$--generalized pseudo-orbit ($\\varepsilon$--gpo):} An {\\em $\\varepsilon$--generalized pseudo-orbit ($\\varepsilon$--gpo)}\nis a sequence $\\underline{v}=\\{v_n\\}_{n\\in\\mathbb{Z}}$ of $\\varepsilon$--double charts\ns.t. $v_n\\overset{\\varepsilon}{\\rightarrow}v_{n+1}$ for all $n\\in\\mathbb{Z}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The parameters $q_\\varepsilon(x),q_\\varepsilon^s(x),q_\\varepsilon^u(x)$}\n\nA transition between Pesin charts only makes sense if their sizes $\\eta,\\eta'$ satisfy $\\tfrac{\\eta}{\\eta'}=e^{\\pm\\varepsilon}$\n(see Theorem \\ref{Thm-non-linear-Pesin-2}). Since the ratio $\\tfrac{Q_\\varepsilon(f(x))}{Q_\\varepsilon(x)}$ \nmight be different from $e^{\\pm\\varepsilon}$, we introduce the parameter $q_\\varepsilon(x)$ below.\n\n\\medskip\n\\noindent\n{\\sc Parameter $q_\\varepsilon(x)$:} For $x\\in{\\rm NUH}_\\chi^*$, let\n$q_\\varepsilon(x):=\\delta_\\varepsilon\\min\\{e^{\\varepsilon|n|}Q_\\varepsilon(f^n(x)):n\\in\\mathbb{Z}\\}$.\n\n\\medskip\nThe above minimum is the greedy way of defining values in $I_\\varepsilon$ smaller than $\\varepsilon Q_\\varepsilon$\nwith the required regularity property.\n\n\\begin{lemma}\\label{Lemma-q}\nFor all $x\\in{\\rm NUH}_\\chi^*$, $0<q_\\varepsilon(x)<\\varepsilon Q_\\varepsilon(x)$\nand $\\tfrac{q_\\varepsilon(f(x))}{q_\\varepsilon(x)}=e^{\\pm\\varepsilon}$.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{Lemma-temperedness}, $\\inf\\{e^{\\varepsilon|n|}Q_\\varepsilon(f^n(x)):n\\in\\mathbb{Z}\\}>0$.\nSince zero is the only accumulation point of $I_\\varepsilon$,\n$q_\\varepsilon(x)$ is well-defined and positive.\nIt is clear that $q_\\varepsilon(x)\\leq \\delta_\\varepsilon Q_\\varepsilon(x)<\\varepsilon Q_\\varepsilon(x)$. Since\n$$\n\\min\\{e^{\\varepsilon|n|}Q_\\varepsilon(f^{n+1}(x)):n\\in\\mathbb{Z}\\}\\leq e^{\\varepsilon}\\min\\{e^{\\varepsilon|n+1|}Q_\\varepsilon(f^{n+1}(x)):n\\in\\mathbb{Z}\\},\n$$\nwe have $q_\\varepsilon(f(x))\\leq e^{\\varepsilon}q_\\varepsilon(x)$. Reversely,\n$$\n e^{-\\varepsilon}\\min\\{e^{\\varepsilon|n+1|}Q_\\varepsilon(f^{n+1}(x)):n\\in\\mathbb{Z}\\}\\leq \\min\\{e^{\\varepsilon|n|}Q_\\varepsilon(f^{n+1}(x)):n\\in\\mathbb{Z}\\}\n$$\ntherefore $e^{-\\varepsilon}q_\\varepsilon(x)\\leq q_\\varepsilon(f(x))$.\n\\end{proof}\n\nWe want to separate the dependence of $q_\\varepsilon(x)$\non the future from its dependence on the past, hence we define the one-sided versions of $q_\\varepsilon(x)$.\n\n\\medskip\n\\noindent\n{\\sc Parameters $q_\\varepsilon^s(x),q_\\varepsilon^u(x)$:}  For $x\\in{\\rm NUH}_\\chi^*$, define\n\\begin{align*}\nq_\\varepsilon^s(x)&:=\\delta_\\varepsilon\\min\\{e^{\\varepsilon|n|}Q_\\varepsilon(f^n(x)):n\\geq 0\\}\\\\\nq_\\varepsilon^u(x)&:=\\delta_\\varepsilon\\min\\{e^{\\varepsilon|n|}Q_\\varepsilon(f^n(x)):n\\leq 0\\}.\n\\end{align*}\n\n\\begin{lemma}\\label{Lemma-q^s}\nFor all $x\\in{\\rm NUH}_\\chi^*$, the following holds:\n\\begin{enumerate}[{\\rm (1)}]\n\\item {\\sc Good definition:} $0<q_\\varepsilon^s(x),q_\\varepsilon^u(x)<\\varepsilon Q_\\varepsilon(x)$ and $q_\\varepsilon^s(x)\\wedge q_\\varepsilon^u(x)=q_\\varepsilon(x)$.\n\\item {\\sc Greedy algorithm:} For all $n\\in\\mathbb{Z}$ it holds\n\\begin{align*}\nq_\\varepsilon^s(f^n(x))&=\\min\\{e^\\varepsilon q_\\varepsilon^s(f^{n+1}(x)),\\delta_\\varepsilon Q_\\varepsilon(f^n(x))\\}\\\\\nq_\\varepsilon^u(f^n(x))&=\\min\\{e^\\varepsilon q_\\varepsilon^u(f^{n-1}(x)),\\delta_\\varepsilon Q_\\varepsilon(f^n(x))\\}.\n\\end{align*}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nAs in the proof of Lemma \\ref{Lemma-q}, $q^s_\\varepsilon(x)$ and $q^u_\\varepsilon(x)$ are well-defined and positive,\nand by definition $q_\\varepsilon^s(x)\\wedge q_\\varepsilon^u(x)=q_\\varepsilon(x)$. This proves (1).\nWe prove the first equality in (2): for a fixed $n\\in\\mathbb{Z}$ we have\n\\begin{align*}\n&q_\\varepsilon^s(f^n(x))=\\delta_\\varepsilon\\min\\{e^{\\varepsilon|m|}Q_\\varepsilon(f^m(f^n(x))):m\\geq 0\\}\\\\\n&=\\min\\{\\delta_\\varepsilon \\min\\{e^{\\varepsilon|m|}Q_\\varepsilon(f^{m+n}(x)):m\\geq 1\\},\\delta_\\varepsilon Q_\\varepsilon(f^n(x))\\}\\\\\n&=\\min\\{e^\\varepsilon\\delta_\\varepsilon\\min\\{e^{\\varepsilon|m|}Q_\\varepsilon(f^m(f^{n+1}(x))):m\\geq 0\\},\\delta_\\varepsilon Q_\\varepsilon(f^n(x))\\}\\\\\n&=\\min\\{e^\\varepsilon q_\\varepsilon^s(f^{n+1}(x)),\\delta_\\varepsilon Q_\\varepsilon(f^n(x))\\}.\n\\end{align*}\nThe second equality is proved similarly.\n\\end{proof}\n\n\n\\medskip\n\\noindent\n{\\sc The set ${\\rm NUH}_\\chi^\\#$:} It is the set of $x\\in{\\rm NUH}_\\chi^*$ s.t.\n$$\n\\limsup_{n\\to\\infty}q_\\varepsilon^s(f^n(x))>0\\text{ and }\\limsup_{n\\to-\\infty}q_\\varepsilon^u(f^n(x))>0.\n$$\n\n\n\n\n\n\n\n\n\n\\subsection{ The graph transform method}\n\nLet $v=\\Psi_x^{p^s,p^u}$ be an $\\varepsilon$--double chart.\n\n\\medskip\n\\noindent\n{\\sc Admissible manifolds:} We define an {\\em $s$--admissible manifold at $v$} as a set\nof the form $\\Psi_x\\{(t,F(t)):|t|\\leq p^s\\}$ where $F:[-p^s,p^s]\\to\\mathbb{R}$ is a $C^{1+\\beta\/3}$ function\ns.t.:\n\\begin{enumerate}\n\\item[(AM1)] $|F(0)|\\leq 10^{-3}(p^s\\wedge p^u)$.\n\\item[(AM2)] $|F'(0)|\\leq \\tfrac{1}{2}(p^s\\wedge p^u)^{\\beta\/3}$.\n\\item[(AM3)] $\\|F'\\|_0+\\Hol{\\beta\/3}(F')\\leq\\tfrac{1}{2}$ where the norms are taken in $[-p^s,p^s]$.\n\\end{enumerate}\nSimilarly, a {\\em $u$--admissible manifold at $v$} is a set\nof the form $\\Psi_x\\{(G(t),t):|t|\\leq p^u\\}$ where $G:[-p^u,p^u]\\to\\mathbb{R}$ is a $C^{1+\\beta\/3}$ function\nsatisfying (AM1)--(AM3), where the norms are taken in $[-p^u,p^u]$.\n\n\\medskip\nThe functions $F,G$ are called the {\\em representing functions}.\nWe let $\\mathfs M^s(v)$ (resp. $\\mathfs M^u(v)$) denote the set of all $s$--admissible\n(resp. $u$--admissible) manifolds at $v$.\n\n\\begin{lemma}\\label{Lemma-admissible-manifolds}\nThe following holds for $\\varepsilon>0$ small enough. If $v=\\Psi_x^{p^s,p^u}$ is an $\\varepsilon$--double chart, then for\nevery $V^s\\in\\mathfs M^s(v)$ and $V^u\\in\\mathfs M^u(v)$ it holds:\n\\begin{enumerate}[{\\rm (1)}]\n\\item $V^s$ and $V^u$ intersect at a single point $P=\\Psi_x(w)$, and $\\|w\\|_\\infty<10^{-2}(p^s\\wedge p^u)$.\n\\item $\\tfrac{\\sin\\angle(V^s,V^u)}{\\sin\\alpha(x)}=e^{\\pm(p^s\\wedge p^u)^{\\beta\/4}}$\nand $|\\cos\\angle(V^s,V^u)-\\cos\\alpha(x)|<2(p^s\\wedge p^u)^{\\beta\/4}$, where\n$\\angle(V^s,V^u)=$ angle of intersection of the tangents to $V^s$ and $V^u$ at $P$.\n\\end{enumerate}\n\\end{lemma}\n\nWhen $M$ is compact and $f$ is a $C^{1+\\beta}$ diffeomorphism, this is \\cite[Prop. 4.11]{Sarig-JAMS}.\nThe same proof works almost verbatim, see the appendix for the necessary adaptations.\n\n\\medskip\nLet $v=\\Psi_x^{p^s,p^u}$, $w=\\Psi_y^{q^s,q^u}$ be $\\varepsilon$--double charts with\n$v\\overset{\\varepsilon}{\\rightarrow}w$. We now define the {\\em graph transforms}: these are two maps\nthat work in different directions of the edge $v\\overset{\\varepsilon}{\\rightarrow}w$, one of them\nsends $u$--admissible manifolds at $v$ to $u$--admissible manifolds at $w$, the other\nsends $s$--admissible manifolds at $w$ to $s$--admissible manifolds at $v$. \n\n\\medskip\n\\noindent\n{\\sc Graph transforms $\\mathfs F_{v,w}^s$ and $\\mathfs F_{v,w}^u$:} The {\\em graph transform}\n$\\mathfs F_{v,w}^s:\\mathfs M^s(w)\\to\\mathfs M^s(v)$ is the map that sends\nan $s$--admissible manifold at $w$ with representing function $F:[-q^s,q^s]\\to\\mathbb{R}$ to the unique\n$s$--admissible  manifold at $v$ with representing function $G:[-p^s,p^s]\\to\\mathbb{R}$ s.t.\n$\\{(t,G(t)):|t|\\leq p^s\\}\\subset f_{x,y}^{-1}\\{(t,F(t)):|t|\\leq q^s\\}$.  \nSimilarly, the {\\em graph transform} $\\mathfs F_{v,w}^u:\\mathfs M^u(v)\\to\\mathfs M^u(w)$\nis the map sending a $u$--admissible manifold at $v$ with representing function $F:[-p^u,p^u]\\to\\mathbb{R}$\nto the unique $u$--admissible  manifold at $w$ with representing function $G:[-q^u,q^u]\\to\\mathbb{R}$ s.t.\n$\\{(G(t),t):|t|\\leq q^u\\}\\subset f_{x,y}\\{(F(t),t):|t|\\leq p^u\\}$.\n\n\\medskip\nIn other words, the representing functions of $s,u$--admissible manifolds\nchange by the application of $f_{x,y}^{-1},f_{x,y}$ respectively.\nFor $V_1,V_2\\in\\mathfs M^s(v)$ with representing functions\n$F_1,F_2$ and for $i\\geq 0$, define $ d_{C^i}(V_1,V_2):=\\|F_1-F_2\\|_i$ where the\nnorm is taken in $[-p^s,p^s]$. A similar definition\nholds in $\\mathfs M^u(v)$.\n\n\\begin{proposition}\\label{Prop-graph-transform}\nThe following holds for $\\varepsilon>0$ small enough. If $v\\overset{\\varepsilon}{\\rightarrow}w$ then\n$\\mathfs F_{v,w}^s$ and $\\mathfs F_{v,w}^u$ are well-defined. Furthermore, if\n$V_1,V_2\\in \\mathfs M^u(v)$ then:\n\\begin{enumerate}[{\\rm (1)}]\n\\item $ d_{C^0}(\\mathfs F_{v,w}^u(V_1),\\mathfs F_{v,w}^u(V_2))\\leq e^{-\\chi\/2} d_{C^0}(V_1,V_2)$.\n\\item $ d_{C^1}(\\mathfs F_{v,w}^u(V_1),\\mathfs F_{v,w}^u(V_2))\\leq\ne^{-\\chi\/2}( d_{C^1}(V_1,V_2)+ d_{C^0}(V_1,V_2)^{\\beta\/3})$.\n\\item $f(V_i)$ intersects every element of $\\mathfs M^u(w)$ at exactly one point.\n\\end{enumerate}\nAn analogous statement holds for $\\mathfs F_{v,w}^s$.\n\\end{proposition}\n\nWhen $M$ is compact and $f$ is a $C^{1+\\beta}$ diffeomorphism,\nthis is \\cite[Prop. 4.12 and 4.14]{Sarig-JAMS}.\nThe proof in our case requires some minor changes, see Appendix B.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Stable and unstable manifolds of $\\varepsilon$--gpo's}\n\nCall a sequence ${\\underline v}^+=\\{v_n\\}_{n\\geq 0}$ a {\\em positive $\\varepsilon$--gpo} if $v_n\\overset{\\varepsilon}{\\to}v_{n+1}$\nfor all $n\\geq 0$. Similarly, a {\\em negative $\\varepsilon$--gpo} is a sequence ${\\underline v}^-=\\{v_n\\}_{n\\leq 0}$\ns.t. $v_{n-1}\\overset{\\varepsilon}{\\to}v_n$ for all $n\\leq 0$.\n\n\\medskip\n\\noindent\n{\\sc Stable\/unstable manifold of positive\/negative $\\varepsilon$--gpo:} The {\\em stable manifold}\nof a positive $\\varepsilon$--gpo ${\\underline v}^+=\\{v_n\\}_{n\\geq 0}$ is \n$$\nV^s[{\\underline v}^+]:=\\lim_{n\\to\\infty}\n(\\mathfs F_{v_0,v_1}^s\\circ\\cdots\\circ\\mathfs F_{v_{n-2},v_{n-1}}^s\\circ\\mathfs F_{v_{n-1},v_n}^s)(V_n)\n$$\nfor some (any) choice of $(V_n)_{n\\geq 0}$ with $V_n\\in\\mathfs M^s(v_n)$.\nThe {\\em unstable manifold} of a negative $\\varepsilon$--gpo ${\\underline v}^-=\\{v_n\\}_{n\\leq 0}$ is \n$$\nV^u[{\\underline v}^-]:=\\lim_{n\\to-\\infty}\n(\\mathfs F_{v_{-1},v_0}^u\\circ\\cdots\\circ\\mathfs F_{v_{n+1},v_{n+2}}^u\\circ\\mathfs F_{v_n,v_{n+1}}^u)(V_n)\n$$\nfor some (any) choice of $(V_n)_{n\\leq 0}$ with $V_n\\in\\mathfs M^u(v_n)$.\n\n\\medskip\nFor an $\\varepsilon$--gpo $\\underline{v}=\\{v_n\\}_{n\\in\\mathbb{Z}}$, let\n$V^s[\\underline v]:=V^s[\\{v_n\\}_{n\\geq 0}]$\nand $V^u[\\underline v]:=V^u[\\{v_n\\}_{n\\leq 0}]$.\n\n\\begin{proposition}\\label{Prop-stable-manifolds}\nThe following holds for all $\\varepsilon>0$ small enough.\n\\begin{enumerate}[{\\rm (1)}]\n\\item {\\sc Admissibility:} $V^s[{\\underline v}^+],V^s[{\\underline v}^-]$ are well-defined admissible manifolds at $v_0$.\n\\item {\\sc Invariance:}\n$$\nf(V^s[\\{v_n\\}_{n\\geq 0}])\\subset V^s[\\{v_n\\}_{n\\geq 1}]\\text{ and }\nf^{-1}(V^u[\\{v_n\\}_{n\\leq 0}])\\subset V^u[\\{v_n\\}_{n\\leq -1}].\n$$\n\\item {\\sc Shadowing:} If ${\\underline v}^+=\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\geq 0}$ then\n$$\nV^s[{\\underline v}^+]=\\{x\\in \\Psi_{x_0}(R[p^s_0]):f^n(x)\\in \\Psi_{x_n}(R[10Q_\\varepsilon(x_n)]),\\,\\forall n\\geq 0\\}.\n$$\nAn analogous statement holds for $V^u[{\\underline v}^-]$.\n\\item {\\sc Hyperbolicity:} If $x,y\\in V^s[{\\underline v}^+]$ then $d(f^n(x),f^n(y))\\xrightarrow[n\\to\\infty]{}0$,\nif $x,y\\in V^u[{\\underline v}^-]$ then $d(f^n(x),f^n(y))\\xrightarrow[n\\to-\\infty]{}0$, and the rates are exponential.\n\\item {\\sc H\\\"older property:} The map $\\underline v^+\\mapsto V^s[\\underline v^+]$ is H\\\"older continuous,\ni.e. there exists $K>0$ and $\\theta<1$ s.t. for all $N\\geq 0$, if $\\underline v^+,\\underline w^+$ are positive $\\varepsilon$--gpo's\nwith $v_n=w_n$ for $n=0,\\ldots,N$\nthen $ d_{C^1}(V^s[\\underline v^+],V^s[\\underline w^+])\\leq K\\theta^N$. The same holds for the map\n$\\underline v^-\\mapsto V^u[\\underline v^-]$.\n\\end{enumerate}\n\\end{proposition}\n\nWhen $M$ is compact and $f$ is a $C^{1+\\beta}$ diffeomorphism,\nthis is \\cite[Prop. 4.15]{Sarig-JAMS}. The same proof works in our case:\nit uses the hyperbolicity of $f_{x,y}$ (Theorem \\ref{Thm-non-linear-Pesin-2}),\nand the contracting properties of the graph transforms (Proposition \\ref{Prop-graph-transform}).\nProposition \\ref{Prop-stable-manifolds} ensures that every $\\varepsilon$--gpo is associated to a unique point.\n\n\\medskip\n\\noindent\n{\\sc Shadowing:} We say that an $\\varepsilon$--gpo $\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}$ {\\em shadows}\na point $x\\in M$ when $f^n(x)\\in \\Psi_{x_n}(R[p^s_n\\wedge p^u_n])$ for all $n\\in\\mathbb{Z}$.\n\n\\begin{lemma}\\label{Lemma-shadowing}\nEvery $\\varepsilon$--gpo shadows a unique point.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\underline v=\\{v_n\\}_{n\\in\\mathbb{Z}}$ be an $\\varepsilon$--gpo. By Proposition \\ref{Prop-stable-manifolds}(3),\nany point shadowed by $\\underline v$ must lie in $V^s[\\{v_n\\}_{n\\geq 0}]\\cap V^u[\\{v_n\\}_{n\\leq 0}]$.\nBy Lemma \\ref{Lemma-admissible-manifolds}(1), this intersection consists of a singleton $\\{x\\}$.\nWrite $v_n=\\Psi_{x_n}^{p^s_n,p^u_n}$. By Proposition \\ref{Prop-stable-manifolds}(2),\nfor all $n\\geq 0$ we have $f^n(x)\\in V^s[\\{v_{n+k}\\}_{k\\geq 0}]\\subset \\Psi_{x_n}(R[10Q_\\varepsilon(x_n)])$,\nand for all $n\\leq 0$ we have $f^n(x)\\in V^u[\\{v_{n+k}\\}_{k\\leq 0}]\\subset\\Psi_{x_n}(R[10Q_\\varepsilon(x_n)])$,\nhence $\\underline v$ shadows $x$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Coarse graining}\\label{Section-coarse-graining}\n\nWe now pass to a countable set of $\\varepsilon$--double charts that define a topological Markov shift that\nshadows all relevant orbits.\n\n\\begin{theorem}\\label{Thm-coarse-graining}\nFor all $\\varepsilon>0$ sufficiently small, there exists a countable family $\\mathfs A$ of $\\varepsilon$--double charts\nwith the following properties:\n\\begin{enumerate}[{\\rm (1)}]\n\\item {\\sc Discreteness}: For all $t>0$, the set $\\{\\Psi_x^{p^s,p^u}\\in\\mathfs A:p^s,p^u>t\\}$ is finite.\n\\item {\\sc Sufficiency:} If $x\\in {\\rm NUH}_\\chi^*$ then there is a sequence $\\underline v\\in{\\mathfs A}^{\\mathbb{Z}}$\nthat shadows $x$.\n\\item {\\sc Relevance:} For all $v\\in \\mathfs A$ there is an $\\varepsilon$--gpo $\\underline{v}\\in\\mathfs A^\\mathbb{Z}$\nwith $v_0=v$ that shadows a point in ${\\rm NUH}_\\chi^*$.\n\\end{enumerate}\n\\end{theorem}\n\nParts (1) and (3) will be crucial to prove the inverse theorem (Theorem \\ref{Thm-inverse}).\nPart (2) says that the $\\varepsilon$--gpo's in $\\mathfs A$ shadow a.e. point with respect to every\n$f$--adapted $\\chi$--hyperbolic measure, see Lemma \\ref{Lemma-adaptedness}.\n\n\\begin{remark}\nIn part (2) we only assume that $x\\in{\\rm NUH}_\\chi^*$, while \\cite{Lima-Sarig,Sarig-JAMS}\nrequire the stronger assumption $x\\in{\\rm NUH}_\\chi^\\#$. The reason of the improvement\nis that here $q_\\varepsilon(x)$ is defined as a minimum instead of a sum,\nand hence Lemma \\ref{Lemma-q^s}(1) holds.\n\\end{remark}\n\n\\begin{proof}\nWhen $M$ is compact and $f$ is a diffeomorphism, the above statement is consequence\nof Propositions 3.5, 4.5 and Lemmas 4.6, 4.7 of \\cite{Sarig-JAMS}. When $M$ is compact (with boundary)\nand $f$ is a local diffeomorphism with bounded derivatives, this is Proposition 4.3 of \\cite{Lima-Sarig}.\nWe follow the same strategy, adapted to our context.\n\n\\medskip\nFor $t>0$, let $M_t=\\{x\\in M: d(x,\\mathfs D)\\geq t\\}$.\nSince $M$ has finite diameter (remember we are even assuming it is smaller than one), each $M_t$\nis precompact\\footnote{$M_t$ might not be compact, since $M$ might have boundaries.}.\nLet $\\mathbb{N}_0=\\mathbb{N}\\cup\\{0\\}$. Fix a countable open cover $\\mathfs P=\\{D_i\\}_{i\\in\\mathbb{N}_0}$ of $M\\backslash\\mathfs D$ s.t.:\n\\begin{enumerate}[$\\circ$]\n\\item $D_i:=D_{z_i}=B(z_i,2\\mathfrak r(z_i))$ for some $z_i\\in M$.\n\\item For every $t>0$, $\\{D\\in\\mathfs P:D\\cap M_t\\neq\\emptyset\\}$ is finite.\n\\end{enumerate}\n\n\n\\medskip\nLet $X:=M^3\\times {\\rm GL}(2,\\mathbb{R})^3\\times (0,1]$.\nFor $x\\in{\\rm NUH}_\\chi^*$, let\n$\\Gamma(x)=(\\underline x,\\underline C,\\underline Q)\\in X$ with\n\\begin{align*}\n\\underline x=(f^{-1}(x),x,f(x)),\\ \\underline C=(C_\\chi(f^{-1}(x)),C_\\chi(x),C_\\chi(f(x))),\\ \\underline Q=Q_\\varepsilon(x).\n\\end{align*}\nLet $Y=\\{\\Gamma(x):x\\in{\\rm NUH}_\\chi^*\\}$. We want to construct a countable dense subset\nof $Y$. Since the maps $x\\mapsto C_\\chi(x),Q_\\varepsilon(x)$ are usually just measurable,\nwe apply a precompactness argument.\nFor each triple of vectors $\\underline{k}=(k_{-1},k_0,k_1)$, $\\underline{\\ell}=(\\ell_{-1},\\ell_0,\\ell_1)$,\n$\\underline a=(a_{-1},a_0,a_1)\\in\\mathbb{N}_0^3$ and $m\\in\\mathbb{N}_0$, define\n$$\nY_{\\underline k,\\underline \\ell,\\underline a,m}:=\\left\\{\\Gamma(x)\\in Y:\n\\begin{array}{cl}\ne^{-k_i-1}\\leq d(f^i(x),\\mathfs D)< e^{-k_i},& -1\\leq i\\leq 1\\\\\ne^{\\ell_i}\\leq\\|C_\\chi(f^i(x))^{-1}\\|<e^{\\ell_i+1},&-1\\leq i\\leq 1\\\\\nf^i(x)\\in D_{a_i},&-1\\leq i\\leq 1\\\\\ne^{-m-1}\\leq Q_\\varepsilon(x)< e^{-m}&\\\\\n\\end{array}\n\\right\\}.\n$$\n\n\\medskip\n\\noindent\n{\\sc Claim 1:} $Y=\\bigcup_{\\underline k,\\underline\\ell,\\underline a\\in\\mathbb{N}_0^3\\atop{m\\in\\mathbb{N}_0}}Y_{\\underline k,\\underline\\ell,\\underline a,m}$, and each\n$Y_{\\underline k,\\underline\\ell,\\underline a,m}$ is precompact in $X$.\n\n\\medskip\n\\noindent\n{\\em Proof of claim $1$.}\nThe first statement is clear. We focus on the second.\nFix $\\underline k,\\underline\\ell,\\underline a\\in \\mathbb{N}_0^3$, $m\\in\\mathbb{N}_0$. Take $\\Gamma(x)\\in Y_{\\underline k,\\underline\\ell,\\underline a,m}$. Then\n$$\n\\underline x\\in M_{e^{-k_{-1}-1}}\\times M_{e^{-k_0-1}}\\times M_{e^{-k_1-1}},$$\na precompact subset of $M^3$.\nFor $|i|\\leq 1$, $C_\\chi(f^i(x))$ is an element of ${\\rm GL}(2,\\mathbb{R})$ with norm $\\leq 1$ and\ninverse norm $\\leq e^{\\ell_i+1}$, hence it belongs to a compact subset of ${\\rm GL}(2,\\mathbb{R})$.\nThis guarantees that $\\underline C$ belongs to a compact subset of ${\\rm GL}(2,\\mathbb{R})^3$. Also,\n$\\underline Q\\in [e^{-m-1},1]$, a compact subinterval of $(0,1]$. Since the product of precompact sets\nis precompact, the claim is proved.\n\n\\medskip\nLet $j\\geq 0$. By claim 1, there exists a finite set\n$Y_{\\underline k,\\underline\\ell,\\underline a,m}(j)\\subset Y_{\\underline k,\\underline\\ell,\\underline a,m}$\ns.t. for every $\\Gamma(x)\\in Y_{\\underline k,\\underline\\ell,\\underline a,m}$\nthere exists $\\Gamma(y)\\in Y_{\\underline k,\\underline\\ell,\\underline a,m}(j)$\ns.t.:\n\\begin{enumerate}[{\\rm (a)}]\n\\item $ d(f^i(x),f^i(y))+\\|\\widetilde{C_\\chi(f^i(x))}-\\widetilde{C_\\chi(f^i(y))}\\|<e^{-8(j+2)}$\nfor $-1\\leq i\\leq 1$.\n\\item $\\tfrac{Q_\\varepsilon(x)}{Q_\\varepsilon(y)}=e^{\\pm \\varepsilon\/3}$.\n\\end{enumerate}\n\n\\medskip\n\\noindent\n{\\sc The alphabet $\\mathfs A$:} Let $\\mathfs A$ be the countable family of $\\Psi_x^{p^s,p^u}$ s.t.:\n\\begin{enumerate}[i i)]\n\\item[(CG1)] $\\Gamma(x)\\in Y_{\\underline k,\\underline\\ell,\\underline a,m}(j)$ for some\n$(\\underline k,\\underline\\ell,\\underline a,m,j)\\in\\mathbb{N}_0^3\\times\\mathbb{N}_0^3\\times\\mathbb{N}_0^3\\times \\mathbb{N}_0\\times \\mathbb{N}_0$.\n\\item[(CG2)] $0<p^s,p^u\\leq \\delta_\\varepsilon Q_\\varepsilon(x)$ and $p^s,p^u\\in I_\\varepsilon$.\n\\item[(CG3)] $e^{-j-2}\\leq p^s\\wedge p^u\\leq e^{-j+2}$.\n\\end{enumerate}\n\n\\medskip\n\\noindent\n{\\em Proof of discreteness.}\nWe will use the following fact, whose proof is in the appendix:\n\\begin{equation}\\label{inequality-C}\n\\|C_\\chi(f^{-1}(x))^{-1}\\|\\leq 2\\rho(x)^{-2a}(1+e^\\chi\\rho(x)^{-a})\\|C_\\chi(x)^{-1}\\|.\n\\end{equation}\n\nFix $t>0$, and let $\\Psi_x^{p^s,p^u}\\in\\mathfs A$ with $p^s,p^u>t$.\nNote that $\\rho(x)>\\rho(x)^{2a}>Q_\\varepsilon(x)>p^s,p^u>t$.\nIf $\\Gamma(x)\\in Y_{\\underline k,\\underline\\ell,\\underline a,m}(j)$ then:\n\\begin{enumerate}[$\\circ$]\n\\item Finiteness of $\\underline k$: for $|i|\\leq 1$, $e^{-k_i}> d(f^i(x),\\mathfs D)\\geq\\rho(x)>t$, hence $k_i< |\\log t|$.\n\\item Finiteness of $\\underline\\ell$: for $i=0,1$, $e^{\\ell_i}\\leq \\|C_\\chi(f^i(x))^{-1}\\|<Q_\\varepsilon(x)^{-1}<t^{-1}$,\nhence $\\ell_i<|\\log t|$. By inequality (\\ref{inequality-C}) above,\n$$\ne^{\\ell_{-1}}\\leq \\|C_\\chi(f^{-1}(x))^{-1}\\|<2t^{-1}(1+e^\\chi t^{-1})t^{-1}<4e^\\chi t^{-3},\n$$\nhence $\\ell_{-1}<\\log 4+\\chi+3|\\log t|=:T_t$, which is bigger than $|\\log t|$.\n\\item Finiteness of $\\underline a$: $f^i(x)\\in D_{a_i}\\cap M_t$, hence $D_{a_i}$ belongs to the finite set\n$\\{D\\in\\mathfs P:D\\cap M_t\\neq\\emptyset\\}$.\n\\item Finiteness of $m$: $e^{-m}>Q_\\varepsilon(x)>t$, hence $m<|\\log t|$.\n\\item Finiteness of $j$: $t<p^s\\wedge p^u\\leq e^{-j+2}$, hence $j\\leq |\\log t|+2$.\n\\item Finiteness of $(p^s,p^u)$: $t< p^s,p^u$, hence $\\#\\{(p^s,p^u):p^s,p^u>t\\}\\leq \\#(I_\\varepsilon\\cap (t,1])^2$ is finite.\n\\end{enumerate}\nThe first five items above give that, for $\\underline a\\in\\mathbb{N}_0^3$ and $t>0$,\n\\begin{align*}\n\\#\\left\\{\\Gamma(x):\n\\begin{array}{c}\n\\Psi_x^{p^s,p^u}\\in\\mathcal A\\text{ s.t. }p^s,p^u>t\\\\\n\\text{and }f^i(x)\\in D_{a_i}, |i|\\leq 1\n\\end{array}\n\\right\\}\\leq\\sum_{j=0}^{\\lceil |\\log t|\\rceil+2}\\sum_{m=0}^{\\lceil |\\log t|\\rceil}\n\\sum_{-1\\leq i\\leq 1\\atop{k_i,\\ell_i=0}}^{T_t}\n\\# Y_{\\underline k,\\underline\\ell,\\underline a,m}(j)\n\\end{align*}\nis the finite sum of finite terms, hence finite. Together with the last item above,\nwe conclude that\n\\begin{align*}\n\\#\\left\\{\\Psi_x^{p^s,p^u}\\in\\mathcal A:p^s,p^u>t\\right\\}&\\leq \n\\sum_{j=0}^{\\lceil |\\log t|\\rceil+2}\\sum_{m=0}^{\\lceil |\\log t|\\rceil}\\sum_{-1\\leq i\\leq 1\\atop{k_i,\\ell_i=0}}^{T_t}\n\\# Y_{\\underline k,\\underline\\ell,\\underline a,m}(j)\\\\\n&\\ \\ \\ \\times (\\#\\{D\\in\\mathfs P:D\\cap M_t\\neq\\emptyset\\})^3\\times (\\#(I_\\varepsilon\\cap (t,1]))^2\n\\end{align*}\nis finite. This proves the discreteness of $\\mathfs A$.\n\n\\medskip\n\\noindent\n{\\em Proof of sufficiency.}\nLet $x\\in {\\rm NUH}_\\chi^*$. Take $(k_i)_{i\\in\\mathbb{Z}},(\\ell_i)_{i\\in\\mathbb{Z}},(m_i)_{i\\in\\mathbb{Z}},(a_i)_{i\\in\\mathbb{Z}},(j_i)_{i\\in\\mathbb{Z}}$  s.t.:\n\\begin{align*}\n& d(f^i(x),\\mathfs D)\\in [e^{-k_i-1},e^{-k_i}), \\|C_\\chi(f^i(x))^{-1}\\|\\in [e^{\\ell_i},e^{\\ell_i+1}),\\\\\n&Q_\\varepsilon(f^i(x))\\in [e^{-m_i-1},e^{-m_i}),f^i(x)\\in D_{a_i}, q_\\varepsilon(f^i(x))\\in[e^{-j_i-1},e^{-j_i+1}).\n\\end{align*}\nFor $n\\in\\mathbb{Z}$, define\n$$\n\\underline k^{(n)}=(k_{n-1},k_n,k_{n+1}),\\ \\underline\\ell^{(n)}=(\\ell_{n-1},\\ell_n,\\ell_{n+1}),\\ \\underline a^{(n)}=(a_{n-1},a_n,a_{n+1}).\n$$\nThen $\\Gamma(f^n(x))\\in Y_{\\underline k^{(n)},\\underline\\ell^{(n)},\\underline a^{(n)},m_n}$.\nTake $\\Gamma(x_n)\\in Y_{\\underline k^{(n)},\\underline\\ell^{(n)},\\underline a^{(n)},m_n}(j_n)$\ns.t.:\n\\begin{enumerate}[aaa)]\n\\item[(${\\rm a}_n$)] $ d(f^i(f^n(x)),f^i(x_n))+\n\\|\\widetilde{C_\\chi(f^i(f^n(x)))}-\\widetilde{C_\\chi(f^i(x_n))}\\|<e^{-8(j_n+2)}$\nfor $|i|\\leq 1$.\n\\item[(${\\rm b}_n$)] $\\tfrac{Q_\\varepsilon(f^n(x))}{Q_\\varepsilon(x_n)}=e^{\\pm\\varepsilon\/3}$.\n\\end{enumerate}\nDefine $p^s_n=\\delta_\\varepsilon\\min\\{e^{\\varepsilon|k|}Q_\\varepsilon(x_{n+k}):k\\geq 0\\}$ and\n$p^u_n=\\delta_\\varepsilon\\min\\{e^{\\varepsilon|k|}Q_\\varepsilon(x_{n+k}):k\\leq 0\\}$.\nWe claim that $\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\in\\mathbb{Z}}$ is an $\\varepsilon$--gpo\nin $\\mathfs A^\\mathbb{Z}$ that shadows $x$.\n\n\\medskip\n\\noindent\n{\\sc Claim 2:} $\\Psi_{x_n}^{p^s_n,p^u_n}\\in\\mathfs A$ for all $n\\in\\mathbb{Z}$.\n\n\\medskip\n\\noindent\n(CG1) By definition, $\\Gamma(x_n)\\in Y_{\\underline k^{(n)},\\underline\\ell^{(n)},\\underline a^{(n)},m_n}(j_n)$.\n\n\\noindent\n(CG2) By (${\\rm b}_n$), $\\inf\\{e^{\\varepsilon|k|}Q_\\varepsilon(x_{n+k}):k\\geq 0\\}=\ne^{\\pm\\varepsilon\/3}\\inf\\{e^{\\varepsilon|k|}Q_\\varepsilon(f^{n+k}(x)):k\\geq 0\\}$ is positive. Since the only accumulation\npoint of $I_\\varepsilon$ is zero, it follows that $p^s_n,p^u_n$ are well-defined and positive.\nThe other conditions are clear from the definition.\n\n\n\\noindent\n(CG3) Again by (${\\rm b}_n$), we have\n$$\n\\min\\{e^{\\varepsilon|k|}Q_\\varepsilon(x_{n+k}):k\\geq 0\\}=e^{\\pm\\varepsilon\/3}\\min\\{e^{\\varepsilon|k|}Q_\\varepsilon(f^{n+k}(x)):k\\geq 0\\}\n$$\nhence $\\tfrac{p^s_n}{q_\\varepsilon^s(f^n(x))}=e^{\\pm\\varepsilon\/3}$, and analogously \n$\\tfrac{p^u_n}{q_\\varepsilon^u(f^n(x))}=e^{\\pm\\varepsilon\/3}$. By Lemma \\ref{Lemma-q^s}(1),\n$p^s_n\\wedge p^u_n=e^{\\pm\\varepsilon\/3}q_\\varepsilon(f^n(x))\\in [e^{-j_n-2},e^{-j_n+2})$.\n\n\\medskip\n\\noindent\n{\\sc Claim 3:} $\\Psi_{x_n}^{p^s_n,p^u_n}\\overset{\\varepsilon}{\\rightarrow}\\Psi_{x_{n+1}}^{p^s_{n+1},p^u_{n+1}}$\nfor all $n\\in\\mathbb{Z}$.\n\n\\medskip\n\\noindent\n(GPO1) We have $f(x_n),x_{n+1}\\in D_{a_{n+1}}$, and by (${\\rm a}_n$) with $i=1$\nand (${\\rm a}_{n+1}$) with $i=0$, we have\n\\begin{align*}\n& d(f(x_n),x_{n+1})+\\|\\widetilde{C_\\chi(f(x_n))}-\\widetilde{C_\\chi(x_{n+1})}\\|\\\\\n&\\leq  d(f^{n+1}(x),f(x_n))+\n\\|\\widetilde{C_\\chi(f^{n+1}(x))}-\\widetilde{C_\\chi(f(x_n))}\\|\\\\\n&\\ \\ \\ \\,+ d(f^{n+1}(x),x_{n+1})+\n\\|\\widetilde{C_\\chi(f^{n+1}(x))}-\\widetilde{C_\\chi(x_{n+1})}\\|\\\\\n&<e^{-8(j_n+2)}+e^{-8(j_{n+1}+2)}\\leq e^{-8}\\left(q_\\varepsilon(f^n(x))^8+q_\\varepsilon(f^{n+1}(x))^8\\right)\\\\\n&\\overset{!}{\\leq} e^{-8}(1+e^{8\\varepsilon})q_\\varepsilon(f^{n+1}(x))^8\\leq e^{-8+8\\varepsilon\/3}(1+e^{8\\varepsilon})(p^s_{n+1}\\wedge p^u_{n+1})^8\n\\overset{!!}{<}(p^s_{n+1}\\wedge p^u_{n+1})^8,\n\\end{align*}\nwhere in $\\overset{!}{\\leq}$ we used Lemma \\ref{Lemma-q} and in $\\overset{!!}{<}$\nwe used that $e^{-8+8\\varepsilon\/3}(1+e^{8\\varepsilon})<1$ when $\\varepsilon>0$ is sufficiently small. This proves\nthat $\\Psi_{f(x_n)}^{p^s_{n+1}\\wedge p^u_{n+1}}\\overset{\\varepsilon}{\\approx}\\Psi_{x_{n+1}}^{p^s_{n+1}\\wedge p^u_{n+1}}$.\nSimilarly, we prove that\n$\\Psi_{f^{-1}(x_{n+1})}^{p^s_n\\wedge p^u_n}\\overset{\\varepsilon}{\\approx}\\Psi_{x_n}^{p^s_n\\wedge p^u_n}$.\n\n\\medskip\n\\noindent\n(GPO2) The very definitions of $p^s_n,p^u_n$ guarantee that\n$p^s_n=\\min\\{e^\\varepsilon p^s_{n+1},\\delta_\\varepsilon Q_\\varepsilon(x_n)\\}$ and\n$p^u_{n+1}=\\min\\{e^\\varepsilon p^u_n,\\delta_\\varepsilon Q_\\varepsilon(x_{n+1})\\}$.\n\n\\medskip\n\\noindent\n{\\sc Claim 4:} $\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\in\\mathbb{Z}}$ shadows $x$.\n\n\\medskip\nBy (${\\rm a}_n$) with $i=0$, we have\n$\\Psi_{f^n(x)}^{p^s_n\\wedge p^u_n}\\overset{\\varepsilon}{\\approx}\\Psi_{x_n}^{p^s_n\\wedge p^u_n}$, hence \nby Proposition \\ref{Lemma-overlap}(3) we have $f^n(x)=\\Psi_{f^n(x)}(0)\\in \\Psi_{x_n}(R[p^s_n\\wedge p^u_n])$,\nthus $\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\in\\mathbb{Z}}$ shadows $x$.\n\n\\medskip\nThis concludes the proof of sufficiency.\n\n\\medskip\n\\noindent\n{\\em Proof of relevance.} The alphabet $\\mathfs A$ might not a priori satisfy\nthe relevance condition, but we can easily reduce it to a sub-alphabet $\\mathfs A'$ satisfying (1)--(3).\nCall $v\\in\\mathfs A$ relevant if there is $\\underline v\\in\\mathfs A^\\mathbb{Z}$ with $v_0=v$ s.t. $\\underline{v}$ shadows\na point in ${\\rm NUH}_\\chi^*$. Since ${\\rm NUH}_\\chi^*$ is $f$--invariant, every $v_i$ is relevant.\nThen $\\mathfs A'=\\{v\\in\\mathfs A:v\\text{ is relevant}\\}$ is discrete\nbecause $\\mathfs A'\\subset\\mathfs A$, it is sufficient and relevant by definition.\n\\end{proof}\n\nLet $\\Sigma$ be the TMS associated to the graph with vertex set $\\mathfs A$ given by\nTheorem \\ref{Thm-coarse-graining} and\nedges $v\\overset{\\varepsilon}{\\to}w$. An element of $\\Sigma$ is an $\\varepsilon$--gpo, hence\nwe define $\\pi:\\Sigma\\to M$ by\n$$\n\\{\\pi[\\{v_n\\}_{n\\in\\mathbb{Z}}]\\}:=V^s[\\{v_n\\}_{n\\geq 0}]\\cap V^u[\\{v_n\\}_{n\\leq 0}].\n$$\nHere are the main properties of the triple $(\\Sigma,\\sigma,\\pi)$.\n\n\\begin{proposition}\\label{Prop-pi}\nThe following holds for all $\\varepsilon>0$ small enough.\n\\begin{enumerate}[{\\rm (1)}]\n\\item Each $v\\in\\mathfs A$ has finite ingoing and outgoing degree, hence $\\Sigma$ is locally compact.\n\\item $\\pi:\\Sigma\\to M$ is H\\\"older continuous.\n\\item $\\pi\\circ\\sigma=f\\circ\\pi$.\n\\item $\\pi[\\Sigma]\\supset{\\rm NUH}_\\chi^*$.\n\\end{enumerate} \n\\end{proposition}\n\nPart (1) follows from (GPO2), part (2) follows from Proposition \\ref{Prop-graph-transform},\npart (3) is obvious, and part (4) follows from Theorem \\ref{Thm-coarse-graining}(2).\nIt is important noting that $(\\Sigma,\\sigma,\\pi)$ does {\\em not} satisfy Theorem \\ref{Thm-main},\nsince $\\pi$ might be (and usually is) infinite-to-one. We use $\\pi$ to induce a locally\nfinite cover of ${\\rm NUH}_\\chi^\\#$, which will then be refined to a partition of ${\\rm NUH}_\\chi^\\#$\nthat will lead to the proof of Theorem \\ref{Thm-main}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The inverse problem}\n\nOur goal is to analyze when $\\pi$ loses injectivity. More specifically, given that\n$\\pi(\\underline{v})=\\pi(\\underline{w})$ we want to compare $v_n$ and $w_n$, and show that they\nare uniquely defined ``up to bounded error''. We do this under the additional assumption\nthat $\\underline{v},\\underline{w}\\in\\Sigma^\\#$. Remind that $\\Sigma^\\#$ is the {\\em recurrent set} of $\\Sigma$:\n$$\n\\Sigma^\\#:=\\left\\{\\underline v\\in\\Sigma:\\exists v,w\\in V\\text{ s.t. }\\begin{array}{l}v_n=v\\text{ for infinitely many }n>0\\\\\nv_n=w\\text{ for infinitely many }n<0\n\\end{array}\\right\\}.\n$$\nThe main result is the following.\n\n\\begin{theorem}[Inverse theorem]\\label{Thm-inverse}\nThe following holds for $\\varepsilon>0$ small enough.\nIf $\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\in\\mathbb{Z}},\\{\\Psi_{y_n}^{q^s_n,q^u_n}\\}_{n\\in\\mathbb{Z}}\\in\\Sigma^\\#$ satisfy\n$\\pi[\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\in\\mathbb{Z}}]=\\pi[\\{\\Psi_{y_n}^{q^s_n,q^u_n}\\}_{n\\in\\mathbb{Z}}]$ then:\n\\begin{enumerate}[{\\rm (1)}]\n\\item $d(x_n,y_n)<25^{-1}\\max\\{p^s_n\\wedge p^u_n,q^s_n\\wedge q^u_n\\}$.\n\\item $\\tfrac{\\sin\\alpha(x_n)}{\\sin\\alpha(y_n)}=e^{\\pm\\sqrt{\\varepsilon}}$ and\n$|\\cos\\alpha(x_n)-\\cos\\alpha(y_n)|<\\sqrt{\\varepsilon}$.\n\\item $\\tfrac{s(x_n)}{s(y_n)}=e^{\\pm 4\\sqrt{\\varepsilon}}$ and $\\tfrac{u(x_n)}{u(y_n)}=e^{\\pm 4\\sqrt{\\varepsilon}}$.\n\\item $\\tfrac{Q_\\varepsilon(x_n)}{Q_\\varepsilon(y_n)}=e^{\\pm \\sqrt[3]{\\varepsilon}}$.\n\\item $\\tfrac{p^s_n}{q^s_n}=e^{\\pm\\sqrt[3]{\\varepsilon}}$ and $\\tfrac{p^u_n}{q^u_n}=e^{\\pm\\sqrt[3]{\\varepsilon}}$.\n\\item $(\\Psi_{y_n}^{-1}\\circ\\Psi_{x_n})(v)=(-1)^{\\sigma_n}v+\\delta_n+\\Delta_n(v)$ for $v\\in R[10Q_\\varepsilon(x_n)]$,\nwhere $\\sigma_n\\in\\{0,1\\}$, $\\delta_n$ is a vector with $\\|\\delta_n\\|<10^{-1}(q^s_n\\wedge q^u_n)$ and\n$\\Delta_n$ is a vector field s.t. $\\Delta_n(0)=0$ and $\\|d\\Delta_n\\|_0<\\sqrt[3]{\\varepsilon}$ on $R[10Q_\\varepsilon(x_n)]$.\n\\end{enumerate}\n\\end{theorem}\n\nThe difference from Theorem \\ref{Thm-inverse} to \\cite[Thm 5.2]{Sarig-JAMS} is\nthat the estimate on our part (6) holds only in the smaller rectangle $R[10Q_\\varepsilon(x_n)]$. Part (1) is proved as\nin \\cite[Prop. 5.3]{Sarig-JAMS}. \nHere is one of its consequences. We have\n$d(x_n,y_n)<25^{-1}(p^s_n\\wedge p^u_n+q^s_n\\wedge q^u_n)<\\varepsilon[d(x_n,\\mathfs D)^a+d(y_n,\\mathfs D)^a]$,\nhence\n$$\nd(x_n,\\mathfs D)=d(y_n,\\mathfs D)\\pm d(x_n,y_n)=d(y_n,\\mathfs D)\\pm\\varepsilon[d(x_n,\\mathfs D)^a+d(y_n,\\mathfs D)^a].\n$$\nThese estimates have two consequences. The first is that\n\\begin{equation}\\label{equation-distances}\n\\frac{1-\\varepsilon}{1+\\varepsilon}\\leq \\frac{d(x_n,\\mathfs D)}{d(y_n,\\mathfs D)}\\leq \\frac{1+\\varepsilon}{1-\\varepsilon}\n\\end{equation}\nand so, for $\\varepsilon>0$ is sufficiently small, it holds\n$\\tfrac{1}{2}\\leq \\tfrac{d(x_n,\\mathfs D)^a}{d(y_n,\\mathfs D)^a}\\leq 2$.\nThe second consequence is that $x_n\\in D_{y_n}$ and $y_n\\in D_{x_n}$, since\n\\begin{align*}\n&d(x_n,y_n)<\\varepsilon[d(x_n,\\mathfs D)^a+d(y_n,\\mathfs D)^a]<3\\varepsilon\\min\\{d(x_n,\\mathfs D)^a,d(y_n,\\mathfs D)^a\\}\\\\\n&<\\min\\{\\mathfrak r(x_n),\\mathfrak r(y_n)\\}.\n\\end{align*}\nTherefore we can take parallel transport with respect to either $x_n$ or $y_n$.\n\n\n\\medskip\nThe proofs of parts (2)--(6) use, as in \\cite{Sarig-JAMS}, some auxiliary facts about admissible manifolds. Let\n$\\underline v^+=\\{v_n\\}_{n\\geq 0}$ be a positive $\\varepsilon$--gpo with $v_n=\\Psi_{x_n}^{p^s_n,p^u_n}$.\nBy Proposition \\ref{Prop-stable-manifolds}, $V^s[\\underline v^+]$ has the following property:\n$f^n(V^s[\\underline v^+])\\subset V^s[\\{v_k\\}_{k\\geq n}]\\subset \\Psi_{x_n}(R[10Q_\\varepsilon(x_n)])$.\nThis motivates the definition of {\\em staying in windows} as in \\cite{Sarig-JAMS}:\ngiven an $\\varepsilon$--double chart, say that $V^s\\in\\mathfs M^s(v)$ stays in windows if\nthere exists a positive $\\varepsilon$--gpo $\\underline v^+$ with $v_0=v$ and $s$--admissible\nmanifolds $W^s_n\\in\\mathfs M^s(v_n)$ s.t. $f^n(V^s)\\subset W^s_n$ for all $n\\geq 0$.\nIn particular, every $V^s[\\underline v^+]$ stays in windows, and the reverse statement is also true.\nAn analogous definition holds for $u$--admissible manifolds. Given $V^s\\in \\mathfs M^s[v]$\nand $x\\in V^s$, let $e^s_x\\in T_xM$ denote the positively oriented vector tangent to $V^s$ at $x$.\n\n\\begin{proposition}\\label{Prop-stay-window}\nThe following holds for all $\\varepsilon>0$ small enough.\n\\begin{enumerate}[{\\rm (1)}]\n\\item  If $V^s\\in\\mathfs M^s[\\Psi_x^{p^s,p^u}]$ stays in windows then for all $y,z\\in V^s$ and $n\\geq 0$: \n\\begin{enumerate}[{\\rm (a)}]\n\\item $d(f^n(y),f^n(z))<6p^s e^{-\\frac{\\chi}{2}n}$.\n\\item $\\|df^n_y e^s_y\\|\\leq 6\\|C_\\chi(x)^{-1}\\|e^{-\\frac{\\chi}{2}n}$.\n\\item $|\\log\\|df^n_y e^s_y\\|-\\log\\|df^n_z e^s_z\\||<Q_\\varepsilon(x)^{\\beta\/4}$.\n\\end{enumerate}\n\\item If $V^s\\in\\mathfs M^s[\\Psi_x^{p^s,p^u}], U^s\\in\\mathfs M^s[\\Psi_x^{q^s,q^u}]$ stay in windows\nthen either $V^s\\subset U^s$ or $U^s\\subset V^s$.\n\\end{enumerate}\nAnalogous statements hold for $u$--admissible manifolds that stay in windows.\n\\end{proposition}\n\nWhen $M$ is compact and $f$ is a $C^{1+\\beta}$ diffeomorphism, this is \\cite[Prop. 6.3 and 6.4]{Sarig-JAMS}.\nThe only adaptation we need to make is in part (1)(c), see Appendix B.\nBecause of part (1)(c), if $y,z\\in V^s$ then $\\tfrac{s(y)}{s(z)}=e^{\\pm Q_\\varepsilon(x)^{\\beta\/4}}$,\ntherefore we can define $s(V^s):=s(\\Psi_x(0,F(0)))$, where $F$ is the representing function of $V^s$.\nNote that $s(V^s)$ might be infinite, in which case $s(y)$ is infinite for all $y\\in V^s$.\nA similar definition holds for $u$--admissible manifolds that stay in windows. \n\n\\medskip\nThe proof of part (2) of Theorem \\ref{Thm-inverse} is analogous to \\cite[Prop. 6.5]{Sarig-JAMS}.\nIn the sequel we adapt the methods of \\cite{Sarig-JAMS} to prove parts (3)--(6).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Control of $s(x_n)$ and $u(x_n)$}\nAs in \\cite{Sarig-JAMS}, the hyperbolicity of $f$ induces an improvement for $s$ and $u$.\nBecause of symmetry, we only state the result for $s$.\n\n\\begin{lemma}[Improvement lemma]\\label{Lemma-improvement}\nThe following holds for all $\\varepsilon>0$ small enough. Let $v\\overset{\\varepsilon}{\\to}w$ with\n$v=\\Psi_x^{p^s,p^u},w=\\Psi_y^{q^s,q^u}$, and assume $V^s\\in\\mathfs M^s[w]$ stays in windows.\n\\begin{enumerate}[{\\rm (1)}]\n\\item If $s(V^s)<\\infty$ then $s[\\mathfs F^s_{v,w}(V^s)]<\\infty$.\n\\item For $\\xi\\geq {\\sqrt{\\varepsilon}}$, if $s(V^s)<\\infty$ and $\\tfrac{s(V^s)}{s(y)}=e^{\\pm\\xi}$\nthen $\\tfrac{s(\\mathfs F^s_{v,w}(V^s))}{s(x)}=e^{\\pm(\\xi-Q_\\varepsilon(x)^{\\beta\/4})}$.\n\\end{enumerate} \n\\end{lemma}\n\nNote that the ratio improves. \n\n\\begin{proof}\nWhen $M$ is compact and $f$ is a $C^{1+\\beta}$ diffeomorphism,\nthis is \\cite[Lemma 7.2]{Sarig-JAMS}, and the proof of part (1) is identical.\nPart (2) requires some finer estimates.\n\n\\medskip\nLet $F,G$ be the representing functions of $V^s,\\mathfs F^s_{v,w}(V^s)$,\nand let $q:=\\Psi_y(0,F(0))$, $p:=\\Psi_x(0,G(0))$. Then\n$\\tfrac{s(\\mathfs F^s_{v,w}(V^s))}{s(x)}=\\tfrac{s(p)}{s(x)}=\\tfrac{s(p)}{s(f^{-1}(q))}\\cdot\\tfrac{s(f^{-1}(q))}{s(f^{-1}(y))}\\cdot\\tfrac{s(f^{-1}(y))}{s(x)}$. We have:\n\\begin{enumerate}[$\\circ$]\n\\item $p,f^{-1}(q)\\in \\mathfs F^s_{v,w}(V^s)$, hence Proposition \\ref{Prop-stay-window}(1)(c) implies\n$\\tfrac{s(p)}{s(f^{-1}(q))}=e^{\\pm Q_\\varepsilon(x)^{\\beta\/4}}$.\n\\item Since $(p^s\\wedge p^u)^3(q^s\\wedge q^u)^3\\ll Q_\\varepsilon(x)^{\\beta\/4}$,\nProposition \\ref{Lemma-overlap}(1) implies $\\tfrac{s(f^{-1}(y))}{s(x)}=e^{\\pm Q_\\varepsilon(x)^{\\beta\/4}}$.\n\\end{enumerate}\nThus it is enough to show that $\\tfrac{s(f^{-1}(q))}{s(f^{-1}(y))}=e^{\\pm(\\xi-3Q_\\varepsilon(x)^{\\beta\/4})}$.\nWe show one side of the inequality (the other is similar).\nNote that this is the term that gives the improvement. As in \\cite[pp. 375]{Sarig-JAMS}, we have\n$$\n\\tfrac{s(f^{-1}(q))^2}{s(f^{-1}(y))^2}\\leq\n\\underbrace{\\left(\\tfrac{2+e^{2\\xi+2\\chi}s(y)^2\\|df e^s_{f^{-1}(y)}\\|^2}{2+e^{2\\chi}s(y)^2\\|df e^s_{f^{-1}(y)}\\|^2}\\right)}_{= \\text{ I}}\\ \n\\underbrace{{\\rm exp}\\left(2|\\log \\|df e^s_{f^{-1}(q)}\\|-\\log \\|df e^s_{f^{-1}(y)}\\||\\right)}_{=\\text{ II}}.\n$$\nWe estimate I as in \\cite[pp. 376]{Sarig-JAMS}: I $\\leq e^{2\\xi-7Q_\\varepsilon(x)^{\\beta\/4}}$.\nTherefore it suffices to show that $\\text{II}\\leq e^{Q_\\varepsilon(x)^{\\beta\/4}}$.\nSince $\\|df e^s_{f^{-1}(z)}\\|=\\|df^{-1}e^s_z\\|^{-1}$,\n$\\text{II}={\\rm exp}(2|\\log \\|df^{-1}e^s_q\\|-\\log \\|df^{-1}e^s_{y}\\||)$, hence\nby the claim in the proof of Proposition \\ref{Prop-stay-window} (Appendix B):\n\\begin{equation}\\label{estimate-II}\n\\log(\\text{II})\\leq2\\mathfrak K\\rho(y)^{-2a}[d(q,y)^\\beta+\\|e^s_q-P_{y,q}e^s_y\\|].\n\\end{equation}\nSince $q=\\Psi_y(0,G(0))$ and $y=\\Psi_y(0,0)$, Lemma \\ref{Lemma-Pesin-chart}(1) implies that\n$d(q,y)\\leq 2|G(0)|\\leq 500^{-1}(q^s\\wedge q^u)\\leq 500^{-1}e^\\varepsilon(p^s\\wedge p^u)$, therefore\n$d(q,y)<Q_\\varepsilon(x),Q_\\varepsilon(y)$. Hence for small $\\varepsilon>0$:\n\\begin{align*}\n&2\\mathfrak K\\rho(y)^{-2a}d(q,y)^\\beta\\leq 2\\mathfrak K\\rho(y)^{-2a}Q_\\varepsilon(y)^{3\\beta\/4}Q_\\varepsilon(x)^{\\beta\/4}\n\\leq 2\\mathfrak K\\rho(y)^{-2a}Q_\\varepsilon(y)^{\\beta\/36}Q_\\varepsilon(x)^{\\beta\/4}\\\\\n&\\leq 2\\mathfrak K\\varepsilon^{1\/12}Q_\\varepsilon(x)^{\\beta\/4}<\\tfrac{1}{2}Q_\\varepsilon(x)^{\\beta\/4}.\n\\end{align*}\nTo bound the second term of (\\ref{estimate-II}), we first estimate $\\sin\\angle(e^s_q,P_{y,q}e^s_y)$.\nSince $e^s_y$ is the unitary vector in the direction of\n$d(\\Psi_y)_0\\colvec{1\\\\0}=d(\\exp{y})_0\\circ C_\\chi(y)\\colvec{1\\\\0}$\nand $e^s_q$ is the unitary vector in the direction of\n$d(\\Psi_y)_{(0,G(0))}\\colvec{1\\\\ G'(0)}=d(\\exp{y})_{C_\\chi(y)\\colvec[.6]{0\\\\G(0)}}\\circ C_\\chi(y)\\colvec{1\\\\ G'(0)}$,\nthe angles they define are the same. In other words, if\n$$\nA=\\widetilde{d(\\exp{y})_0\\circ C_\\chi(y)},B=\\widetilde{d(\\exp{y})_{C_\\chi(y)\\colvec[.6]{0\\\\G(0)}}\\circ C_\\chi(y)},\nv_1=\\colvec{1\\\\0},v_2=\\colvec{1\\\\ G'(0)}\n$$\nthen $\\sin\\angle(e^s_q,P_{y,q}e^s_y)=\\sin\\angle(Av_1,Bv_2)$. Using (\\ref{gen-ineq-angles}) \nwith $L=A$, $v=v_1$, $w=A^{-1}Bv_2$, we get\n\\begin{align*}\n&|\\sin\\angle(Av_1,Bv_2)|\\leq \\|A\\|\\|A^{-1}\\||\\sin\\angle(v_1,A^{-1}Bv_2)|\\\\\n&\\leq \\|C_\\chi(y)^{-1}\\|[|\\sin\\angle(v_1,v_2)|+|\\sin\\angle(v_2,A^{-1}Bv_2)|].\n\\end{align*}\nWe have $|\\sin\\angle(v_1,v_2)|\\leq |G'(0)|\\leq \\tfrac{1}{2}(q^s\\wedge q^u)^{\\beta\/3}\\leq\n\\tfrac{e^{\\frac{\\beta\\varepsilon}{3}}}{2}(p^s\\wedge p^u)^{\\beta\/3}$, therefore for small $\\varepsilon>0$ it holds\n$|\\sin\\angle(v_1,v_2)|\\leq Q_\\varepsilon(x)^{\\beta\/3},Q_\\varepsilon(y)^{\\beta\/3}$. In particular\n$|\\sin\\angle(v_1,v_2)|\\leq Q_\\varepsilon(y)^{\\beta\/12}Q_\\varepsilon(x)^{\\beta\/4}$. Also, by (A3):\n\\begin{align*}\n&\\|A^{-1}B-{\\rm Id}\\|\\leq \\|A^{-1}\\|\\|A-B\\|\\leq\n\\|C_\\chi(y)^{-1}\\| \\left\\|\\widetilde{d(\\exp{y})_0}-\\widetilde{d(\\exp{y})_{C_\\chi(y)\\colvec[.6]{0\\\\ G(0)}}}\\right\\|\\\\\n&\\leq \\|C_\\chi(y)^{-1}\\|\\rho(y)^{-a}|G(0)|\\leq  \\|C_\\chi(y)^{-1}\\|\\rho(y)^{-a}Q_\\varepsilon(y)^{1-\\frac{\\beta}{4}}Q_\\varepsilon(x)^{\\beta\/4}\\\\\n&\\leq Q_\\varepsilon(y)^{1-\\frac{11\\beta}{36}}Q_\\varepsilon(x)^{\\beta\/4}<\\tfrac{1}{4}Q_\\varepsilon(y)^{\\beta\/12}Q_\\varepsilon(x)^{\\beta\/4}\\ll 1.\n\\end{align*}\nThis implies that $v_2,A^{-1}Bv_2$ are almost unitary vectors, therefore\n$$\n|\\sin\\angle(v_2,A^{-1}Bv_2)|\\leq 2\\|v_2-A^{-1}Bv_2\\|\\leq 4\\|A^{-1}B-{\\rm Id}\\|<Q_\\varepsilon(y)^{\\beta\/12}Q_\\varepsilon(x)^{\\beta\/4},$$\nthus $|\\sin\\angle(e^s_q,P_{y,q}e^s_y)|<2\\|C_\\chi(y)^{-1}\\|Q_\\varepsilon(y)^{\\beta\/12}Q_\\varepsilon(x)^{\\beta\/4}$.\nSince $\\|e^s_q\\|=\\|P_{y,q}e^s_y\\|=1$ and the angle between them is small,\n$\\|e^s_q-P_{y,q}e^s_y\\|\\leq 2|\\sin\\angle(e^s_q,P_{y,q}e^s_y)|<4\\|C_\\chi(y)^{-1}\\|Q_\\varepsilon(y)^{\\beta\/12}Q_\\varepsilon(x)^{\\beta\/4}$.\nThe conclusion is that for small $\\varepsilon>0$:\n\\begin{align*}\n&2\\mathfrak K\\rho(y)^{-2a}\\|e^s_q-P_{y,q}e^s_y\\|\\leq\n8\\mathfrak K\\|C_\\chi(y)^{-1}\\|\\rho(y)^{-2a}Q_\\varepsilon(y)^{\\beta\/12}Q_\\varepsilon(x)^{\\beta\/4}\\\\\n&\\leq 8\\mathfrak K\\|C_\\chi(y)^{-1}\\|Q_\\varepsilon(y)^{\\beta\/24}\\rho(y)^{-2a}Q_\\varepsilon(y)^{\\beta\/36}Q_\\varepsilon(x)^{\\beta\/4}\\\\\n&\\leq 8\\mathfrak K\\varepsilon^{5\/24}Q_\\varepsilon(x)^{\\beta\/4}<\\tfrac{1}{2}Q_\\varepsilon(x)^{\\beta\/4}.\n\\end{align*}\nHence (\\ref{estimate-II}) implies that $\\text{II}<e^{Q_\\varepsilon(x)^{\\beta\/4}}$.\n\\end{proof}\n\nWe are now ready to prove part (3) of Theorem \\ref{Thm-inverse}.\n\\begin{proposition}\nThe following holds for all $\\varepsilon>0$ small enough.\nIf $\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\in\\mathbb{Z}}$, $\\{\\Psi_{y_n}^{q^s_n,q^u_n}\\}_{n\\in\\mathbb{Z}}\\in\\Sigma^\\#$ satisfy\n$\\pi[\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\in\\mathbb{Z}}]=\\pi[\\{\\Psi_{y_n}^{q^s_n,q^u_n}\\}_{n\\in\\mathbb{Z}}]$ then for all $n\\in\\mathbb{Z}$:\n$$\n\\tfrac{s(x_n)}{s(y_n)}=e^{\\pm 4\\sqrt{\\varepsilon}}\\text{ and }\\tfrac{u(x_n)}{u(y_n)}=e^{\\pm 4\\sqrt{\\varepsilon}}.\n$$\n\\end{proposition}\n\nWhen $M$ is compact and $f$ is a $C^{1+\\beta}$ diffeomorphism,\nthis is \\cite[Prop. 7.3]{Sarig-JAMS}, and the proof is identical.\nLet $\\underline v=\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\in\\mathbb{Z}}$ and $\\underline w=\\{\\Psi_{y_n}^{q^s_n,q^u_n}\\}_{n\\in\\mathbb{Z}}$.\nWe sketch the proof for the first estimate:\n\\begin{enumerate}[$\\circ$]\n\\item If $\\pi(\\underline v)=x$ then $s(x)<\\infty$: this follows from the relevance of $\\mathfs A$ \n(Thm. \\ref{Thm-coarse-graining}(3)).\n\\item Apply Lemma \\ref{Lemma-improvement} along $\\underline v$ and the orbit of $x$: if\n$v_n=v$ for infinitely many $n>0$, then the ratio improves at each of these indices.\nThe conclusion is that $\\tfrac{s(V^s[\\{v_k\\}_{k\\geq n}])}{s(x_n)}=e^{\\pm\\sqrt{\\varepsilon}}$, and\nanalogously $\\tfrac{s(V^s[\\{w_k\\}_{k\\geq n}])}{s(y_n)}=e^{\\pm\\sqrt{\\varepsilon}}$.\n\\item Since $f^n(x)\\in V^s[\\{v_k\\}_{k\\geq n}]\\cap V^s[\\{w_k\\}_{k\\geq n}]$, Proposition \\ref{Prop-stay-window}(1)(c)\nimplies that $\\tfrac{s(V^s[\\{v_k\\}_{k\\geq n}])}{s(f^n(x))}=e^{\\pm\\sqrt{\\varepsilon}}$\nand $\\tfrac{s(V^s[\\{w_k\\}_{k\\geq n}])}{s(f^n(x))}=e^{\\pm\\sqrt{\\varepsilon}}$.\n\\end{enumerate}\nHence $\\tfrac{s(x_n)}{s(y_n)}=\\tfrac{s(x_n)}{s(V^s[\\{v_k\\}_{k\\geq n}])}\\cdot\\tfrac{s(V^s[\\{v_k\\}_{k\\geq n}])}{s(f^n(x))}\n\\cdot\\tfrac{s(f^n(x))}{s(V^s[\\{w_k\\}_{k\\geq n}])}\\cdot\\tfrac{s(V^s[\\{w_k\\}_{k\\geq n}])}{s(y_n)}=e^{\\pm4\\sqrt{\\varepsilon}}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Control of $Q_\\varepsilon(x_n)$}\n\nRemind that $Q_\\varepsilon(x):=\\max\\{q\\in I_\\varepsilon:q\\leq \\widetilde Q_\\varepsilon(x)\\}$ where\n$$\n\\widetilde Q_\\varepsilon(x)=\\varepsilon^{3\/\\beta}\n\\min\\left\\{\\|C_\\chi(x)^{-1}\\|_{\\rm Frob}^{-24\/\\beta},\\|C_\\chi(f(x))^{-1}\\|^{-12\/\\beta}_{\\rm Frob}\\rho(x)^{72a\/\\beta}\\right\\},\n$$\nso we first control $\\widetilde Q_\\varepsilon(x_n)$.\nBy parts (2)--(3), $\\tfrac{\\|C_\\chi(x_n)^{-1}\\|_{\\rm Frob}}{\\|C_\\chi(y_n)^{-1}\\|_{\\rm Frob}}=e^{\\pm 5\\sqrt{\\varepsilon}}$.\nUsing that $\\Psi_{f(x_n)}^{p^s_{n+1}\\wedge p^u_{n+1}}\\overset{\\varepsilon}{\\approx}\\Psi_{x_{n+1}}^{p^s_{n+1}\\wedge p^u_{n+1}}$,\nProposition \\ref{Lemma-overlap}(1)--(2) implies that\n$\\tfrac{\\|C_\\chi(f(x_n))^{-1}\\|_{\\rm Frob}}{\\|C_\\chi(x_{n+1})^{-1}\\|_{\\rm Frob}}=e^{\\pm\\sqrt{\\varepsilon}}$,\nand similarly $\\tfrac{\\|C_\\chi(f(y_n))^{-1}\\|_{\\rm Frob}}{\\|C_\\chi(y_{n+1})^{-1}\\|_{\\rm Frob}}=e^{\\pm\\sqrt{\\varepsilon}}$.\nHence\n$$\\tfrac{\\|C_\\chi(f(x_n))^{-1}\\|_{\\rm Frob}}{\\|C_\\chi(f(y_n))^{-1}\\|_{\\rm Frob}}=\n\\tfrac{\\|C_\\chi(f(x_n))^{-1}\\|_{\\rm Frob}}{\\|C_\\chi(x_{n+1})^{-1}\\|_{\\rm Frob}}\\cdot\n\\tfrac{\\|C_\\chi(x_{n+1})^{-1}\\|_{\\rm Frob}}{\\|C_\\chi(y_{n+1})^{-1}\\|_{\\rm Frob}}\\cdot\n\\tfrac{\\|C_\\chi(y_{n+1})^{-1}\\|_{\\rm Frob}}{\\|C_\\chi(f(y_n))^{-1}\\|_{\\rm Frob}}=e^{\\pm 7\\sqrt{\\varepsilon}}.$$\n\n\\medskip\nWe now estimate the ratio $\\tfrac{\\rho(x_n)}{\\rho(y_n)}$. For that we obtain estimates\nsimilar to (\\ref{equation-distances}) for $f^{\\pm 1}(x_n),f^{\\pm 1}(y_n)$. By symmetry,\nwe only need to get the inequalities for $f(x_n),f(y_n)$. Start by noting that\n$d(f(x_n),x_{n+1})\\leq (p^s_{n+1}\\wedge p^u_{n+1})^8<\\varepsilon d(x_{n+1},\\mathfs D)$, hence\n$d(f(x_n),\\mathfs D)=d(x_{n+1},\\mathfs D)\\pm d(f(x_n),x_{n+1})=(1\\pm\\varepsilon)d(x_{n+1},\\mathfs D)$\nand thus $d(f(x_n),x_{n+1})<2\\varepsilon d(f(x_n),\\mathfs D)$. Similarly $d(f(y_n),y_{n+1})<2\\varepsilon d(f(y_n),\\mathfs D)$.\nUsing part (1), $d(x_{n+1},y_{n+1})<\\varepsilon[d(x_{n+1},\\mathfs D)+d(y_{n+1},\\mathfs D)]<\n2\\varepsilon[d(f(x_n),\\mathfs D)+d(f(y_n),\\mathfs D)]$, therefore\n\\begin{align*}\nd(f(x_n),f(y_n))&\\leq d(f(x_n),x_{n+1})+d(x_{n+1},y_{n+1})+d(y_{n+1},f(y_n))\\\\\n&<4\\varepsilon[d(f(x_n),\\mathfs D)+d(f(y_n),\\mathfs D)].\n\\end{align*}\nThis implies that $d(f(x_n),\\mathfs D)=d(f(y_n),\\mathfs D)\\pm 4\\varepsilon[d(f(x_n),\\mathfs D)+d(f(y_n),\\mathfs D)]$\nand so\n$\\tfrac{1-4\\varepsilon}{1+4\\varepsilon}\\leq \\tfrac{d(f(x_n),\\mathfs D)}{d(f(y_n),\\mathfs D)}\\leq\\tfrac{1+4\\varepsilon}{1-4\\varepsilon}$.\nThe same estimate holds for $f^{-1}$. Together with (\\ref{equation-distances}), we get that\n$\\tfrac{1-4\\varepsilon}{1+4\\varepsilon}\\leq\\tfrac{\\rho(x_n)}{\\rho(y_n)}\\leq\\tfrac{1+4\\varepsilon}{1-4\\varepsilon}$.\nIf $\\varepsilon>0$ is small enough then\n$e^{-\\sqrt{\\varepsilon}}<\\left(\\tfrac{1-4\\varepsilon}{1+4\\varepsilon}\\right)^{\\frac{72a}{\\beta}}\n<\\left(\\tfrac{1+4\\varepsilon}{1-4\\varepsilon}\\right)^{\\frac{72a}{\\beta}}<e^{\\sqrt{\\varepsilon}}$, hence\n$\\tfrac{\\rho(x_n)^{72a\/\\beta}}{\\rho(y_n)^{72a\/\\beta}}=e^{\\pm\\sqrt{\\varepsilon}}$.\nThe conclusion is that\n$\\tfrac{\\widetilde Q_\\varepsilon(x_n)}{\\widetilde Q_\\varepsilon(y_n)}={\\rm exp}[\\pm(\\tfrac{120}{\\beta}\\sqrt{\\varepsilon})]$,\nwhich implies that $\\tfrac{Q_\\varepsilon(x_n)}{Q_\\varepsilon(y_n)}={\\rm exp}[\\pm(\\frac{2}{3}\\varepsilon+\\tfrac{120}{\\beta}\\sqrt{\\varepsilon})]$.\nHence if $\\varepsilon>0$ is small enough it holds $\\tfrac{Q_\\varepsilon(x_n)}{Q_\\varepsilon(y_n)}=e^{\\pm\\sqrt[3]{\\varepsilon}}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Control of $p^s_n$ and $p^u_n$}\n\nAs in \\cite[Prop. 8.3]{Sarig-JAMS}, (GPO2) implies the lemma below.\n\n\\begin{lemma}\\label{Lemma-maximality}\nIf $\\underline v=\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\in\\mathbb{Z}}\\in\\Sigma^\\#$ then \n$p^s_n=\\delta_\\varepsilon Q_\\varepsilon(x_n)$ for infinitely many $n>0$ and\n$p^u_n=\\delta_\\varepsilon Q_\\varepsilon(x_n)$ for infinitely many $n<0$.\n\\end{lemma}\n\nWe now prove the first half of part (5) (the other half is analogous).\nBy symmetry, it is enough to prove that $p^s_n\\geq e^{-\\sqrt[3]{\\varepsilon}}q^s_n$ for all $n\\in\\mathbb{Z}$.\n\\begin{enumerate}[$\\circ$]\n\\item If $p^s_n=\\delta_\\varepsilon Q_\\varepsilon(x_n)$ then part (4) gives\n$p^s_n=\\delta_\\varepsilon Q_\\varepsilon(x_n)\\geq e^{-\\sqrt[3]{\\varepsilon}}\\delta_\\varepsilon Q_\\varepsilon(y_n)\n\\geq e^{-\\sqrt[3]{\\varepsilon}}q^s_n$.\n\\item If $p^s_n\\geq e^{-\\sqrt[3]{\\varepsilon}}q^s_n$ then (GPO2) and part (4) give:\n$$p^s_{n-1}=\\min\\{e^\\varepsilon p^s_n,\\delta_\\varepsilon Q_\\varepsilon(x_{n-1})\\}\\geq\ne^{-\\sqrt[3]{\\varepsilon}}\\min\\{e^\\varepsilon q^s_n,\\delta_\\varepsilon Q_\\varepsilon(y_{n-1})\\}=e^{-\\sqrt[3]{\\varepsilon}}q^s_{n-1}.$$\n\\end{enumerate}\nBy Lemma \\ref{Lemma-maximality}, it follows that $p^s_n\\geq e^{-\\sqrt[3]{\\varepsilon}}q^s_n$ for all $n\\in\\mathbb{Z}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Control of $\\Psi_{y_n}^{-1}\\circ\\Psi_{x_n}$}\n\nFor $z_n=x_n,y_n$, the calculations in the\nproof of Lemma \\ref{Lemma-linear-reduction} give that\n$$\n\\widetilde{C_\\chi(z_n)}=R_{z_i}\\left[\\begin{array}{cc}\\tfrac{1}{s(z_n)}& \\tfrac{\\cos\\alpha(z_n)}{u(z_n)}\\\\\n0 & \\tfrac{\\sin\\alpha(z_n)}{u(z_n)}\\end{array}\\right]\n$$\nwhere $R_{z_n}$ is the rotation that takes $e_1$ to $\\iota_{z_n}e^s_{z_n}$.\n\n\\begin{lemma}\\label{Lemma-rotations}\nUnder the conditions of Theorem \\ref{Thm-inverse}, for all $n\\in\\mathbb{Z}$ it holds\n$$\nR_{y_n}^{-1}R_{x_n}=(-1)^{\\sigma_n}{\\rm Id}+\n\\left[\\begin{array}{cc}\\varepsilon_{11}&\\varepsilon_{12}\\\\ \\varepsilon_{21}&\\varepsilon_{22}\\end{array}\\right]\n$$\nwhere $\\sigma_n\\in\\{0,1\\}$ and $|\\varepsilon_{jk}|<(p^s_n\\wedge p^u_n)^{\\beta\/5}+(q^s_n\\wedge q^u_n)^{\\beta\/5}<\\sqrt{\\varepsilon}$.\n\\end{lemma}\n\nWhen $M$ is compact and $f$ is a $C^{1+\\beta}$ diffeomorphism,\nthis is \\cite[Prop. 6.7]{Sarig-JAMS}. See Apendix B for the proof in our context.\n\n\\medskip\nNow we establish part (6). It is enough to prove the case $n=0$.\nWrite $\\Psi_{x_0}^{p^s_0,p^u_0}=\\Psi_{x}^{p^s,p^u}$,\n$\\Psi_{y_0}^{q^s_0,q^u_0}=\\Psi_{y}^{q^s,q^u}$, $p=p^s\\wedge p^u$, $q=q^s\\wedge q^u$,\n$\\sigma=\\sigma_0$.\nWrite $\\widetilde{C_\\chi(x)}=R_xC_x$, $\\widetilde{C_\\chi(y)}=R_yC_y$.\nAs in \\cite[\\S9]{Sarig-JAMS}, Lemma \\ref{Lemma-rotations} gives\n$\\|C_y^{-1}C_x-(-1)^\\sigma{\\rm Id}\\|<14\\sqrt{\\varepsilon}$ and hence for small $\\varepsilon>0$:\n\\begin{align*}\n&\\|\\widetilde{C_\\chi(x)}-\\widetilde{C_\\chi(y)}\\|\\leq \\|R_xC_x-(-1)^\\sigma R_xC_y\\|+\\|R_xC_y-(-1)^\\sigma R_yC_y\\|\\\\\n&\\leq \\|C_y^{-1}\\|\\|C_y^{-1}C_x-(-1)^\\sigma{\\rm Id}\\|+\\|R_y^{-1}R_x-(-1)^\\sigma{\\rm Id}\\|<\n16\\sqrt{\\varepsilon}\\|C_y^{-1}\\|<\\|C_y^{-1}\\|.\n\\end{align*}\nWe use this to show that $\\Psi_y^{-1}\\circ\\Psi_x$ is well-defined in $R[10Q_\\varepsilon(x)]$.\nThe argument is very similar to the proof of Proposition \\ref{Lemma-overlap}(3).\nFor $v\\in R[10Q_\\varepsilon(x)]$, (A2) and part (4) imply that for small $\\varepsilon>0$:\n\\begin{align*}\n&d(\\Psi_x(v),\\Psi_y(v))\\leq 2d_{\\rm Sas}(C_\\chi(x)v,C_\\chi(y)v)\\leq 4(d(x,y)+\\|\\widetilde{C_\\chi(x)}-\\widetilde{C_\\chi(y)}\\|\\|v\\|)\\\\\n&< 4(q+\\|C_y^{-1}\\|\\|v\\|)<100\\|C_y^{-1}\\|Q_\\varepsilon(y).\n\\end{align*}\nhence $\\Psi_x(v)\\in B(\\Psi_y(v),100\\|C_y^{-1}\\|Q_\\varepsilon(y))\\subset \\Psi_y[B]$ where\n$B\\subset\\mathbb{R}^2$ is the ball with center $v$ and radius $200\\|C_y^{-1}\\|^2Q_\\varepsilon(y)$.\nIf $\\varepsilon>0$ is small then for $w\\in B$ we have\n\\begin{align*}\n&\\|w\\|\\leq \\|v\\|+200\\|C_y^{-1}\\|^2Q_\\varepsilon(y)<20Q_\\varepsilon(y)+200\\varepsilon^{1\/4}Q_\\varepsilon(y)^{1-\\beta\/12}\\\\\n&<20\\varepsilon^{3\/\\beta}d(y,\\mathfs D)^a+200\\varepsilon^{1\/4}d(y,\\mathfs D)^a<d(y,\\mathfs D)^a<2\\mathfrak r(y),\n\\end{align*}\ntherefore $\\Psi_y^{-1}\\circ\\Psi_x$ is well-defined in $R[10Q_\\varepsilon(x)]$.\n\n\\medskip\nIt remains to estimate $\\Psi_y^{-1}\\circ\\Psi_x-(-1)^\\sigma{\\rm Id}$. Write\n$\\Psi_y^{-1}\\circ\\Psi_x=(-1)^\\sigma{\\rm Id}+\\delta+\\Delta$, where $\\delta\\in\\mathbb{R}^2$ is a constant vector\nand $\\Delta:R[10Q_\\varepsilon(x)]\\to\\mathbb{R}^2$. Let $v\\in R[10Q_\\varepsilon(x)]$.\nProceeding as in \\cite[pp. 382]{Sarig-JAMS} and applying (A4) we get for small $\\varepsilon>0$ that:\n\\begin{align*}\n&\\|d(\\Delta)_v\\|\\leq 2\\|C_y^{-1}\\|d(y,\\mathfs D)^{-a}d(x,y)+14\\sqrt{\\varepsilon}\n<2\\|C_y^{-1}\\|d(y,\\mathfs D)^{-a}Q_\\varepsilon(y)+14\\sqrt{\\varepsilon}\\\\\n&<2\\sqrt{\\varepsilon}\\|C_y^{-1}\\|Q_\\varepsilon(y)^{\\beta\/24}d(y,\\mathfs D)^{-a}Q_\\varepsilon(y)^{\\beta\/72}+14\\sqrt{\\varepsilon}<16\\sqrt{\\varepsilon}<\\sqrt[3]{\\varepsilon}.\n\\end{align*}\nThe estimate of $\\|\\delta\\|$ is identical to \\cite[pp. 383]{Sarig-JAMS}. This completes the proof\nof part (6), and hence of Theorem \\ref{Thm-inverse}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Symbolic dynamics}\n\n\\subsection{A countable Markov partition}\n\nLet $(\\Sigma,\\sigma)$ be the TMS\nconstructed in Theorem \\ref{Thm-coarse-graining}, and let\n$\\pi:\\Sigma\\to M$ as defined in the end of section \\ref{Section-coarse-graining}.\nIn the sequel we use Theorem \\ref{Thm-inverse} to construct a cover of ${\\rm NUH}_\\chi^\\#$\nthat is locally finite and satisfies a (symbolic) Markov property.\n\n\\medskip\n\\noindent\n{\\sc The Markov cover $\\mathfs Z$:} Let $\\mathfs Z:=\\{Z(v):v\\in\\mathfs A\\}$, where\n$$\nZ(v):=\\{\\pi(\\underline v):\\underline v\\in\\Sigma^\\#\\text{ and }v_0=v\\}.\n$$\n\n\\medskip\nIn other words, $\\mathfs Z$ is the family defined by the natural partition of $\\Sigma^\\#$ into\ncylinder at the zeroth position. Admissible manifolds allow us to\ndefine {\\em invariant fibres} inside each $Z\\in\\mathfs Z$. Let $Z=Z(v)$.\n\n\\medskip\n\\noindent\n{\\sc $s$\/$u$--fibres in $\\mathfs Z$:} Given $x\\in Z$, let $W^s(x,Z):=V^s[\\{v_n\\}_{n\\geq 0}]\\cap Z$\nbe the {\\em $s$--fibre} of $x$ in $Z$ for some (any) $\\underline v=\\{v_n\\}_{n\\in\\mathbb{Z}}\\in\\Sigma^\\#$\ns.t. $\\pi(\\underline v)=x$ and $v_0=v$. Similarly, let $W^u(x,Z):=V^u[\\{v_n\\}_{n\\leq 0}]\\cap Z$ be\nthe {\\em $u$--fibre} of $x$ in $Z$.\n\n\\medskip\nBy Proposition \\ref{Prop-stay-window}(2), the definitions above do not depend on the choice of $\\underline v$, \nand any two $s$--fibres ($u$--fibres) either coincide or are disjoint. We also\ndefine $V^s(x,Z):=V^s[\\{v_n\\}_{n\\geq 0}]$ and $V^u(x,Z):=V^u[\\{v_n\\}_{n\\leq 0}]$.\nBelow we collect the main properties of $\\mathfs Z$.\n\n\\begin{proposition}\\label{Prop-Z}\nThe following are true.\n\\begin{enumerate}[{\\rm (1)}]\n\\item {\\sc Covering property:} $\\mathfs Z$ is a cover of ${\\rm NUH}_\\chi^\\#$.\n\\item {\\sc Local finiteness:} For every $Z\\in\\mathfs Z$, $\\#\\{Z'\\in\\mathfs Z:Z\\cap Z'\\neq\\emptyset\\}<\\infty$.\n\\item {\\sc Product structure:} For every $Z\\in\\mathfs Z$ and every $x,y\\in Z$, the intersection\n$W^s(x,Z)\\cap W^u(y,Z)$ consists of a single point of $Z$.\n\\item {\\sc Symbolic Markov property:} If $x=\\pi(\\underline v)$ with $\\underline v\\in\\Sigma^\\#$, then\n$$\nf(W^s(x,Z(v_0)))\\subset W^s(f(x),Z(v_1))\\, \\text{ and }\\, f^{-1}(W^u(f(x),Z(v_1)))\\subset W^u(x,Z(v_0)).\n$$\n\\end{enumerate}\n\\end{proposition}\n\nPart (1) follows from Theorem \\ref{Thm-coarse-graining}(2),\npart (2) follows from Theorem \\ref{Thm-inverse}(5), part (3) follows from\nLemma \\ref{Lemma-admissible-manifolds}(1), and part (4) is proved as in \\cite[Prop. 10.9]{Sarig-JAMS}.\nFor $x,y\\in Z$, let $[x,y]_Z:=$ intersection point of $W^s(x,Z)$ and $W^u(y,Z)$, and\ncall it the {\\em Smale bracket} of $x,y$ in $Z$.\n\n\\begin{lemma}\nThe following holds for all $\\varepsilon>0$ small enough.\n\\begin{enumerate}[{\\rm (1)}]\n\\item {\\sc Compatibility:} If $x,y\\in Z(v_0)$ and $f(x),f(y)\\in Z(v_1)$ with\n$v_0\\overset{\\varepsilon}{\\to} v_1$ then $f([x,y]_{Z(v_0)})=[f(x),f(y)]_{Z(v_1)}$.\n\\item {\\sc Overlapping charts properties:} If $Z=Z(\\Psi_x^{p^s,p^u}),Z'=Z(\\Psi_y^{q^s,q^u})\\in\\mathfs Z$\nwith $Z\\cap Z'\\neq \\emptyset$ then:\n\\begin{enumerate}[{\\rm (a)}]\n\\item $Z\\subset \\Psi_y(R[q^s\\wedge q^u])$.\n\\item If $x\\in Z\\cap Z'$ then $W^{s\/u}(x,Z)\\subset V^{s\/u}(x,Z')$. \n\\item If $x\\in Z,y\\in Z'$ then $V^s(x,Z)$ and $V^u(y,Z')$ intersect at a unique point. \n\\end{enumerate}\n\\end{enumerate}\n\\end{lemma}\n\nWhen $M$ is compact and $f$ is a diffeomorphism, part (1) is \\cite[Lemma 10.7]{Sarig-JAMS}\nand part (2) is \\cite[Lemmas 10.8 and 10.10]{Sarig-JAMS}. The same proofs work in our case,\nsince all calculations are made in the rectangle $R[10Q_\\varepsilon(x)]$, and in this domain\nwe have Theorem \\ref{Thm-inverse}(6). \n\n\\medskip\nNow we apply a refinement method to destroy non-trivial intersections in $\\mathfs Z$. \nThe result is a partition of ${\\rm NUH}_\\chi^\\#$ with the (geometrical) Markov property.\nThis idea, originally developed by Sina{\\u\\i} and Bowen\nfor finite covers \\cite{Sinai-Construction-of-MP,Sinai-MP-U-diffeomorphisms,Bowen-LNM},\nworks equally well for countable covers with the local finiteness property \\cite{Sarig-JAMS}.\nWrite $\\mathfs Z=\\{Z_1,Z_2,\\ldots\\}$.\n\n\\medskip\n\\noindent\n{\\sc The Markov partition $\\mathfs R$:} For every $Z_i,Z_j\\in\\mathfs Z$, define a partition of $Z_i$ by:\n\\begin{align*}\nT_{ij}^{su}&=\\{x\\in Z_i: W^s(x,Z_i)\\cap Z_j\\neq\\emptyset,\nW^u(x,Z_i)\\cap Z_j\\neq\\emptyset\\}\\\\\nT_{ij}^{s\\emptyset}&=\\{x\\in Z_i: W^s(x,Z_i)\\cap Z_j\\neq\\emptyset,\nW^u(x,Z_i)\\cap Z_j=\\emptyset\\}\\\\\nT_{ij}^{\\emptyset u}&=\\{x\\in Z_i: W^s(x,Z_i)\\cap Z_j=\\emptyset,\nW^u(x,Z_i)\\cap Z_j\\neq\\emptyset\\}\\\\\nT_{ij}^{\\emptyset\\emptyset}&=\\{x\\in Z_i: W^s(x,Z_i)\\cap Z_j=\\emptyset,\nW^u(x,Z_i)\\cap Z_j=\\emptyset\\}.\n\\end{align*}\nLet $\\mathfs T:=\\{T_{ij}^{\\alpha\\beta}:i,j\\geq 1,\\alpha\\in\\{s,\\emptyset\\},\\beta\\in\\{u,\\emptyset\\}\\}$,\nand let $\\mathfs R$ be the partition generated by $\\mathfs T$.\n\n\n\\medskip\nSince $T_{ii}^{su}=Z_i$, $\\mathfs R$ is a partition of ${\\rm NUH}_\\chi^\\#$.\nClearly, $\\mathfs R$ is a refinement of $\\mathfs Z$. Theorem \\ref{Thm-inverse}\nimplies two local finiteness properties for $\\mathfs R$:\n\\begin{enumerate}[$\\circ$]\n\\item For every $Z\\in\\mathfs Z$, $\\#\\{R\\in\\mathfs R:R\\subset Z\\}<\\infty$.\n\\item For every $R\\in\\mathfs R$, $\\#\\{Z\\in\\mathfs Z:Z\\supset R\\}<\\infty$.\n\\end{enumerate}\n\n\\medskip\nNow we show that $\\mathfs R$ is a Markov partition in the sense of Sina{\\u\\i} \\cite{Sinai-MP-U-diffeomorphisms}. \n\n\\medskip\n\\noindent\n{\\sc $s$\/$u$--fibres in $\\mathfs R$:} Given $x\\in R\\in\\mathfs R$, we define the {\\em $s$--fibre}\nand {\\em $u$--fibre} of $x$ by:\n\\begin{align*}\nW^s(x,R):=\n\\bigcap_{T_{ij}^{\\alpha\\beta}\\in\\mathfs T\\atop{T_{ij}^{\\alpha\\beta}\\supset R}} W^s(x,Z_i)\\cap T_{ij}^{\\alpha\\beta}\n\\, \\text{ and }\\, W^u(x,R):=\n\\bigcap_{T_{ij}^{\\alpha\\beta}\\in\\mathfs T\\atop{T_{ij}^{\\alpha\\beta}\\supset R}}  W^u(x,Z_i)\\cap T_{ij}^{\\alpha\\beta}.\n\\end{align*}\n\nAny two $s$--fibres ($u$--fibres) either coincide or are disjoint.\n\n\\begin{proposition}\\label{Prop-R}\nThe following are true.\n\\begin{enumerate}[{\\rm (1)}]\n\\item {\\sc Product structure:} For every $R\\in\\mathfs R$ and every $x,y\\in R$, the intersection\n$W^s(x,R)\\cap W^u(y,R)$ consists of a single point of $R$. Denote it by $[x,y]$.\n\\item {\\sc Hyperbolicity:} If $z,w\\in W^s(x,R)$ then $d(f^n(z),f^n(w))\\xrightarrow[n\\to\\infty]{}0$, and\nif $z,w\\in W^u(x,R)$ then $d(f^n(z),f^n(w))\\xrightarrow[n\\to-\\infty]{}0$. The rates are exponential.\n\\item {\\sc Geometrical Markov property:} Let $R_0,R_1\\in\\mathfs R$. If $x\\in R_0$ and $f(x)\\in R_1$ then \n$$\nf(W^s(x,R_0))\\subset W^s(f(x),R_1)\\, \\text{ and }\\, f^{-1}(W^u(f(x),R_1))\\subset W^u(x,R_0).\n$$\n\\end{enumerate}\n\\end{proposition}\n\nWhen $M$ is compact and $f$ is a diffeomorphism, this is \\cite[Prop. 11.5 and 11.7]{Sarig-JAMS}\nand the same proof works in our case.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{A finite-to-one Markov extension}\n\nWe construct a new symbolic coding of $f$.\nLet $\\widehat{\\mathfs G}=(\\widehat V,\\widehat E)$ be the oriented graph with vertex set\n$\\widehat V=\\mathfs R$ and edge set $\\widehat E=\\{R\\to S:R,S\\in\\mathfs R\\text{ s.t. }f(R)\\cap S\\neq\\emptyset\\}$,\nand let $(\\widehat\\Sigma,\\widehat\\sigma)$ be the TMS induced by $\\widehat{\\mathfs G}$.\nThe ingoing and outgoing degree of every vertex in $\\widehat\\Sigma$ is finite.\n\n\\medskip\nFor $\\ell\\in\\mathbb{Z}$ and a path $R_m\\to\\cdots\\to R_n$ on $\\widehat{\\mathfs G}$ define\n$_\\ell[R_m,\\ldots,R_n]:=f^{-\\ell}(R_m)\\cap\\cdots\\cap f^{-\\ell-(n-m)}(R_n)$, the set of points whose itinerary\nfrom $\\ell$ to $\\ell+(n-m)$ visits the rectangles $R_m,\\ldots,R_n$. The crucial property that\ngives the new coding is that $_\\ell[R_m,\\ldots,R_n]\\neq\\emptyset$. This follows by induction, using the\nMarkov property of $\\mathfs R$ (Proposition \\ref{Prop-R}(3)).\n\n\\medskip\nThe map $\\pi$ defines similar sets: for $\\ell\\in\\mathbb{Z}$ and a path\n$v_m\\overset{\\varepsilon}{\\to}\\cdots\\overset{\\varepsilon}{\\to}v_n$ on $\\Sigma$ let\n$\nZ_\\ell[v_m,\\ldots,v_n]:=\\{\\pi(\\underline w):\\underline w\\in\\Sigma^\\#\\text{ and }w_\\ell=v_m,\\ldots,w_{\\ell+(n-m)}=v_n\\}$.\nThere is a relation between $\\Sigma$ and $\\widehat\\Sigma$ in terms of these sets:\nif $\\{R_n\\}_{n\\in\\mathbb{Z}}\\in\\widehat\\Sigma$ then there exists $\\{v_n\\}_{n\\in\\mathbb{Z}}\\in\\Sigma$\ns.t. $_{-n}[R_{-n},\\ldots,R_n]\\subset Z_{-n}[v_{-n},\\ldots,v_n]$ for all $n\\geq 0$ (in particular $R_n\\subset Z(v_n)$\nfor all $n\\in\\mathbb{Z}$). This fact is proved as in \\cite[Lemma 12.2]{Sarig-JAMS}.\nBy Proposition \\ref{Prop-R}(2), $\\bigcap_{n\\geq 0}\\overline{_{-n}[R_{-n},\\ldots,R_n]}$\nis the intersection of a descending chain of nonempty closed sets with\ndiameters converging to zero.\n\n\\medskip\n\\noindent\n{\\sc The map $\\widehat\\pi:\\widehat\\Sigma\\to M$:} Given $\\underline R=\\{R_n\\}_{n\\in\\mathbb{Z}}\\in\\widehat\\Sigma$,\n$\\widehat\\pi(\\underline R)$ is defined by the identity\n$$\n\\{\\widehat\\pi(\\underline R)\\}:=\\bigcap_{n\\geq 0}\\overline{_{-n}[R_{-n},\\ldots,R_n]}.\n$$\n\n\\medskip\nThe triple $(\\widehat\\Sigma,\\widehat\\sigma,\\widehat\\pi)$ is the one that satisfies Theorem \\ref{Thm-main}.\n\n\\begin{theorem}\\label{Thm-widehat-pi}\nThe following holds for all $\\varepsilon>0$ small enough.\n\\begin{enumerate}[{\\rm (1)}]\n\\item $\\widehat\\pi:\\widehat\\Sigma\\to M$ is H\\\"older continuous.\n\\item $\\widehat\\pi\\circ\\widehat\\sigma=f\\circ\\widehat\\pi$.\n\\item $\\widehat\\pi[\\widehat\\Sigma^\\#]\\supset {\\rm NUH}_\\chi^\\#$, hence\n$\\pi[\\widehat\\Sigma^\\#]$ carries all $f$--adapted $\\chi$--hyperbolic measures. \n\\item Every point of $\\widehat\\pi[\\widehat\\Sigma^\\#]$ has finitely many pre-images in $\\widehat\\Sigma^\\#$.\n\\end{enumerate}\n\\end{theorem}\n\nWhen $M$ is compact and $f$ is a diffeomorphism, parts (1)--(3) are \\cite[Thm. 12.5]{Sarig-JAMS}\nand part (4) is \\cite[Thm. 5.6(5)]{Lima-Sarig}.\nThe same proofs work in our case, and the bound\non the number of pre-images is exactly the same: there is a function \n$N:\\mathfs R\\to\\mathbb{N}$ s.t. if $x=\\widehat\\pi(\\underline R)$ with $R_n=R$ for infinitely many $n>0$ and $R_n=S$\nfor infinitely many $n<0$ then $\\#\\{\\underline S\\in\\widehat\\Sigma^\\#:\\widehat\\pi(\\underline S)=x\\}\\leq N(R)N(S)$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Appendix A: Underlying assumptions}\n\nRemember the definition of $\\widetilde{A}\\in\\mathfs L_{x,x'}$ for $A\\in\\mathfs L_{y,z}$ and\n$y\\in D_x,z\\in D_{x'}$. Remember also the definition of $\\tau=\\tau_x:D_x\\times D_x\\to \\mathfs L_x$\nby $\\tau(y,z)=\\widetilde{d(\\exp{y}^{-1})_z}$.\nThroughout the text, we assume that there are constants $\\mathfrak K,a>1$ s.t. for all\n$x\\in M\\backslash\\mathfs D$ there is $d(x,\\mathfs D)^a<\\mathfrak r(x)<1$ \ns.t. for $D_x:=B(x,2\\mathfrak r(x))$ it holds:\n\\begin{enumerate}[ii]\n\\item[(A1)] If $y\\in D_x$ then ${\\rm inj}(y)\\geq 2\\mathfrak r(x)$, $\\exp{y}^{-1}:D_x\\to T_yM$\nis a diffeomorphism onto its image, and\n$\\tfrac{1}{2}(d(x,y)+\\|v-P_{y,x}w\\|)\\leq d_{\\rm Sas}(v,w)\\leq 2(d(x,y)+\\|v-P_{y,x} w\\|)$ for all $y\\in D_x$ and\n$v\\in T_xM,w\\in T_yM$ s.t. $\\|v\\|,\\|w\\|\\leq 2\\mathfrak r(x)$, where \t\n$P_{y,x}:=P_\\gamma$ is the radial geodesic $\\gamma$ joining $y$ to $x$.\n\\item[(A2)] If $y_1,y_2\\in D_x$ then\n$d(\\exp{y_1}v_1,\\exp{y_2}v_2)\\leq 2d_{\\rm Sas}(v_1,v_2)$ for $\\|v_1\\|$, $\\|v_2\\|\\leq 2\\mathfrak r(x)$,\nand $d_{\\rm Sas}(\\exp{y_1}^{-1}z_1,\\exp{y_2}^{-1}z_2)\\leq 2[d(y_1,y_2)+d(z_1,z_2)]$\nfor $z_1,z_2\\in D_x$ where the expression makes sense.\nIn particular $\\|d(\\exp{x})_v\\|\\leq 2$ for $\\|v\\|\\leq 2\\mathfrak r(x)$,\nand $\\|d(\\exp{x}^{-1})_y\\|\\leq 2$ for $y\\in D_x$.\n\\item[(A3)] If $y_1,y_2\\in D_x$ then\n$$\n\\|\\widetilde{d(\\exp{y_1})_{v_1}}-\\widetilde{d(\\exp{y_2})_{v_2}}\\|\n\\leq d(x,\\mathfs D)^{-a}d_{\\rm Sas}(v_1,v_2)\\leq \\rho(x)^{-a}d_{\\rm Sas}(v_1,v_2)\n$$\nfor all $\\|v_1\\|,\\|v_2\\|\\leq 2\\mathfrak r(x)$ and \n\\begin{align*}\n\\|\\tau(y_1,z_1)-\\tau(y_2,z_2)\\|&\\leq d(x,\\mathfs D)^{-a}[d(y_1,y_2)+d(z_1,z_2)]\\\\\n&\\leq \\rho(x)^{-a}[d(y_1,y_2)+d(z_1,z_2)]\n\\end{align*}\nfor all $z_1,z_2\\in D_x$.\n\\item[(A4)] If $y_1,y_2\\in D_x$ then the map $\\tau(y_1,\\cdot)-\\tau(y_2,\\cdot):D_x\\to \\mathfs L_x$\nhas Lipschitz constant $\\leq d(x,\\mathfs D)^{-a}d(y_1,y_2)\\leq \\rho(x)^{-a}d(y_1,y_2)$.\n\\item[(A5)] If $y\\in D_x$ then $\\|df_y^{\\pm 1}\\|\\leq d(x,\\mathfs D)^{-a}\\leq \\rho(x)^{-a}$.\n\\item[(A6)] If $y_1,y_2\\in D_x$ and $f(y_1),f(y_2)\\in D_{x'}$ then\n$\\|\\widetilde{df_{y_1}}-\\widetilde{df_{y_2}}\\|\\leq \\mathfrak Kd(y_1,y_2)^\\beta$,\nand if $y_1,y_2\\in D_x$ and $f^{-1}(y_1),f^{-1}(y_2)\\in D_{x''}$ then\n$\\|\\widetilde{df_{y_1}^{-1}}-\\widetilde{df_{y_2}^{-1}}\\|\\leq \\mathfrak Kd(y_1,y_2)^\\beta$.\n\\item[(A7)] $\\|df^{\\pm 1}_x\\|\\geq m(df^{\\pm 1}_x)\\geq \\rho(x)^a$.\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Appendix B: Standard proofs and adaptations of \\cite{Sarig-JAMS}}\\label{Appendix-standard-proofs}\n\nIn this appendix we prove some statements claimed throughout the text, most of them consisting\nof adaptations of proofs in \\cite{Sarig-JAMS}. The main issue is the lack of higher\nregularity of the exponential map. The results of \\cite{Sarig-JAMS} are technical but extremely\nwell-written, so rewriting it to our context would probably increase the technicalities and decrease\nthe clarity. Hence we decided to write this appendix as a tutorial:\nwe follow the proofs of \\cite{Sarig-JAMS} as most as possible, mentioning the necessary changes. \nThe main changes are in the geometrical estimates on $M$:\nsome Lipschitz constants of \\cite{Sarig-JAMS} are substituted by terms of\nthe form $d(x,\\mathfs D)^{-a}$. We then show that our\ndefinition of $Q_\\varepsilon(x)$ is strong enough to cancel out these terms.\nSince the proofs of \\cite{Sarig-JAMS} have freedom in the choice of exponents,\nwe obtain the same final results and therefore (almost always) the same statements of \\cite{Sarig-JAMS}.\n\n\\begin{proof}[Proof of Lemma \\ref{Lemma-admissible-manifolds}.]\nPart (1) is proved exactly as in \\cite[Prop. 4.11(1)--(2)]{Sarig-JAMS}.\nWe concentrate on part (2). Let $\\eta=p^s\\wedge p^u$.\nThe estimate of $\\tfrac{\\sin\\angle(V^s,V^u)}{\\sin\\alpha(x)}$ in \\cite{Sarig-JAMS} is\ndivided into the analysis of four factors. The estimate of the first\ntwo factors is identical; the difference is in the estimates of the remaining two factors.\n\n\\medskip\nBy (A3), if $x\\in M\\backslash\\mathfs D$ and $\\|v\\|\\leq 2\\mathfrak r(x)$ then\n$|{\\rm det}[d(\\exp{x})_v]-1|\\leq 4d(x,\\mathfs D)^{-a}\\|v\\|$,\ni.e. we substitute $K_1$ in \\cite[pp. 407]{Sarig-JAMS} by $4d(x,\\mathfs D)^{-a}$.\nWith this notation,\n$K_1\\eta<4d(x,\\mathfs D)^{-a}Q_\\varepsilon(x)^{\\beta\/72}\\eta^{1-\\beta\/72}<4\\varepsilon^{1\/24}\\eta^{1-\\beta\/72}<\\eta^{2\\beta\/3}$\nfor $\\varepsilon>0$ small, then the third factor is $e^{\\pm 2\\eta^{2\\beta\/3}}$. To estimate the fourth\nfactor, note that again by (A3) if $x\\in M\\backslash\\mathfs D$ and $\\|v\\|\\leq 2\\mathfrak r(x)$\nthen $\\|\\widetilde{d(\\exp{x})_v}-{\\rm Id}\\|\\leq d(x,\\mathfs D)^{-a}\\|v\\|$,\ni.e. we substitute $K_2$ in \\cite[pp. 407]{Sarig-JAMS} by $d(x,\\mathfs D)^{-a}$. Noting as above\nthat $3K_2\\eta<\\eta^{2\\beta\/3}$, we get that the fourth factor is $e^{\\pm\\tfrac{1}{3}\\eta^{\\beta\/4}}$\nas in \\cite[pp. 408]{Sarig-JAMS}.\n\n\\medskip\nThe estimates of $|\\cos\\angle(V^s,V^u)-\\cos\\alpha(x)|$ work as in \\cite{Sarig-JAMS} after using again\nthat $K_2\\eta<\\eta^{2\\beta\/3}$, in which case $K_3=24$. \n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{Prop-graph-transform}.]\nWe follow the proofs of \\cite[Prop. 4.12 and 4.14]{Sarig-JAMS},\nwith the modifications below.\n\\begin{enumerate}[$\\circ$]\n\\item Pages 411--412: in claim 3, it is enough to have $|G'(0)|<\\tfrac{1}{2}(q^s\\wedge q^u)^{\\beta\/3}$.\nProceed as in \\cite{Sarig-JAMS} to get that\n$$\n|G'(0)|< e^{-\\chi+\\varepsilon}\\left[|A||F'(0)|+\\tfrac{2}{3}\\varepsilon^{\\beta\/3}(p^s\\wedge p^u)^{\\beta\/3}\n+6\\varepsilon (p^s\\wedge p^u)^{\\beta\/3}\\right]\n$$\nand then note that for $\\varepsilon>0$ small enough this is at most\n\\begin{align*}\n&e^{-\\chi+\\varepsilon}\\left[\\tfrac{1}{2}e^{-\\chi}+\\tfrac{2}{3}\\varepsilon^{\\beta\/3}+6\\varepsilon\\right](p^s\\wedge p^s)^{\\beta\/3}\\\\\n&\\leq e^{-\\chi+\\varepsilon+\\varepsilon\\beta\/3}\\left[\\tfrac{1}{2}e^{-\\chi}+\\tfrac{2}{3}\\varepsilon^{\\beta\/3}+6\\varepsilon\\right](q^s\\wedge q^s)^{\\beta\/3}\n<\\tfrac{1}{2}(q^s\\wedge q^u)^{\\beta\/3}.\n\\end{align*}\n\\item Page 412: in claim 4, it is enough to have $\\|G'\\|_0+\\Hol{\\beta\/3}(G')<\\tfrac{1}{2}$.\nProceed as in \\cite{Sarig-JAMS} to get that\n$\\|G'\\|_0+\\Hol{\\beta\/3}(G')<e^{-\\chi+3\\varepsilon}\\left[\\tfrac{1}{2}e^{-\\chi}+\\tfrac{3}{2}\\varepsilon\\right]$.\nThis is $<\\tfrac{1}{2}$ when $\\varepsilon>0$ is small.\n\\item Pages 414--415: in the proof of part 2, proceed as in \\cite{Sarig-JAMS} to get that\n$$\n\\|G_1-G_2\\|_0\\leq (|A|+3\\varepsilon^2)(1+\\varepsilon^2+3\\varepsilon^3)\\|F_1-F_2\\|_0\n$$\nand note that $ (|A|+3\\varepsilon^2)(1+\\varepsilon^2+3\\varepsilon^3)<(e^{-\\chi}+3\\varepsilon^2)(1+\\varepsilon^2+3\\varepsilon^3)<e^{-\\chi\/2}$ when $\\varepsilon>0$ is small enough.\n\\end{enumerate}\n\\end{proof}\n\n\n\\begin{proof}[Proof of inequality {\\rm (\\ref{inequality-C})}.]\nWe will use assumption (A3) as stated in section \\ref{Section-introduction}:\n\\begin{enumerate}[ii]\n\\item[(A3)] $\\|df_x\\|< d(x,\\mathfs D)^{-a}$ and\n$\\|df^{-1}_x\\|< d(x,\\mathfs D)^{-a}$ for all $x\\in M\\backslash\\mathfs D$.\n\\end{enumerate}\nWe have:\n\\begin{align*}\n&s(f^{-1}(x))^2=2\\sum_{n\\geq 0}e^{2n\\chi}\\|df^ne^s_{f^{-1}(x)}\\|^2=\n2+2e^{2\\chi}\\|dfe^s_{f^{-1}(x)}\\|^2\\sum_{n\\geq 0}e^{2n\\chi}\\|df^ne^s_x\\|^2\\\\\n&=\n2+e^{2\\chi}\\|dfe^s_{f^{-1}(x)}\\|^2s(x)^2\\leq (1+e^{2\\chi}\\|dfe^s_{f^{-1}(x)}\\|^2)s(x)^2.\n\\end{align*}\nBy (A3),\n$\\tfrac{s(f^{-1}(x))^2}{s(x)^2}\\leq 1+e^{2\\chi} d(f^{-1}(x),\\mathfs D)^{-2a}\\leq 1+e^{2\\chi}\\rho(x)^{-2a}$.\nWe also have that\n\\begin{align*}\n&u(f^{-1}(x))^2=2\\sum_{n\\geq 0}e^{2n\\chi}\\|df^{-n}e^u_{f^{-1}(x)}\\|^2=2\\|df^{-1}e^u_x\\|^{-2}\n\\sum_{n\\geq 0}e^{2n\\chi}\\|df^{-(n+1)}e^u_x\\|^2\\\\\n&=2e^{-2\\chi}\\|df^{-1}e^u_x\\|^{-2}\\sum_{n\\geq 1}e^{2n\\chi}\\|df^{-n}e^u_x\\|^2=\ne^{-2\\chi}\\|df^{-1}e^u_x\\|^{-2}(u(x)^2-2)\\\\\n&<\\|df^{-1}e^u_x\\|^{-2} u(x)^2,\n\\end{align*}\nhence by (A6) we get that\n$\\tfrac{u(f^{-1}(x))^2}{u(x)^2}\\leq \\rho(x)^{-2a}<1+e^{2\\chi}\\rho(x)^{-2a}$.\nFinally, applying (\\ref{gen-ineq-angles}) for $L=df^{-1}_x$, $v=e^s_x$, $w=e^u_x$ and\nusing (A3), we have\n$$\n\\tfrac{\\sin\\alpha(x)}{\\sin\\alpha(f^{-1}(x))}=\\tfrac{\\sin\\angle(e^s_x,e^u_x)}{\\sin\\angle(df^{-1}_x e^s_x,df^{-1}_xe^u_x)}\n\\leq \\|df^{-1}_x\\|\\|df_{f^{-1}(x)}\\|<\\rho(x)^{-2a}.\n$$\nSince $\\|\\cdot \\|\\leq \\|\\cdot\\|_{\\rm Frob}\\leq \\sqrt{2}\\|\\cdot\\|$, the above inequalities and\nLemma \\ref{Lemma-linear-reduction} give that\n\\begin{align*}\n&\\|C_\\chi(f^{-1}(x))^{-1}\\|\\leq \\|C_\\chi(f^{-1}(x))^{-1}\\|_{\\rm Frob}\n\\leq\\rho(x)^{-2a}\\sqrt{1+e^{2\\chi}\\rho(x)^{-2a}}\\|C_\\chi(x)^{-1}\\|_{\\rm Frob}\\\\\n&\\leq 2\\rho(x)^{-2a}(1+e^\\chi\\rho(x)^{-a})\\|C_\\chi(x)^{-1}\\|.\n\\end{align*}\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{Prop-stay-window}.]\nThe proof of part (2) is identical to the proof of \\cite[Prop. 6.4]{Sarig-JAMS},\nand the proof of part (1)(a)--(b) is identical to the proof of \\cite[Prop. 6.3(1)--(2)]{Sarig-JAMS}.\nTo prove (1)(c), we make some modifications in the proof of \\cite[Prop. 6.3(3))]{Sarig-JAMS}.\nWe start with the claim below.\n\n\\medskip\n\\noindent\n{\\sc Claim:} If $y,z\\in D_x$ and $v\\in T_yM,w\\in T_zM$ with $\\|v\\|=\\|w\\|=1$ then\n\\begin{align*}\n&|\\|df_y^{\\pm 1}(v)\\|-\\|df_z^{\\pm 1}(w)\\||\\leq \\mathfrak K\\rho(x)^{-a}[d(y,z)^\\beta+\\|v-P_{z,y}w\\|]\\hspace{.2cm}\\text{and}\\\\\n&\\left|\\frac{\\|df_y^{\\pm 1}(v)\\|}{\\|df_z^{\\pm 1}(w)\\|}-1\\right|\\leq \\mathfrak K\\rho(x)^{-2a}[d(y,z)^\\beta+\\|v-P_{z,y}w\\|].\n\\end{align*}\nIn particular\n$\\left|\\log\\|df_y^{\\pm 1}(v)\\|-\\log\\|df_z^{\\pm 1}(w)\\|\\right|\\leq \\mathfrak K\\rho(x)^{-2a}[d(y,z)^\\beta+\\|v-P_{z,y}w\\|]$.\n\n\\medskip\n\\noindent\n{\\em Proof of the claim.} The inequalities are consequences of (A5)--(A7). Since\nthese assumptions are symmetric on $f$ and $f^{-1}$, we only prove the claim for $f$.\nNote that:\n\\begin{align*}\n&|\\|df_y(v)\\|-\\|df_z(w)\\||\\leq \\|\\widetilde{df_y}(P_{y,x}v)-\\widetilde{df_z}(P_{z,x}w)\\|\\\\\n&\\leq \\|\\widetilde{df_y}-\\widetilde{df_z}\\|+\\|\\widetilde{df_z}\\|\\|v-P_{z,y}w\\|\\leq\n\\mathfrak Kd(y,z)^\\beta+\\rho(x)^{-a}\\|v-P_{z,y}w\\|\\\\\n&\\leq \\mathfrak K\\rho(x)^{-a}[d(y,z)^\\beta+\\|v-P_{z,y}w\\|].\n\\end{align*}\nThe second inequality follows from the first one and from (A7).\n\n\\medskip\nLet us now prove part (1)(c). Write $V^s=V^s[\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\geq 0}]$. By the claim,\n\\begin{align*}\n&|\\log\\|df^n e^s_y\\|-\\log\\|df^n e^s_z\\||\\leq \\sum_{k=0}^{n-1}|\\log\\|df e^s_{f^k(y)}\\|-\\log\\|df e^s_{f^k(z)}\\||\\\\\n&\\leq \\sum_{k=0}^{n-1}\\mathfrak K\\rho(x_k)^{-2a}[d(f^k(y),f^k(z))^\\beta+\n\\|e^s_{f^k(y)}-P_{f^k(z),f^k(y)}e^s_{f^k(z)}\\|].\n\\end{align*}\nBy part (1)(a) and the definition of $Q_\\varepsilon(x_k)$,\n\\begin{align*}\n&\\rho(x_k)^{-2a}d(f^k(y),f^k(z))^\\beta<\\varepsilon^{1\/12}Q_\\varepsilon(x_k)^{-\\beta\/36}6e^{-\\frac{\\beta\\chi}{2} k}(p^s_0)^\\beta\\\\\n&<6\\varepsilon^{1\/12}(p^s_k)^{-\\beta\/36}e^{-\\frac{\\beta\\chi}{2} k}(p^s_0)^\\beta.\n\\end{align*}\nBy (GPO2) we have $p^s_0\\leq e^{\\varepsilon k}p^s_k$, then for small $\\varepsilon>0$ the last expression above is\n\\begin{align*}\n\\leq 6\\varepsilon^{1\/12}(p^s_0)^{-\\beta\/36}e^{-\\frac{\\beta\\chi}{2}k+\\frac{\\beta\\varepsilon}{36}k}(p^s_0)^\\beta\n< 6\\varepsilon^{1\/12}e^{-\\frac{\\beta\\chi}{3}k}(p^s_0)^{\\beta\/4}\n\\end{align*}\nand thus\n$$\n\\sum_{k=0}^{n-1}\\mathfrak K\\rho(x_k)^{-2a}d(f^k(y),f^k(z))^\\beta\\leq\n\\tfrac{6\\mathfrak K\\varepsilon^{1\/12}}{1-e^{-\\frac{\\beta\\chi}{3}}}(p^s_0)^{\\beta\/4}<\\tfrac{1}{2}(p^s_0)^{\\beta\/4}.\n$$\nWe now estimate the second sum. Call $N_k:=\\|e^s_{f^k(y)}-P_{f^k(z),f^k(y)}e^s_{f^k(z)}\\|$.\nWrite $f^k(y)=\\Psi_{x_k}(\\underline y_k)=\\Psi_{x_k}(y_k,F_k(y_k))$ and\n$f^k(z)=\\Psi_{x_k}(\\underline z_k)=\\Psi_{x_k}(z_k,F_k(z_k))$, where $F_k$ is the representing function\nof $V^s[\\{\\Psi_{x_n}^{p^s_n,p^u_n}\\}_{n\\geq k}]$. In part (1), it is proved\nthat $\\|\\underline y_k-\\underline z_k\\|\\leq 3p^s_0e^{-\\frac{\\chi}{2}k}$.\nAs in \\cite[pp. 418--419]{Sarig-JAMS}, \n\\begin{align*}\n&N_k\\leq 2\\|C_\\chi(x_k)^{-1}\\|\\|\\underline y_k-\\underline z_k\\|^{\\beta\/3}\\\\\n&\\hspace{.85cm}+4\\|C_\\chi(x_k)^{-1}\\|\n\\left\\|\\widetilde{d(\\exp{x_k})_{C_\\chi(x_k)\\underline y_k}\\circ C_\\chi(x_k)}-\\widetilde{d(\\exp{x_k})_{C_\\chi(x_k)\\underline z_k}\\circ C_\\chi(x_k)}\\right\\|\n\\end{align*}\nwhich, by (A3), is \n$\\leq 2\\|C_\\chi(x_k)^{-1}\\|\\|\\underline y_k-\\underline z_k\\|^{\\beta\/3}+4\\|C_\\chi(x_k)^{-1}\\|\\rho(x_k)^{-a}\\|\\underline y_k-\\underline z_k\\|$.\nFor $\\varepsilon>0$ small enough\n\\begin{align*}\n&4\\rho(x_k)^{-a}\\|\\underline y_k-\\underline z_k\\|^{\\beta\/72}\\leq 12\\rho(x_k)^{-a}(p^s_0)^{\\beta\/72}e^{-\\frac{\\beta\\chi}{144}k}\\\\\n&\\leq 12\\rho(x_k)^{-a}(p^s_k)^{\\beta\/72}e^{-\\frac{\\beta\\chi}{144}k+\\frac{\\beta\\varepsilon}{72}k}\n\\leq 12\\varepsilon^{1\/24}e^{-\\frac{\\beta\\chi}{144}k+\\frac{\\beta\\varepsilon}{72}k}<1,\n\\end{align*}\nthus $N_k\\leq 3\\|C_\\chi(x_k)^{-1}\\|\\|\\underline y_k-\\underline z_k\\|^{\\beta\/3}$. Hence for small $\\varepsilon>0$\n\\begin{align*}\n&\\rho(x_k)^{-2a}N_k\\leq 3\\|C_\\chi(x_k)^{-1}\\|\\rho(x_k)^{-2a}\\|\\underline y_k-\\underline z_k\\|^{\\beta\/3}\\\\\n&\\leq 9\\|C_\\chi(x_k)^{-1}\\|\\rho(x_k)^{-2a}(p^s_0)^{\\beta\/3}e^{-\\frac{\\beta\\chi}{6}k}\\\\\n&\\leq 9\\|C_\\chi(x_k)^{-1}\\|\\rho(x_k)^{-2a}(p^s_0)^{\\beta\/12}e^{-\\frac{\\beta\\chi}{6}k}(p^s_0)^{\\beta\/4}\\\\\n&\\leq 9\\|C_\\chi(x_k)^{-1}\\|\\rho(x_k)^{-2a}(p^s_k)^{\\beta\/12}e^{-\\frac{\\beta\\chi}{6}k+\\frac{\\beta\\varepsilon}{12}k}(p^s_0)^{\\beta\/4}\\\\\n&\\leq 9\\|C_\\chi(x_k)^{-1}\\|(p^s_k)^{\\beta\/24}\\rho(x_k)^{-2a}(p^s_k)^{\\beta\/36}e^{-\\frac{\\beta\\chi}{6}k+\\frac{\\beta\\varepsilon}{12}k}(p^s_0)^{\\beta\/4}\\\\\n&\\leq 9\\varepsilon^{5\/24}e^{-\\frac{\\beta\\chi}{7}k}(p^s_0)^{\\beta\/4}\n\\end{align*}\nand therefore\n$$\n\\sum_{k=0}^{n-1}\\mathfrak K\\rho(x_k)^{-2a}\\|e^s_{f^k(y)}-P_{f^k(z),f^k(y)}e^s_{f^k(z)}\\|\\leq\n\\tfrac{9\\mathfrak K\\varepsilon^{5\/24}}{1-e^{-\\beta\\chi\/7}}(p^s_0)^{\\beta\/4}<\\tfrac{1}{2}(p^s_0)^{\\beta\/4}.\n$$\nThe conclusion is that $|\\log\\|df^n e^s_y\\|-\\log\\|df^n e^s_z\\||<(p^s_0)^{\\beta\/4}<Q_\\varepsilon(x)^{\\beta\/4}$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma \\ref{Lemma-rotations}]\nIt is enough to prove the case $n=0$. Write $\\Psi_{x_0}^{p^s_0,p^u_0}=\\Psi_{x}^{p^s,p^u}$,\n$\\Psi_{y_0}^{q^s_0,q^u_0}=\\Psi_{y}^{q^s,q^u}$, $p=p^s\\wedge p^u$, $q=q^s\\wedge q^u$.\nWrite $\\widetilde{C_\\chi(x)}=R_xC_x$, $\\widetilde{C_\\chi(y)}=R_yC_y$.\nSince $R_y^{-1}R_x$ is a rotation matrix, it is enough to estimate its angle.\nAs in \\cite[pp. 372]{Sarig-JAMS}, $\\exists\\lambda\\neq 0$ s.t.\n$C_x\\underline a=\\lambda[\\widetilde{d(\\exp{x})_{C_\\chi(x)\\underline\\zeta}}]^{-1}\n[\\widetilde{d(\\exp{y})_{C_\\chi(y)\\underline\\eta}}]C_y\\underline b$ where:\n\\begin{enumerate}[$\\circ$]\n\\item $\\underline\\zeta\\in R[10^{-2}p]$, $\\underline a=\\colvec{1 \\\\ a}$ and $|a|<p^{\\beta\/3}$.\n\\item $\\underline\\eta\\in R[10^{-2}q]$, $\\underline b=\\colvec{1 \\\\ b}$ and $|b|<q^{\\beta\/3}$.\n\\end{enumerate}\nThe proof is based on three claims. Write $\\vec{v}\\propto\\vec{w}$ if $\\vec{v}=t\\vec{w}$ for some $t\\neq 0$.\n\n\\medskip\n\\noindent\n{\\sc Claim 1:} $C_x\\underline a\\propto R_x\\colvec{1\\pm p^{\\beta\/4}\\\\ \\pm p^{\\beta\/4}}$ and\n$C_y\\underline a\\propto R_y\\colvec{1\\pm q^{\\beta\/4}\\\\ \\pm q^{\\beta\/4}}$.\n\n\\medskip\nThe proof is the same as in \\cite[pp. 372]{Sarig-JAMS}.\n\n\\medskip\n\\noindent\n{\\sc Claim 2:} If $x,y\\in D_z$ and $\\|v\\|,\\|w\\|\\leq \\mathfrak r(z)$ then\n$$\n\\|[\\widetilde{d(\\exp{x})_v}]^{-1}[\\widetilde{d(\\exp{y})_w}]-{\\rm Id}\\|\n<2d(z,\\mathfs D)^{-a}d_{\\rm Sas}(v,w).\n$$\n\n\\medskip\nThe proof is a direct consequence of (A2)--(A3). In particular, if we write\n$E:=[\\widetilde{d(\\exp{x})_{C_\\chi(x)\\underline\\zeta}}]^{-1}[\\widetilde{d(\\exp{y})_{C_\\chi(y)\\underline\\eta}}]-{\\rm Id}$\nthen\n\\begin{align*}\n&\\|E\\|<2d(y,\\mathfs D)^{-a}d_{\\rm Sas}(C_\\chi(x)\\underline\\zeta,C_\\chi(y)\\underline\\eta)\n\\leq 4d(y,\\mathfs D)^{-a}[d(x,y)+\\|\\underline\\zeta-\\underline\\eta\\|]\\\\\n&<4d(y,\\mathfs D)^{-a}(p+q)<8d(x,\\mathfs D)^{-a}p+8d(y,\\mathfs D)^{-a}q\\ll p^{\\beta\/3}+q^{\\beta\/3}\n\\end{align*}\nsince $d(x,y)<25^{-1}(p+q)$ and $\\|\\underline\\zeta\\|+\\|\\underline\\eta\\|<10^{-2}(p+q)$.\n\n\\medskip\n\\noindent\n{\\sc Claim 3:} $R_x\\colvec{1\\\\0}+\\underline\\varepsilon_1\\propto R_y\\colvec{1\\\\0}+\\underline\\varepsilon_2$ where\n$\\|\\underline\\varepsilon_1\\|,\\|\\underline\\varepsilon_2\\|<3(p^{\\beta\/4}+q^{\\beta\/4})\\leq 6\\varepsilon^{3\/4}$.\n\n\\medskip\nTo see this, note that since $C_x\\underline a\\propto (E+I)C_y\\underline b$, claim 1 gives that\n$$\nR_x\\colvec{1\\\\0}+\\underbrace{R_x\\colvec{\\pm p^{\\beta\/4}\\\\ \\pm p^{\\beta\/4}}}_{=\\underline\\varepsilon_1}\n\\propto R_x\\colvec{1\\\\0}+\n\\underbrace{R_y\\colvec{\\pm q^{\\beta\/4}\\\\ \\pm q^{\\beta\/4}}+EC_y\\underline b}_{=\\underline\\varepsilon_2}\n$$\nand that $\\|\\underline\\varepsilon_1\\|\\leq 2p^{\\beta\/4}$ and\n$\\|\\underline\\varepsilon_2\\|\\leq 2q^{\\beta\/4}+2(p^{\\beta\/3}+q^{\\beta\/3})<3(p^{\\beta\/4}+q^{\\beta\/4})$.\nThe remainder of the proof is identical to \\cite[pp. 373]{Sarig-JAMS}.\n\\end{proof}\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introduction}\n\n\nStarburst galaxies (SBs) host the most powerful star-formation events in the Universe, with star-formation rates (SFR) that, following to the definition of \\citet{rodighiero11}, are at least $4$ times higher compared to the average galaxy population at a given redshift, identified by the star-forming Main Sequence (MS) \\citep{brinchmann04,noeske07,elbaz07,daddi07a}.\nIn the Local Universe these systems are called Ultra Luminous Infrared galaxies (ULIRGs) and have a global infrared luminosity higher than $10^{12}$ L$_\\odot$, due to their high stellar production. \nULIRGs are characterized by peculiar and rather homogeneous properties: they show very compact starbursting cores of dense gas, with typical diameter sizes of $0.5$-$2$ kpc, and may contain double nuclei \\citep{genzel98,diazsantos10}. As a consequence of the enhanced SFR density, these cores are also highly dust-rich and completely obscured in the optical\/UV, with A$_\\text{V}$ ranging $5$-$50$ mag in a foreground dust screen model \\citep{genzel98,goldader02}. \n\nSBs represent a small fraction of the whole star-forming (SF) galaxy population ($2$-$3\\%$), almost constant with redshift, according to \\citet{sargent12,sargent14} and \\citet{schreiber15}, but they constitute a key event in the life of a galaxy. Indeed, at $z\\sim0$ the only viable option to explain these concentrations of gas, dust and SFR is through highly dissipative major merger events \\citep{sanders96}, in which the gas of colliding galaxies loses angular momentum and energy during the interaction, falling into the coalescing center of the system. Here it serves as fuel for the starburst and for the growth of a supermassive black hole in the center. This scenario is sustained by studies of both local ULIRGs and higher redshift merging\/starburst galaxies, showing the presence of an AGN with optical \\citep{ellison11}, mid-IR \\citep{daddi07b,weston17,satyapal17,goulding18}, X-ray \\citep{brusa10,aird18} and radio \\citep{best12,chiaberge15} diagnostics.\nThe starburst activity and subsequent AGN feedback can cause gas consumption and removal through powerful winds \\citep{sanders88,silk98}, leading to a passively evolving elliptical galaxy \\citep{kormendy92,springel05b,hopkins08a}.\n\nHydrodynamical simulations indicate that the morphology of merger remnants is consistent with early type galaxies (ETGs), suggesting the scenario that gas-rich major mergers are primarily responsible for the formation of the massive ellipticals \\citep{barnes91,mihos94,mihos96}.\nThis evolutionary sequence is strengthened by observations of low surface brightness tidal tales and residual interacting features in local ETGs \\citep{duc13}, which may come from major mergers. Other studies have shown that ULIRGs lie on or close to the fundamental plane (effective radius-velocity dispersion-surface brightness) of intermediate-mass ellipticals, S0 galaxies and bulges of local spirals, suggesting that these systems are the final step of gas-rich merger episodes \\citep{genzel01}.\n\nThe morphological transformation during the merger happens in different phases that are well represented by the so-called Toomre sequence \\citep{toomre72}. Following the result of their simulations, several studies have tried to characterize this sequence by looking at different physical properties of the nuclear regions of merger galaxies, but found no correlations with evolutionary stages \\citep[e.g.,][]{laine03}, apart from noticing that latest interaction times, during or after the coalescence, have among the highest infrared luminosities (L$_\\text{IR}$) \\citep{laine03,haan13}.\nThe absence of clear correlations in these works can be interpreted either with the difficulty of identifying correctly the merger phase from the optical morphology (which becomes even more problematic beyond the Local Universe) and\/or that other parameters are driving and tracing this transformation. \nFor example, \\citet{gao99} and \\citet{leech10} studied a sample of local (U)LIRGs with double merging nuclei. They found lower molecular gas masses and higher star-formation efficiency and gas excitation (probed by the CO(3-2)\/CO(1-0) line ratio) with decreasing separation between merger components, i.e., toward more advanced merger stages. However, since their sample only includes interacting pairs more distant than $\\sim1$ kpc, this result is limited to relatively early stage mergers and does not necessarily apply also to coalesced systems. Furthermore, some spatially resolved studies on local (U)LIRGs have found that there is a higher contribution of shocks accompanied by an increased velocity dispersion of the gas toward later merger stages  \\citep{monrealibero06,monrealibero10,rich14,rich15}.\n\nWhile in the Local Universe the relation between starbursts and mergers is well settled observationally \\citep[e.g.][]{murphy96,luo14}, at higher redshift the interpretation becomes less clear. Due to the enhanced gas fractions and disk instabilities of high redshift star-forming galaxies, mergers as a result might increase SFRs less dramatically \\citep{fensch17}, and starburst galaxies may be triggered by anomalous gas accretion events \\citep{scoville16}. However, other studies have shown that the most extreme starbursts are still merger-driven, displaying disturbed morphologies \\citep{elbaz03} and increased star-formation efficiencies compared to MS galaxies \\citep{sargent14,silverman18a,silverman18b,silv15}. \nSimilarly, the connection between mergers and AGNs is still debated. Even though it may still hold for the most luminous cases \\citep{combes03}, some studies (focused especially on optical wavelengths) do not find systematic differences in merger fraction and galaxy distortions between active and non-active systems \\citep{cisternas11,kocevski12}, and not all AGNs are triggered by mergers, according to, e.g., \\citet{draper12} and \\citet{ellison15}. \n\nThe solution to these long-standing problems is further complicated due to the intrinsic faintness of interacting features (e.g., tidal tales, bridges) and their elevated obscurations, which hamper their identification and physical characterization. \\citet{puglisi17} have selected a sample of starbursts ($\\times 4$ above the MS) at z $\\sim1.6$, showing that optical lines, including H$\\alpha$ and H$\\beta$, only probe a small nearly-unattenuated component of the galaxies, accounting for $\\sim 10\\%$ of the total SFR$_\\text{IR}$. Therefore, a possible solution to study the properties of the dusty starburst population at high redshift is to observe at longer wavelengths, targeting near-infrared rest-frame lines, such as the Paschen line series of hydrogen. Current ground-based spectrographs can observe Pa$\\beta$ (the second brightest Paschen recombination line) in K band up to a redshift of $\\sim 0.9$. Motivated by this idea, we followed-up of a sample of $25$ SBs at $0.5<z<0.9$ with FIRE (Folded Port InfraRed Echellette), a near-IR spectrograph mounted at the Magellan telescope, in order to detect optical\/near-IR lines ranging $0.6<\\lambda<1.3$ \\AA\\ rest-frame. This sample was presented in \\citet{calabro18}, which hereafter is referred as Paper I.  \n\nBy comparing H$\\alpha$, Pa$\\beta$ fluxes and bolometric IR luminosities, we found that the attenuation properties of intermediate redshift SBs are not consistent with the predictions of dust-foreground extinction models, but rather with those of a geometrical model in which dust and stars are homogeneously mixed (Paper I). We also found that they are highly obscured on average, with median A$_{\\text{V,tot}}$= $9$ mag in a mixed model, while independently derived dust-column densities suggest A$_{\\text{V,tot}}$ even higher for a large fraction of them, up to $75$ magnitudes. This means that they have extremely obscured cores and that optical-near-IR lines only probe an external skin containing a fraction ($\\sim10$-$30 \\%$) of the total SFR. Even more, we argued that the presence of optically thick cores are themselves striking evidence of the merger origin of intermediate-z SBs (like in local ULIRGs), as no other mechanisms are known to produce such large obscuring column densities of gas and dust. This can be used to identify other mergers in the high-redshift Universe \\citep{calabro18}, where other methods based on morphology become unfeasible. \n\nDespite their higher average obscurations compared to MS galaxies, we found that our sample, while showing a nearly constant Pa$\\beta$\/H$\\alpha$ ratios, spans a large range of A$_\\text{V,tot}$ between $2$ and $30$ (see Figure~4 in Paper I), thus forming a sequence of increasing attenuations.\nThis indicates that a substantial intrinsic diversity should exist among them, possibly related to different phases of the merger, to the gas properties (i.e., total gas mass, gas fraction) or to the morphology of pre-existing colliding galaxies. In alternative, the sequence may be driven also by the orbital geometry of the merger. For example, \\citet{dimatteo08} have shown that the inclination of the two encounters affects both the duration and intensity of the star formation, regulating the amount of gas that is funneled toward the coalescing center. \n\nIn this paper we investigate the origin of the obscuration sequence that was found in Paper I, analyzing the attenuations A$_\\text{V,tot}$ of our starbursts in relation with other physical properties derived from our Magellan spectra and from available multiwavelength images. In Section~2 we describe how the starbursts are selected from the parent photometric catalog along with our Magellan-FIRE observations, spectra reduction (which includes also the analysis of public optical spectra available for a fraction galaxies) and subsequent emission line fluxes (and equivalent width) measurements. This will be followed by a description of the morphological properties, radio size measurements and AGN identification within our sample. In Section \\ref{results} we will show the main results of this work, which are discussed extensively in Section~4. A summary with conclusions will be presented in the last Section~5. We adopt the \\citet{chabrier03} Initial Mass Function, AB magnitudes, and standard cosmology ($H_{0}=70$ $\\rm km s^{-1}Mpc^{-1}$, $\\Omega_{\\rm m} = 0.3$, $\\Omega_\\Lambda = 0.7$). We also assume by convention a positive equivalent width (EW) for emission lines and a negative EW for lines in absorption.\n\n\\section{Methodology}\\label{methodology}\n\nIn this section we outline the general starburst selection procedure, and then describe the observations, the spectral reduction and the emission line measurements that were performed for a representative subset of $25$ of them. We will then present their morphological classification, radio size measurements and AGN identification procedure.\n\nAs described in Paper I, we have already calculated for this subset the absolute dust attenuation in V band towards the center ($=$ A$_{\\text{V,tot}}$) in two steps. First, we considered the ratio (SFR$_\\text{IR}$\/SFR$_{Pa\\beta,\\text{obs}}$), where SFR$_\\text{IR}$ is the intrinsic SFR from the infrared and $\\text{SFR}_{Pa\\beta ,\\text{obs}}$ is derived from the observed Pa$\\beta$ luminosity, adopting an intrinsic ratio Pa$\\beta$\/H$\\alpha=0.057$ and a standard \\citet{kennicutt94} calibration, valid for case B recombination and T$_e=10^4K$. A$_{Pa\\beta,\\text{IRX}}$ can be eventually inferred from that quantity as $=2.5 \\times log_{10}(1+\\text{SFR}_{\\text{IR}}\/\\text{SFR}_{Pa\\beta,\\text{obs}}$).  \nIn the final step, the attenuation in V band is obtained by solving for A$_{\\text{V,tot}}$ the following equation valid for a mixed geometry model of dust and ionizing stars:\n\\begin{equation}\\label{eqAV}\n\\frac{SFR_{Pa\\beta,\\text{obs}}}{SFR_{IR}} = \\frac{log_{10}(e)}{0.4} \\times \\left(\\frac{1-10^{-0.8 A_{\\text{tot}}(Pa\\beta)}}{2 A_{\\text{tot}}(Pa\\beta)}\\right)\n\\end{equation}\nwhere $A_{\\text{tot}}(Pa\\beta)$ is the total absolute attenuation at Pa$\\beta$ towards the center defined as k(Pa$\\beta$) $\\times$ A$_\\text{V,tot}$\/R$_\\text{V}$. In the last expression, k(Pa$\\beta$) corresponds to the local extinction at Pa$\\beta$, assuming a \\citet{cardelli89} law (R$_\\text{V}=3.1$). More details on the derivation and the implications of Eq.\\ref{eqAV} can be found in Paper I. \n\n\\subsection{Sample selection and Magellan FIRE observations}\\label{sample_selection}\n\nThe starburst candidates were selected from the IR+(sub)mm catalog of \\citet{jin18} by first requiring the spectroscopic redshift (coming from optical surveys, Salvato et al. in preparation) to be in the range between $0.5$ and $0.9$. This guarantees that Pa$\\beta$ and H$\\alpha$ fall, respectively, in the K and YJ bands, thus within the wavelength coverage of FIRE ($0.82$ $\\mu$m - $2.4$ $\\mu$m). For two galaxies, we found that their previous spectroscopic redshift measurements were not correct. Even though they are outside of our selection range, we did not remove them since they satisfy the other selection criteria. \n\nSecondly, we required SFR$_{\\text{norm}}$ to be more than a factor of $4$ higher than SFR$_{\\text{MS,z=0.73}}$, as in \\citet{rodighiero11}\\footnote{Two starbursts that we have previously selected and observed turned out to have slightly lower SFRs than the $\\times4$ limit, because of subsequent updates of the photometric catalog.}, where SFR$_{\\text{norm}}$ is the total SFR (SFR$_\\text{tot}$=SFR$_\\text{IR}$+SFR$_{\\text{UV,unobscured}}$), normalized to the median redshift of the sample ($0.73$) using the evolution of \\citet{sargent14}: \n\\begin{equation}\nSFR_{\\text{norm}}=(SFR_\\text{IR}+SFR_{\\text{UV,unobs}}) \\times \\left(\\frac{1+0.73}{1+z}\\right)^{2.8},\n\\end{equation}\\label{SFRmed}\nwhile SFR$_{\\text{MS,z=0.73}}$ is the SFR of the Main Sequence at redshift $0.73$ from \\citet{schreiber15}. This latter was found in Paper I to agree well with the MS derived independently for our parent sample through a running median on $10$ stellar mass bins from $10^{10}$ to $10^{12}$ M$_\\odot$. We chose this selection procedure in order to compare all the galaxies with a single MS (cf. Fig.~1 of \\citet{calabro18}), instead of considering different relations for each galaxy redshift. However, the two procedures yield the same number of starbursts among the whole sample. For the targets studied in this paper, the mean distance from the MS ($\\text{dist}_\\text{MS}=$ log$_{10}$(SFR$_\\text{tot}$\/SFR$_\\text{MS}$)) would change by $1.8\\%$, thus it would not affect in any case the results.\n\nThe SFR$_{\\text{IR}}$  was derived following the methodology of \\citet{jin18} and \\citet{liu18}, which is based on fitting the photometry from IRAC to radio $1.4$ GHz with four components: a stellar \\citet{bruzual03} SED (with age $200$ Myr, constant star-formation history, solar metallicity Z$_\\odot$, Chabrier IMF and Calzetti attenuation law), a warm+cold dust emission template from \\citet{draine07} and \\citet{magdis12}, and a mid-infrared AGN SED from \\citet{mullaney11} to separate the IR contributions of the AGN and SF.\nThe unobscured UV-based SFRs (SFR$_{\\text{UV,unobscured}}$) were calculated instead from the u-band magnitudes \\citep{laigle16}, which probes the UV rest-frame at our redshift, following \\citet{heinis14}. Overall, the contributions of the AGN and of the UV-unobscured SFR to the total SFR are on average small (3$\\%$ and 1$\\%$, respectively).\n\nFinally, as a last requirement, we asked for M$_\\ast$ to be greater than $10^{10}$ M$_\\odot$, where M$_\\ast$ comes from \\citet{laigle16} and is computed at the photometric redshift. This condition ensures that, within the mass and redshift constraints adopted here, all SBs would be IR-detected with a S\/N$_{\\text{IR}}>5$ (cf. Fig.~13 in \\citet{jin18}), thus we have a SFR complete sample of SBs. \nThen, for the whole sample, we considered the spectroscopic redshifts (when available) instead of the photometric values, however this does not significantly affect the stellar masses: the two estimates for the parent sample are in agreement within a 1$\\sigma$ scatter of $0.11$ dex, compatible with the uncertainties reported by \\citet{laigle16}. \nOnly for one SB analyzed in this work (ID $685067$, z$_\\text{spec}=0.37$), the new stellar mass was remarkably lower ($-0.56$ dex), due to the large difference with previous photometric redshift (z$_\\text{phot}=0.71$). Therefore, $\\text{dist}_\\text{MS}$ was even higher, strengthening its starburst selection.\n\nThis criteria yielded 152 SBs, 25 of which were observed during 4 nights at the Magellan 6.5 $m$ Baade Telescope ($17$-$18$\/$03$\/$2017$ and $22$-$23$\/$03$\/$2018$). The observed $25$ galaxies were chosen from the pre-selected SB sample according to a priority list. \nWe preferentially observed sources close to bright stars (J < $19$-$20$ mag), so as to facilitate target acquisition, although we eventually avoided blind offsets, since our galaxies are already sufficiently bright (peak magnitudes $<19$ mag) to be detected in $\\sim 20$-$60$ s in the good seeing conditions of those observations. In addition, we targeted intrinsically brighter sources first, maximizing SFR\/D$_L^2$(z) ratio (D$_L$ is the luminosity distance), and assuming no prior knowledge about the dust attenuation of the system, which was set to 0 in all cases. This introduces a small bias in our selection toward the more massive objects. However, our galaxies span the full range of stellar masses above $10^{10} M_\\odot$. We refer to Fig.~1 in Paper I, where we presented the redshift, the SFR and the stellar mass distribution of our observed starbursts and our parent galaxy sample.\n\nOur targets were observed with the single-slit echelle spectrograph FIRE \\citep{simcoe13}, which has a wavelength range of $0.82$-$2.4 \\mu$m. We refer to \\citet{simcoe13} for the full technical description of the instrument.\nWe chose a slit width of $1''$, (yielding a spectral resolution of R$=3000$) to minimize slit losses (the average intrinsic FWHM angular size in Ks-band for our sample is $\\sim 0.6''$) and reduce the impact of OH sky emission. In all the cases, the slits were oriented along the semi-major axis of each galaxy, as determined from HST i-band images. Additionally, we benefited from good seeing conditions over all the four nights, with an average of $0.7''$ and a minimum of $0.45''$. \nThe majority of our starbursts were observed in AB sequence, with single exposure times of $15$-$20$ minutes \\footnote{We chose single frame integrations of $20$ minutes during the first run and $15$ minutes in the second run, which significantly reduces saturation of OH lines in K band, thus helping spectral reduction in that band.}. We decided to double the integration times (completing the ABBA sequence)\\footnote{In practice, doing an AB sequence is irrelevant for the spectral reduction, as the pipeline reduces each frame separately (See Section \\ref{spectroscopic_reduction}), though it allows us to easily derive 2D emission line maps with the standard IRAF tasks \\textsc{imarith} and \\textsc{imcombine}.} for galaxies with a lower S\/N of the H$\\alpha$+[\\ion{N}{II}] complex (based on real-time reduction), to improve the detection of fainter lines. \n\n\\subsection{Spectroscopic reduction}\\label{spectroscopic_reduction}\n\nThe spectra were fully reduced using the publicly available IDL-based FIREHOSE pipeline \\citep{gagne15}. For each exposure, we used internal quartz lamps (one for each observing session) to trace the $21$ orders of the echelle spectra and to apply the flat field correction. Then, the wavelength calibration was performed by fitting a low (1-5) order polynomial (depending on the spectral order) to ThAr lines of lamp exposures (taken close in time to the corresponding science frames). We checked that the residuals of the fitted lines to the best-fit wavelength solution are less than $1$ pixel in all the cases, and is $< 0.1$ pixel for the majority of the orders. This translates into an average wavelength accuracy of $\\Delta \\lambda \/ \\lambda$ $\\simeq 5 \\times 10^{-5}$, nearly constant across the entire spectral range. Finally, the sky subtraction was applied independently for each single frame following the method of \\citet{kelson03}. In this step, the OH lines in the spectra are used to refine the wavelength calibration.\n\nThe 1-D spectra are extracted from the 2-D frames using an optimal extraction method \\citep{horne86}. However, this procedure cannot be applied when there is a rapidly varying spatial profile of the object flux \\citep{horne86}, as in the presence of spatially extended and tilted emission lines. We used in these cases a boxcar extraction procedure, with a sufficiently large extraction aperture (always $> 1.3''$) in order to include all the line emission from the 2-D exposure. We adopted the boxcar extraction for $3$ galaxies in our sample, which are the IDs $245158$, $493881$ and $470239$. Given the agreement within the uncertainties between the fluxes measured with the two approaches for the remaining galaxies, the use of the boxcar procedure does not appear to introduce a systematic flux bias. \n\nAfter the spectral extraction, we applied the flux calibration to each 1-D extracted spectrum, using telluric spectra derived from the observations of A0V stars. Before dividing the object and telluric spectra in the pipeline, we could interactively refine the wavelength matching between the two by minimizing the $rms$ of the product. However, at infrared wavelengths slightly different times and\/or airmasses between science and standard star observations can produce non-negligible telluric line residuals, affecting the subsequent analysis. We found that this problem was more relevant in K-band, where strong telluric features are present in the observed wavelength ranges $19950$-$20250$ \\AA\\ and $20450$-$20800$ \\AA. The residuals in these regions can produce artificial variations of the real flux up to a factor of $2$, while it is less significant at shorter wavelengths (Y to H). In order to remove these artifacts, we followed the procedure described in \\citet{mannucci01}: we first considered a standard star at an airmass of $\\sim1.5$ and calibrated it with two different stars observed at significantly lower and higher airmasses (e.g., $1.2$ and $1.9$). Then the two obtained spectra are divided, yielding a global correction function (which is different from $1$ only in the regions of strong telluric features defined above) that applies to all the single-exposure spectra, each of them with a different multiplicative factor until the telluric line residuals disappear\n\\footnote{We fitted a linear function in nearby regions free of telluric regions and emission lines, and then determined the correction function through minimizing the rms of the difference between the corrected spectrum and the afore-mentioned continuum fit}.\nFinally, we combined (with a weighted-average) all the 1-D calibrated spectra of the same object. \n\nThe error on the flux density f$_\\lambda$ obtained from the FIRE pipeline was checked over all the spectral range, analyzing the continuum of each galaxy in spectral windows of $200$ \\AA\\ and steps of $100$ \\AA, masking emission lines. In each window, we rescaled the rms noise so as to have the $\\chi_{\\text{reduced}}^2 =$ 1 when fitting the continuum with a low-order ($\\lesssim 1$) polynomial\\footnote{A spline of order $1$ spanning the whole wavelength range was used as a correction function}. This criterion, equivalently, ensures that the noise level matches the $1$-$\\sigma$ dispersion of the object spectrum in each window. Typical corrections are within a factor of $2$, variable across the spectral bands. \n\nDue to slit losses, variable seeing conditions and the spatial extension of our objects, which are typically larger than the slit width ($1''$), part of the total flux of the galaxies is lost. In order to recover the total absolute flux, we matched the whole spectrum to the photometric SED. This was done by applying a $5\\sigma$ clipping and error-weighted average to the Magellan spectrum inside z++, Y, J, H and K$_s$ photometric bands, and comparing the obtained mean $f_\\lambda$ in each filter to the corresponding broad-band photometry \\citep{laigle16}. Since the SED shapes derived from the spectra are generally in agreement with the photometric SED shapes, we rescaled our spectra with a constant factor, determined through a least squares minimization procedure. The aperture correction factors for our sample range between $0.8$ and $3$, with a median of $1.4$. They are subject to multiple contributions, i.e., slit position with respect to the object, seeing conditions during the target and the standard stars observations. The few cases in which the aperture correction was lower than $1$ could be due indeed to a much better seeing of the standard star compared to the target observation.\nWe remind that this procedure assumes that lines and continuum are equally extended, which is clearly an approximation. Spatially resolved near-IR line maps (e.g., with SINFONI) would be required to test possible different gradients of the two emission components, and to derive better total flux corrections.\n\n\\subsection{Complementary optical spectra}\\label{complementary_optical_spectra}\n\n\\begin{figure*}[t!]\n  \\centering\n  \\raggedright{\\textbf{HST F814W} \\quad \\qquad \\textbf{UltraVISTA H} \\qquad \\textbf{VLA 3 GHz}}\n  \\includegraphics[angle=0,width=17.5cm,trim={0cm 0.cm 0cm 127cm},clip]{test_mosaic_all_compressed.pdf}\n  \\caption{\\small For each of the galaxies observed in our second observing run at Magellan we show (from left to right): HST-ACS F814W images (FWHM$_{\\text{res}}=0.095''$), H-band UltraVISTA cutouts (same f.o.v. and FWHM$_{res} \\sim 0.75''$) and 3 GHz radio images from VLA-COSMOS 3GHz Large Project \\citep{smolcic17} (FWHM$\\sim 0.75''$).\n  Cutout images for the sample of the first observing run were presented in Paper I.}\\label{stamps}\n\\end{figure*}\n\nA subset of our Magellan sample also has publicly available optical spectra: $10$ starbursts in our sample have been observed with the VIMOS spectrograph \\citep{lefevre03} by the zCOSMOS survey \\citep{lilly07}, and their optical spectra are publicly available through the LAM website (\\url{cesam.lam.fr\/zCOSMOS}). They span the range $5550<\\lambda<9650$ \\AA, which includes [\\ion{O}{II}]$\\lambda 3727$ \\AA, H$\\gamma$, H$\\beta$ and [\\ion{O}{III}]$\\lambda 5007$ \\AA\\ lines for our galaxies. The spectral resolution is on average $R=600$, constant across the whole range, while the noise level increases from the blue to the red part of the spectrum, due to the increased OH sky emission at longer wavelengths. Due to the absence of the noise spectrum in the public zCOSMOS release, we used instead a sky spectrum at the same resolution of VIMOS, rescaled with a spline to match the flux standard deviation in spectral regions free of emission lines, similarly to what has been done before on the Magellan spectra.\nEven though it is a first approximation, this procedure allows to reproduce the increasing noise in correspondence of OH lines, i.e., where strong sky-subtraction residuals are expected, and take into account the higher average noise level of the red part of the spectrum. \nFor one galaxy in our sample (ID $493881$), we took its optical spectrum from SDSS-III DR9 \\citep{ahn12}, which spans a wider wavelength range $3600<\\lambda<10400$ \\AA\\ at higher resolution (R$\\sim 2000$), thus allowing a better sky subtraction and a higher SNR. \nIn both cases, the spectra were already fully reduced, wavelength and flux calibrated, as described in the respective papers. We apply only an aperture correction by matching the observed spectrum to the photometric SED \\citep{laigle16} with the same methodology adopted for the Magellan spectra. However, we warn that there could be some mis-matches compared to our Magellan observations in the slit \ncentering and orientation, as also in the seeing conditions, thus the spectra may not be exactly representative of the same regions.\n\n\\subsection{Line measurements}\\label{line_measurements}\n\nWe measured emission line fluxes, line widths and uncertainties on fully calibrated and aperture corrected spectra using the python anaconda distribution (Mark Rivers, 2002\\footnote{GitHub Repository: \\href{stsci.tools\/lib\/stsci\/tools\/nmpfit.py}{stsci.tools\/lib\/stsci\/tools\/nmpfit.py}}) of the IDL routine MPFIT (Markwardt 2009).\nGiven the FWHM resolution of FIRE for $1''$ slit width ($=100$ km\/s), all our emission lines are resolved, owing to intrinsically higher velocity widths. \n\nWe fit the lines with a single Gaussian on top of an underlying order-1 polynomial continuum. In each case, we require a statistical significance of the fit of $95\\%$, as determined from the residual $\\chi^2$. When a single Gaussian does not satisfy the above condition, we use a double Gaussian (with varying flux ratio and same FWHM, in km\/s, for the two components), which instead provides a better fit, placing its $\\chi^2$ within the asked confidence level. The flux uncertainties were derived by MPFIT itself, and they were always well behaved, with best-fit $\\chi_\\text{reduced}^2 \\simeq 1$. For non detected lines (i.e., SNR $<2$ in our case), we adopt a $2\\sigma$ upper limit \\footnote{We remark that we are guided by the wavelength position, line width and flux ratio (for double line components) of H$\\alpha$, which is always detected at $>4\\sigma$. The Gaussian amplitude remains thus the only variable to constrain the fit for the other lines.}. However, we highlight that our detected emission lines have always high S\/N ratios: H$\\alpha$, [\\ion{N}{II}]$\\lambda$6584$\\AA$\\ and Pa$\\beta$ are identified on average at 9.3, 8.4 and 7.4 $\\sigma$, respectively (lowest SNRs are 4, 5.3 and 3.3 for the same lines). \n\nWe fitted a double Gaussian for $12$ galaxies in our sample.\nAs we will see later in Section \\ref{pre-coalescence} by combining the informations of their 1D and 2D spectra, in $6$ of them we interpret the two Gaussians as coming from different merger components. For the remaining galaxies, in $2$ cases the lines are consistent with global rotation, while for the last $4$ we were not able to derive firm conclusions, even though we favour the contribution of multiple system parts to their emission. In Appendix \\ref{emissionlinefits}, we show the 1D emission line fits for all our $25$ starbursts, and we discuss in more detail the origin of double Gaussians line components.\n\nWe applied the stellar absorption correction on Balmer and Paschen emission lines, rescaling upwards their fluxes. In order to determine the level of absorption for these lines, we adopted \\citet{bruzual03} synthetic spectra with solar metallicity and constant star-formation history for $200$-$300$ Myr, which are the typical merger-triggered starburst timescales \\citep{dimatteo08}.\nThe current starburst activity imposed by our selection suggests that the final coalescence of the major merger occurred relatively recently, certainly within the last 200 Myr. \nNumerical simulations of major mergers with different masses and dynamical times indicate indeed that star formation stops within 100-200 Myr after the coalescence, even without AGN quenching \\citep{springel05a,bournaud11,powell13,moreno15}.\nAveraging the results over this interval, we applied an EW$_{abs}$ of $5$, $2.5$, $2.5$ and $2$ \\AA\\ for H$\\beta$, H$\\alpha$, Pa$\\gamma$ and Pa$\\beta$, respectively\\footnote{The EW$_{abs}$ of H$\\beta$ and H$\\alpha$ are consistent with those adopted in previous works \\citep[e.g.][]{valentino15}, while it was not possible to make comparisons for Pa$\\gamma$ and Pa$\\beta$.}. In the same order, we estimated for these lines an average absorption correction of $35\\%$, $7\\%$, $26\\%$, $13\\%$ of the total flux. If we allow an uncertainty of $\\pm 1$ \\AA\\ on the EW$^{abs}$ correction of either Pa$\\gamma$ and Pa$\\beta$, this will produce variations on the final fluxes that are $6\\%$ on median average, and thus it will not significantly affect our results. \n\nAll the lines in the Magellan spectra, either in emission or in absorption, were analyzed based on the following steps. Firstly, H$\\alpha$ and [\\ion{N}{II}]$\\lambda\\lambda$ $6548$,$6583$, which are the lines with the highest S\/N, were fitted together assuming a common linear continuum and a fixed ratio of the [\\ion{N}{II}] doublet of $3.05$ \\citep{storey88}. From this fit we derived the redshift of the galaxy, the intrinsic FWHM of H$\\alpha$ (in terms of velocity), and the flux ratio of the two H$\\alpha$ components for double gaussian fits. The instrinsic total line widths were obtained by subtracting in quadrature the FIRE resolution width ($100$ km\/s) from the best-fit observed FWHM. For double Gaussians, the total FWHM was calculated adding the single FWHM and the separation between the two component peaks, as this quantity is more representative of the whole system.\n\nThen, the three parameters defined above were fixed and used to fit all other emission lines, including those in the optical zCOSMOS and SDSS spectra, after rescaling the FWHM to account for the different spectral resolutions. For the galaxies with the highest S\/N of Pa$\\beta$ ($>8\\sigma$), we verified that even without fixing its FWHM a-priori, the fit yields a line velocity width consistent within the errors with the value found from H$\\alpha$, indicating that our assumption is generally valid.\nGiven the \\textit{rms} wavelength calibration accuracy (see Section \\ref{spectroscopic_reduction}), we allowed the line central wavelength to vary in the fit by $3\\sigma$, corresponding to $1.5$ \\AA\\ at $10000$ \\AA, and $3$ \\AA\\ at $20000$ \\AA.\nFor each measured flux, we also added in quadrature an error due to the uncertainty of the absolute flux calibration. This additional uncertainty ranges between $5\\%$ and $10\\%$, and is determined as the maximum residual difference between the average fluxes estimated from the photometry and from the aperture corrected spectra, among all the bands ranging from z++ to K$_S$. \nFinally, the equivalent widths of the lines were derived following its definition ($=\\int (F(\\lambda)-F_{cont}(\\lambda))$\/$F_{cont}(\\lambda)$), where $F(\\lambda)$ is the best-fit gaussian flux distribution and $F_{cont}$ is the fitted underlying continuum.  \nSince the fluxes of H$\\alpha$ and Pa$\\beta$ were presented in Paper I, here we show in Table \\ref{table2} the FWHM of the lines, the EWs of H$\\alpha$, H$\\delta$ and Pa$\\beta$, the fluxes of [\\ion{O}{III}]$5007$ and H$\\beta$ that have been used in the BPT diagram.\n\n\\subsection{Ancillary data}\\label{ancillary_data}\n\nAlmost all of our starbursts (24) were observed by HST-ACS in the F814W filter \\citep{koekemoer07} at an angular resolution of $0.095''$ ($\\sim0.7$ kpc at z$=0.7$). The UltraVISTA survey \\citep{mccracken12} observed our galaxies in YJHK bands at a spatial FWHM resolution of $\\sim0.75''$, comparable to the average seeing during our Magellan observations. Two galaxies in the subset have higher resolution ($0.19''$) F160W HST images from the DASH program \\citep{momcheva16}.\nFinally, all our galaxies are well detected in radio $3$ GHz VLA images \\citep{smolcic17} (owing a similar a spatial resolution of $\\simeq 0.75''$), with an average S\/N of $18$. \nIn \\citet{calabro18} we showed the HST F814W, H-band UltraVISTA and radio $3$ GHz VLA cutout images for only $10$ galaxies in our whole sample, for reason of space. Therefore, we include here in Fig.\\ref{stamps} the same types of images for the remaining sample of $14$ starbursts, all of which have been observed during the second Magellan run (we remind that galaxy ID 578239 was not observed by HST, so we did not show it).\n\n\\subsection{Morphological classification}\\label{morphological_classification}\n\nEven though the light emission of dusty starbursts at optical\/NIR wavelengths might be still severely affected by dust, the high resolutions offered by HST F814W images allows us to investigate the global structure of these systems. In Paper I we showed that the morphology of $18$ galaxies has been already classified by \\citet{kartaltepe10} (K10), revealing a merger origin for the majority of them. Adopting the same criteria of K10, we have classified visually the remaining $6$ galaxies (one has no HST coverage), but the results do not change significantly: $61\\%$ of our total sample are major mergers, as revealed by their highly disturbed morphology, tidal tales and bridges, $23\\%$ are classified as minor mergers from the presence of only slightly perturbed structures (e.g., warped disks, asymmetric spiral arms, etc.) without large companions, $11\\%$ are classified as spheroidal\/S0 galaxies and the remaining $5\\%$ as spirals. The major merger subset is additionally divided in five smaller classes according to their merger state (I: First approach, II: First contact, III: pre-merger, IV: Merger, V: Old-merger\/merger remnant), following K10. \n\nHowever, we remind that the merger recognition and, even more, the merger stage classification from the optical morphology is very uncertain and more difficult at higher redshifts, due to lower resolution and to surface brightness dimming, which hampers the detection of faint tidal tales\/interacting features, especially after the coalescence. The galaxy ID $245158$ represents a show-case example of this uncertainty: it has been classified as spiral\/minor merger from its global morphology, but it clearly shows a double nucleus in the central region, further confirmed by a double component H$\\alpha$ in the 2-D and 1-D spectrum, indicating rather an ongoing merger system.\n\n\\subsection{Radio size measurements}\\label{radio_size_measurements}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[angle=0,width=\\linewidth,trim={0.cm 0.cm 0.cm 0cm},clip]{Radio_fit_PaperII_compressed.pdf}\n   \n    \\caption{\\small GALFIT fitting of the radio VLA image (3 GHz) of the galaxy ID 685067 ($z = 0.37$) with the VLA synthesised beam in the \\textit{upper} row and with a Gaussian profile (convolved with the beam) in the \\textit{bottom} row. In horizontal sequence are shown, from left to right, the original image, the fitted model and the residual (original-model). For this galaxy, we derive an angular size of $0.20 \\pm 0.04''$ (the pixel scale is $0.2''$\/pixel), which corresponds to a physical size of $1.06 \\pm 0.19$ kpc. We notice that here GALFIT converges when fixing the position angle (PA) and axis ratio (q) parameters, to 0 and 1, respectively (Table \\ref{table2}). This example illustrates the possibility to reliably measure the radio sizes of our objects even when they are smaller than the FWHM resolution of VLA ($0.75''$). In this case, the difference between the two models is recognized by looking at the residual images (i.e., original-model).}\\label{galfit_residuals}\n\\end{figure}\n\nThe radio continuum emission has been used as a dust-free tracer of the SFR in galaxies in the absence of contamination from an AGN \\citep[e.g.,][]{condon92}. Since all our galaxies do not show either radio jets or radio flux excess compared to that expected from the SFR only (as we will show in Section \\ref{AGN_identification}), we used $3$ GHz VLA images for measuring the FWHM size of their starburst cores, where the bulk of star-formation is taking place. For each SB, we used GALFIT \\citep{peng10} to fit a 2D function, convolved with the VLA synthesised beam, to their radio emission. We tested several 2D profiles, which include a Gaussian, a S\\'ersic function and the VLA beam itself, requiring a significance of the fit (probed by the $\\chi^2$) of at least $95\\%$ confidence level, as for the emission line measurements. In addition to the $\\chi^2$ analysis, all the GALFIT residuals of the fit (original-model) were checked by eye inspection, and the excluded fittings have always worst residuals. \nThe best-fit profiles obtained for our sample are summarized as follows:\n\\begin{itemize}\n    \\item A single 2D Gaussian with varying FWHM, axis ratio and position angle provides the best fit for $13$ starbursts. In one case (ID 470239), the required conditions are obtained only by fitting a single S\\'ersic profile with varying parameters, but its half light diameter (calculated as $2 \\times r_e$, with $r_e$ the effective radius) is only $3\\%$ different from the total FWHM of a single gaussian fit, thus we assume the latter as the final value. \\footnote{We also tried to fit a single and double S\\'ersic profile for all the other sources. However, given the larger number of parameters of this profile and the limited VLA resolution, we do not obtain convergence for the majority of them, or the resulting $\\chi_\\text{reduced}^2$ are too high.}\n    \\item A double 2D Gaussian is required by $3$ galaxies (ID 245158, 412250 and 519651), allowing to resolve them and measure single components FWHM and their relative separation.\n    \\item Fitting the VLA synthesised beam yields the best solution for $6$ galaxies, which are then unresolved with current resolution ($0.75''$)\n    \\item A single 2D Gaussian with fixed axis ratio and position angle (to $1$ and $0$, respectively) is used for $2$ sources (ID 578239 and 685067). We remark that, in case the $95\\%$ significance level of the fit is satisfied with either this or the previous approach (as in the case of some very compact sources), we adopt the Gaussian solution only if the associated $\\chi^2$ probability level is at least double compared to the fit with the VLA beam.\n\\end{itemize}\n\nAs shown later, for a few galaxies we have measured angular sizes that are much smaller than the synthesised beam FWHM ($\\sim 0.75''$), down to $\\sim0.2''$ and to a physical scale of $1$ kpc. To demonstrate that it is possible to reliably determine the sizes even for these extreme, compact sources, we show in Fig. \\ref{galfit_residuals} a comparison between the GALFIT residuals obtained when fitting the image with a Gaussian (convolved to the VLA beam) (upper row) and with the radio synthesised beam itself (bottom row). It is evident that a Gaussian provides a better fit of the original source profile and a cleaner residual compared to the VLA beam alone. \n\nThe uncertainties on the sizes were recalculated for all the starbursts from their radio SNRs, using the fact that better detected radio sources also have the smallest radio size uncertainties, as shown, e.g., in \\citet{coogan18}. We used then the same formulation as:   \n\\begin{equation}\n    FWHM_{err}\\simeq 1 \\times \\frac{FWHM_{beam}}{SNR},\n\\end{equation}\\label{errorsize} \nwhere FWHM$_{beam}$ is the circularized FWHM of the VLA synthetized beam, and the multiplying coefficient was determined from simulations, following \\citet{coogan18}. All the size measurements with relative uncertainties and the method used for their determination are included in Table \\ref{table2}.\n\nSince our galaxies are well detected in radio band (average SNR of $18$) and the VLA synthetized beam is well known, we always obtain a good fit for the resolved sources. Among them, we were able to fit a double Gaussian for $3$ objects. In these cases, their total FWHM (adopted throughout the paper) were determined as the sum of the average single FWHM sizes and the separation between the two components. However, we will also consider the single sizes in some cases, e.g., in Fig. \\ref{Mass_Size}. This finding suggests that also some other galaxies may represent double nuclei that are blended in $0.75''$ resolution VLA images. \nFor the unresolved galaxies instead (i.e. those fitted with the VLA beam, as explained above), we adopted a $3\\sigma$ upper limit on their FWHM. Within the most compact starbursts, some of them may be affected by pointlike emission from an AGN, which decreases artificially the observed size. However, we tend to discard this possibility since, as we will see later, none of our AGN candidates show a radio-excess compared to the radio emission due to their SFR.  \n\n\\subsection{AGN identification}\\label{AGN_identification}\n \nWe started searching for AGN components in the mid-IR. Through the multi-component SED fitting of IR+(sub)mm photometry (described in Section \\ref{sample_selection}) we detected at $>3\\sigma$ the dusty torus emission component for a subset of $12$ SBs. The significance of the detection was derived from the ratio between the total best-fit dusty torus luminosity ($=L_\\text{AGN,IR}$) and its $1\\sigma$ uncertainty, inferred as the luminosity range (symmetrized) yielding a variation of the $\\chi_{red}^2$ $\\leq1$ with respect to the minimum value of best-fit \\citep{avni76}. More details about the torus estimation method are described in \\citet{liu18} and \\citet{jin18}. Among the $12$ mid-IR AGNs, we detected the dusty torus emission at high significance level ($> 5\\sigma$) for $6$ starbursts (ID 777034, 519651, 222723, 232171, 466112, 894779), while for the remaining objects we obtained a lower significance ranging $3\\sigma<$ $L_{AGN,IR}$ $<5\\sigma$ (see Table \\ref{table2}). The SED fitting of all the galaxies can be found in the Appendix \\ref{additional_plots}.  \n\nWithin the sample of IR-detected AGNs, $6$ galaxies (ID 777034, 222723, 232171, 635862, 578239, 911723) are also detected in X-rays at more than $3\\sigma$ by XMM-Newton, Chandra or NuStar \\citep{cappelluti09,marchesi16,civano15}. Throughout the paper, we will consider the X-ray luminosities L$_X$ measured by \\citet{lanzuisi17}, integrated over the energy range $2$-$10$ keV. \nTo estimate the contribution of star-formation to the total intrinsic L$_\\text{X}$, we used the relation between SFR and L$_\\text{X,SFR}$ of \\citet{mineo14}, rescaled to a Chabrier IMF and applying a correction factor of $0.6761$ to convert the X-ray luminosity from the $0.5$-$8$ keV to the $2$-$10$ keV band.\n\nWe remind that the column densities N$_H$ inferred from their hardness ratios \\citep{lanzuisi17} are consistent with those derived from the dust attenuations (toward the centers) assuming a mixed model (Paper I), suggesting that also the X-ray emission is coming from the nucleus, where the AGN is expected to be located. \nFurthermore, all our starbursts do not show radio jets in VLA images, and do not have significant radio excess than expected from their SFR, assuming a typical IR-radio correlation with q$_\\text{IR} = 2.4$, as in \\citet{ibar08}, \\citet{ivison10} and \\citet{liu18}. \n\n\n\\section{Results}\\label{results}\n\n\\begin{figure*}[h!]\n    \\centering\n    \\includegraphics[angle=0,width=0.48\\linewidth,trim={0.cm 3cm 0.cm 0cm},clip]{Quadruple_plot_paperII1.pdf}\n    \\includegraphics[angle=0,width=0.48\\linewidth,trim={0.cm 3cm 0.cm 0cm},clip]{Quadruple_plot_paperII2.pdf}\n   \n    \\caption{\\small \\textit{Left:} Correlations of A$_\\text{V,tot}$ with the radio size (top), and with the N2 index (bottom). We show with filled blue diamonds all the Magellan SBs that were used to derive the best-fit linear relation (blue continuous line) and the $\\pm$1$\\sigma$ dispersion (blue shaded area), while empty diamonds represent galaxies excluded from those calculations. The latter comprise the $4$ outliers discussed in the text (ID 303305, 345018, 472775, 545185) and all the upper limits in the A$_\\text{V,tot}$ vs FWHM$_\\text{radio}$ plot.\n    In the corners, we show in black the equation of the linear fit (which includes $1\\sigma$ error of the two best-fit parameters), and in gray the Spearman correlation coefficient (R) with the corresponding p-value (p), the reduced chi-square of the fit ($\\chi_\\text{red}^2$), and the $1\\sigma$ scatter of our SBs around the best-fit line, all of which do not take into account upper limits and the $4$ outliers mentioned above.\n    For comparison, the linear fit and $1\\sigma$ dispersion including the $4$ outlier galaxies are highlighted with a gray continuous dashed line and two dotted lines of the same color. \\textit{Right:} Correlations of A$_\\text{V,tot}$ with the line velocity width (top), and with the EW(Pa$\\beta$) (bottom). In the last diagram, $4$ galaxies without EW(Pa$\\beta$) measurements are not considered.}\\label{correlations}\n\\end{figure*}\n\nThe spectra that we have obtained at the Magellan telescope, along with longer wavelength radio images, provide us key information\nto understand both the attenuation sequence and the variety of morphological classes of our starbursts. First of all, since dissipative mergers are able to funnel the gas from the large scales of Milky-way-like disks ($\\sim10$ kpc) to sizes that are more than one order of magnitude smaller \\citep{dimatteo05}, it is useful to analyze the characteristic star-forming sizes of our starbursts.\nBesides this, from the galaxy integrated Magellan spectra we can study together the excitation and kinematic state of the gas, and the aging of the stellar population in the outer starburst cores, traced respectively by the [\\ion{N}{II}]\/H$\\alpha$ ratios, the intrinsic (resolution corrected) line velocity widths of single 1-D Gaussian components ($=$FWHM$_{\\text{line}}$, which is a proxy for the velocity dispersion in the system) and the Balmer\/Paschen line equivalent widths ($=$EW$_{H\\alpha,Pa\\beta}$).\n\nIn Fig. \\ref{correlations} we present the main result of this analysis, showing that the FWHM$_{\\text{radio}}$, the N2 parameter, the FWHM$_{\\text{line}}$ and the EW$_{Pa\\beta}$ are all correlated to the total dust attenuation A$_\\text{V,tot}$, which is used here as the reference quantity for comparison. \nThis suggests that our starbursts can be described as, at first order, a one-parameter sequence: similar correlations at different significance levels are indeed found also when comparing on a single basis each pair of the above parameters.\n\nWe tested these correlations using three different approaches with all the available data, excluding from the calculations only the upper limits and missing EW(Pa$\\beta$) measurements.\nFirstly, we calculated the Spearman rank correlation coefficient R (the higher R, the stronger the correlation) and the corresponding p-value, which represents the probability of obtaining an equal (or stronger) R if no correlation exists. We defined a threshold probability of $5\\%$ to accept the correlation. Overall, we found that the p-values are nearly always lower than $0.05$, meaning that the correlations are significant according to our criteria. In only one case (EW$_{Pa\\beta}$ vs FWHM$_{\\text{radio}}$) we determined a slightly higher p-value of $0.1$ (thus a higher probability of no correlation), which could be partly affected by the lowest number of data (i.e. lowest statistics) available here compared to the other diagrams. However, the other methods indicate instead a stronger physical connection between the two quantities.\n\nIn the second approach, we fitted the galaxies in each diagram with a linear relation (in log-log space, except for the last diagram where the y-axis is in linear scale), by using an Orthogonal Distance Regression procedure (ODR), which allows to take into account measurement uncertainties in both axis (we discuss later possible outliers or different fitting functions). We determined the SNR of the angular coefficient (i.e., how much it differs from 0), finding significant correlations at more than $3\\sigma$ in 8 cases, while they are less strong ($2<$ SNR $<3$) for the remaining two diagrams. In the four correlations shown in Fig.\\ref{correlations}, we obtained a significance of $5.8$, $5$, $4.3$ and $3.65\\sigma$ for A$_\\text{V,tot}$ vs N2, FWHM$_{\\text{line}}$ and EW$_{Pa\\beta}$, respectively. With this method, we also determined the 1$\\sigma$ dispersion of our data with respect to the best-fit linear relation. \n\n\\begin{table*}[!htb]\n    \\centering\n    \\begin{tabular}{||l||l||l||l||l||}\n    \\hlineB{2}\n    & \\bfseries FWHM$_{\\text{radio}}$ & \\bfseries N2 & \\bfseries FWHM$_{\\text{line}}$ & \\bfseries EW$_{\\text{Pa}\\bm{\\beta}}$ \\\\\n    \\hlineB{2}\n    \\bfseries A$_{\\text{V,tot}}$ & -0.6 (0.007) & 0.48 (0.027) & 0.61 (0.0037)  & -0.52 (0.026) \\\\\n    \\rowcolor{SeaGreen3!30!} & 5.8$\\sigma$  & 6$\\sigma$  & 4.3$\\sigma$ & 4.3$\\sigma$ \\\\\n    \\rowcolor{Tan3!30!} & 0.028\\% & $<0.001\\%$  & $2.9\\%$ & $0.12\\%$ \\\\\n    \\hlineB{1.5}\n    \\bfseries FWHM$_{\\text{radio}}$  &  & -0.71 (0.0006)  & -0.46 (0.049) & 0.45 (0.1) \\\\\n    \\rowcolor{SeaGreen3!30!} &  & 5.74$\\sigma$  & 2.93$\\sigma$ & 4.9$\\sigma$ \\\\\n    \\rowcolor{Tan3!30!} &  & $<0.001\\%$  & $0.033\\%$  & $3\\%$ \\\\\n    \\hlineB{1.5}\n    \\bfseries N2 &  &  & 0.67 (0.0003) & -0.43 (0.05)  \\\\\n    \\rowcolor{SeaGreen3!30!}  &  &  &  5.45$\\sigma$ & 3.85$\\sigma$ \\\\\n    \\rowcolor{Tan3!30!}  &  &  &  $<0.001\\%$ & $0.15\\%$ \\\\\n    \\hlineB{1.5}\n    \\bfseries FWHM$_{\\text{line}}$  &  &  &  & -0.46 (0.05) \\\\\n    \\rowcolor{SeaGreen3!30!}  &  &  &  &  2.5$\\sigma$  \\\\\n    \\rowcolor{Tan3!30!}  &  &  &  &  $12.9\\%$  \\\\    \n    \\hlineB{2}\n    \\end{tabular}\n\\caption{\\small Correlation coefficients among the total attenuation towards the center in a mixed model (A$_\\text{V,tot}$), the 3GHz radio FWHM size (FWHM$_{\\text{radio}}$), the line velocity width (FWHM$_{\\text{line}}$) and the equivalent width of Pa$\\beta$ (which tightly correlates also with the EW of H$\\alpha$, H$\\beta$ and H$\\delta$). In each case we show in three colored lines: \\textbf{(white)} the Spearman correlation coefficient and corresponding p-value; \\textbf{(green)} the significance of the correlation derived from the ratio of the linear best-fit angular coefficient and its uncertainty; \\textbf{(orange)} the probability of having a significance lower than 2$\\sigma$ if a random $20\\%$ of the sample is removed. For the calculations we excluded the upper limits, missing EW(Pa$\\beta$) measurements, and the $4$ outlier starbursts (ID 303305, 345018, 472775, 545185) in the three diagrams relating A$_\\text{V,tot}$ to N2, FWHM$_{\\text{line}}$ and EW(Pa$\\beta$).}\\label{table1}\n\\end{table*}\n\nFinally, we also performed Monte Carlo simulations: for each relation, we run 100k simulations, removing each time at random $20\\%$ of the points, recalculating the significance of the correlation using our second approach. We then estimated the rate ($\\sim$ probability) at which such correlations completely disappear with a significance falling below $2\\sigma$. This analysis allows to test the systematics and scatter of the correlations, ensuring they are robust and not driven by a few outliers. Overall, we find low probabilities (less than $5\\%$) to obtain a less than $2\\sigma$ significance when removing a random $20\\%$ of the galaxies, indicating that our correlations do not cancel out and are not found by chance. In the four diagrams of Fig.\\ref{correlations}, we obtained probabilities of $0.028\\%$, $0.001\\%$, $4.7\\%$ and $0.7\\%$, in the same order as above.\n\nWe remind that A$_\\text{V,tot}$ are determined from the Pa$\\beta$ observed fluxes and the bolometric L$_{IR}$ (assuming a mixed model geometry) \\footnote{As explained in Paper I, for $4$ galaxies in our sample where Pa$\\beta$ resides in opaque atmospheric regions or out of the FIRE coverage, we estimated the attenuation (ID 245158) or its upper limit (ID 303305, 500929, 893857) through the Pa$\\gamma$ line, adopting a flux ratio Pa$\\beta$\/Pa$\\gamma=2.2$. This is the average expected observed ratio for all the attenuation values in our range, assuming either a mixed model or a foreground dust-screen geometry, and it is verified by $9$ starbursts with simultaneous detection of Pa$\\gamma$ and Pa$\\beta$.}. However, for $4$ galaxies in the sample (ID 303305, 500929, 893857 and 232171) we do not detect either Pa$\\beta$ or Pa$\\gamma$, thus in these cases we derived A$_\\text{V,tot}$ in a similar way from their H$\\alpha$ fluxes (so to avoid upper limits), adding a representative error of $0.1$ dex determined from the remaining sample as the scatter of the correlation between Pa$\\beta$ and H$\\alpha$ based A$_\\text{V,tot}$.\nWe also verified that including the upper limits in the calculations does not significantly alter the fitted trends.  \nHereafter, we discuss in detail on a single basis the most important findings.\n\nIn the first (top-left) panel of Fig. \\ref{correlations}, the FWHM radio sizes, while spanning a wide range from less than $600$ pc to $\\sim$12 kpc, are tightly anti-correlated to the dust obscuration level A$_\\text{V,tot}$ (R=-0.6, p-value=0.007, and a scatter of $0.26$ dex). Towards the smaller sizes and higher obscurations (A$_\\text{V,tot}>20$ mag), three galaxies are unresolved with VLA, thus they may be actually closer to the best-fit linear relation derived from the remaining sample. In this diagram, X-ray detected AGNs are found both at small and large radii, and have a similar distribution compared to the other galaxies, suggesting that radio size measurements and hence the result in Fig. \\ref{correlations} are not contaminated by AGNs.\n\nIn the last three panels of Fig. \\ref{correlations}, the [\\ion{N}{II}]\/H$\\alpha$ ratio, the line velocity width (FWHM$_\\text{line}$) and the EW of Pa$\\beta$\\footnote{We use this line for comparison since, being at longer wavelength, it is more representative of the whole system, allowing to probe a larger fraction of starburst cores if a mixed geometry holds. However, in the Appendix \\ref{additional_plots} we show that EW(Pa$\\beta$) is tightly correlated to the EW of H$\\alpha$, H$\\beta$ and H$\\delta$, all of them being strongly sensitive to the age of the stellar population (at fixed SFH), thus similar results are obtained also if choosing a different line for the EW.} are also correlated to the total attenuation at more than 3$\\sigma$ significance level (R coefficients and p-values are $0.51$(0.009), $0.48$(0.015) and $-0.46$(0.034),  respectively). \nHowever, we notice that $4$ galaxies (ID$=$ 303305, 472775, 345018 and 545185) are outside the 1$\\sigma$ dispersion of the best-fit relations in all the three diagrams (gray dashed and dotted thin lines). They show lower N2, FWHM$_\\text{line}$, and higher EW than expected from their dust obscuration level. Alternatively, they have a larger A$_\\text{V,tot}$ for their N2, FWHM$_\\text{line}$ and EW values. \n\nIn order to understand the nature of these galaxies, we simulated $100$k different realizations of the last three diagrams of Fig.\\ref{correlations}, with N2, FWHM$_\\text{line}$, EW(Pa$\\beta$) and A$_\\text{V,tot}$ of $25$ galaxies distributed according to the best-fit relations and the corresponding $1\\sigma$ dispersions.\nThen we computed the probability of having at least $4$ galaxies ($3$ for the last plot) with an orthogonal distance from the best-fit relation (gray dashed line) equal or greater than the $4$ (or $3$) outliers described above. \nWe found, in the same order presented above, a probability of $0.2\\%$, $0.025\\%$ and $0.005\\%$, indicating that those $4$ galaxies are real outliers and cannot be simply explained by the $1\\sigma$ scatter of the best-fit lines. \n\nGiven their deviant behavior, we excluded these outliers and derived again the best-fit relations, which are shown in Fig.\\ref{correlations} with a blue continuous line. We found on average a reduction of the $1\\sigma$ dispersion (shown with a light blue shaded area) by $\\sim0.1$ dex and a slight improvement of the correlation significance compared to the previous calculations. However, the best-fit linear equations are not significantly different, thus we give only the analytic expressions of this second fit where the outliers are not considered. The new results for the three diagrams, and all the diagnostics for the remaining $7$ correlations are presented in Table \\ref{table1}. We notice that the $4$ divergent starbursts have an upper limit on their radio size, and are not outliers in other diagrams that do not involve A$_\\text{V,tot}$, thus the latter are not affected by this analysis. \nA possible physical explanation of the diverging behavior of these $4$ galaxies will be discussed in Section \\ref{outliers}. \n\nFinally, if we look at all the correlations in Table \\ref{table1}, we can notice the presence of a subset of quantities that correlate better than others. Apart from the previously discussed A$_\\text{V,tot}$ vs. FWHM$_{\\text{size}}$, the line width, N2 and radio size are tightly and robustly correlated with each other. Indeed, from bootstrapping analysis,\nthe probability that there is no correlation is less than $0.033\\%$. As we will see in Section \\ref{velocity_enhancement}, this result hides a deeper physical link among them. \n\n\\section{Discussion}\n\nThe results presented in the previous section show that the wide range of attenuations measured in Paper I translate into a wide range of other physical properties, i.e., radio sizes, N2, velocity width, Balmer\/Paschen EW, and even more, all these quantities appear to be connected to each other, defining a one-parameter sequence. In this Section we propose a physical interpretation of this sequence, and show that the correlating properties considered before are consistent with being primarily reflecting different evolutionary merger stages. Then we discuss the role played by each parameter into this sequence.\n\n\\subsection{Identification of early-phase, pre-coalescence mergers}\\label{pre-coalescence}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[angle=0,width=\\linewidth,trim={0.cm 7.4cm 1.8cm 1cm},clip]{2Dspectra_last_revised_compressed.pdf}\n    \\caption{\\small Close-up view of the H$\\alpha$ emission line profiles for the galaxies satisfying one of the two pre-coalescence criteria defined in Section \\ref{pre-coalescence}. \n    In each panel, as a consequence of the sky-subtraction procedure applied to the 2D spectra, the lines appear twice in different slit positions, in the first with a positive flux (in yellow) and in the second with a negative flux (in black). For each cutout, we show with two white arrows the slit position (s) and the dispersion direction ($\\lambda$), which are slightly rotated due to the curved Magellan spectra.\n    In order to clarify the classification criteria adopted in this work for finding pre-coalescence SBs, we highlight: (1) with continuous lines the different tilting angles of H$\\alpha$ line profiles (first $3$ SBs); (2) with dotted ellipses the spatially separated H$\\alpha$ lines coming from different merger components (last $4$ SBs). \n    }\\label{2Dspectra}\n\\end{figure}\n\nA first guess for a physical understanding of what is guiding the large spread of properties comes from the morphology. Indeed we have already seen that our sample comprises mergers at different stages of evolution (MI to MV), though this classification is very uncertain and sometimes misleading, as shown in Section \\ref{morphological_classification}: the faintness of tidal tails and residual interacting features make systems at the coalescence difficult to recognize, while multiple optical components and double nuclei in HST images may just reflect the dust attenuation pattern rather than the true distribution of SFR and M$_\\ast$.  \nAs shown in Section \\ref{radio_size_measurements}, a double gaussian component fit on radio images allowed to resolve three sources, suggesting that they may be composed of two interacting nuclei. However, the limited resolution of VLA ($0.75''$ of FWHM) does not allow us to derive solid conclusions on the remaining sample, which might contain more close pairs. New maps and ALMA follow-ups would increase the resolution and hopefully resolve these blended pre-coalescence systems. \n\nA complementary way to find close interacting pairs in early merger stages comes from the analysis of their 2D spectra.\nWith that aim, we performed a crude sky-subtraction procedure: we subtracted the 2D spectra taken for the same object but at different positions along the slit (A and B, separated by $2.2''$), in order to remove the sky lines and allow a visual inspection of the emission line profiles. \nFor construction, the lines appear twice in each sky-subtracted frame and exactly with the same shape: one time with a positive flux (when the object is in position A), and the other with a negative flux (when the galaxy is in position B). \nBy looking at these line profiles, we identified interacting pairs by requiring one of the following conditions:\n\\begin{enumerate}\n    \\item detached H$\\alpha$ line components along the spatial direction, coming from separated merger components located at different slit positions (e.g., ID 223715, 519651, 545185, 668738 in Fig. \\ref{2Dspectra}).  \n    \\item tilted H$\\alpha$ line with two different inclination angles (based on visual inspection), indicating the presence of two emitting regions with independent kinematic properties, inconsistent with a single rotating disk (e.g., ID 245158, 493881, 470239 in Fig. \\ref{2Dspectra}).\n   \n\\end{enumerate}\n\nIn our sample, we identified from the two above conditions $7$ close-pair pre-coalescence starbursts, which are shown in Fig.\\ref{2Dspectra}. For an additional source with a double radio emission component (ID $412250$), one of the two nuclei was not falling inside the slit, thus it was not observable with FIRE. However, this SB should be considered a merging pair at the same level of the others. \n\nAs we can see in Fig. \\ref{correlations}, the selected pre-coalescence starbursts are preferentially found at larger half-light radii, and all the systems with FWHM$_\\text{radio}>6$ kpc belong to this category. This result has two main implications. \nFirstly, the sizes measured in radio are not necessarily those of single merger components, but they should be interpreted primarily as separation between the two pre-coalescence starburst units \n(e.g., for all the three systems resolved in radio (ID 245158, 412250, 519651), their separation is larger than the size of single nuclei). Secondly, the early evolutionary phases are also characterized by lower dust obscurations, suggesting that the merger induced gas compaction (i.e., the increase of hydrogen column density in the center) has not yet completed.\n\nThis pre-coalescence subset identification provides an immediate physical interpretation for $6$ galaxies of those that were simultaneously fitted with a double Gaussian in the 1D spectrum (Section \\ref{line_measurements}), explaining this profile as coming from different merger components.\nHowever, we warn that these diagnostics are not identical and the connection between the line profiles in the 1D and in the 2D spectrum is not straightforward. Starbursts with multiple spatial emission lines do not necessarily display double Gaussians in the 1D spectrum, because this is subject to projection effects and depends on the distribution in wavelength of each spatial component. Indeed, the lines of one of the galaxies shown in Fig. \\ref{2Dspectra} (ID 519651) were still fitted with a single Gaussian in the 1D.\n\nFurthermore, our subset of $6$ pre-coalescence starbursts identified from the 2D spectra is not necessarily complete, as many galaxies (e.g., ID 635862, 777034, 472775, 685067) have sky-subtracted 2D spectra with low S\/N, not allowing to apply the visual criteria 1) and 2) presented above in this Section. We would have required longer integration times or spatially resolved observations to build a complete sample of starbursts before the coalescence. Similarly, if the two merger nuclei are too close, it would be impossible to detect them even in the 2D spectra, and would need a significant improvement of spatial resolution to identify the pair.\n\n\n\\subsection{Velocity enhancement and shocks toward the coalescence}\\label{velocity_enhancement}\n\n\\subsubsection{BPT diagram and shocks}\n\nThe second (bottom-left) panel of Fig.\\ref{correlations} shows that more obscured starbursts tend to have higher N2 relative to H$\\alpha$, reaching [\\ion{N}{II}]\/H$\\alpha$ ratios higher than 1, which are more typical of AGN and LINERs. Indeed, the classical BPT diagnostic diagram in Figure \\ref{BPT1} \\footnote{Two variants of the BPT using the [SII]6717+6731\/H$\\alpha$ or the [OIII]$\\lambda$5007\/[OII]$\\lambda$3727+3729 ratios (S2BPT or O2BPT, respectively) are shown in Fig. \\ref{BPTdiagram} in the Appendix. We remind that, due to an enhanced ionization parameter and lower metallicity (at fixed mass) in the ISM at higher redshifts, the average star-forming galaxies population at z$=0.7$ occupies a region in the BPT diagram which is shifted rightwards by $\\lesssim +0.1$ dex compared to z$=0.1$ \\citep{faisst18,masters16}. However, there are currently no studies addressing how this will affect the separation lines among SB, AGN and LINERs. If we suppose that at z$\\sim0.7$ the same shift applies also to these lines, galaxies at intermediate obscurations and line widths would still fall in the composite region with dominant LINER\/AGN-like properties. Also, this would not affect our subsequent conclusions based on the comparison with shock models.}, \nperformed on $9$ galaxies with [\\ion{O}{III}] and H$\\beta$ available measurements, confirms that SBs with higher obscuration and line velocity width are found in the composite, AGN or LINER classification regions, according to empirical separation lines derived in the local Universe \\citep{kauffmann03,kewley01,cidfernandes10,veilleux87}.\n\nNotably, the location of this subset of galaxies (which are shifted to the right compared to the purely SF region) is consistent with the predictions of shock models, with varying shock contribution and velocity (compare with Fig. 10 and Fig. 2 of \\citet{rich11,rich14}, respectively). Additionally, \\citet{lutz99} argue that LINER-like spectra in infrared selected galaxies are due to shocks, possibly related to galactic superwinds. \nThe presence of increasing widespread shocks provides the most immediate interpretation for the spectra in our sample with enhanced [\\ion{N}{II}]\/H$\\alpha$, given that AGN emission would be highly suppressed (Paper I). \n\nHowever, we cannot exclude some residual influence by an AGN. Hydrodynamical simulations performed by \\citet{roos15} show that even in the case of high obscuration\nan AGN can ionize the gas very far from the nucleus, reaching kpc scales and the circum-galactic medium. Furthermore, the accreting black hole might not be in the center, but that sounds unplausible: the attenuations towards the center derived independently from the X-ray detected AGNs are consistent with those derived from the mixed model (see Paper I) and, even further, \\citet{rujopakarn18} show that the AGN position correlates with that of active star forming regions. Finally, we also notice that two galaxies (which simultaneously have X-rays and mid-IR dusty torus detection) were fitted with broad H$\\alpha$ components (line width of $\\sim 1000$ km\/s). Such large velocity widths have been observed in both shock-dominated regions (possibly supernova driven, \\citet{ghavamian17}) and AGNs \\citep{peterson97,gaskell09,netzer15}. IFU data would be needed to disentangle shock or AGN emission, as we expect the latter to be much more concentrated in the central part of the system.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[angle=0,width=8.4cm,trim={0.1cm 3.3cm 2.6cm 5.5cm},clip]{BPT_N2_double.pdf}\n    \\caption{\\small \\textit{Top:} BPT diagram of $9$ starbursts in our sample with optical spectra available (for one galaxy included in zCOSMOS, we did not detect both [\\ion{O}{III}]5007 and H$\\beta$). While $3$ sources lie in the SF excitation region, the remaining galaxies are not consistent with SF, and their spectra show a mixture of composite, AGN and LINER properties. The color coding indicates that galaxies with higher N2 which are closer to the AGN\/LINER regions also have increasingly higher line velocity widths ($\\sigma_{\\text{line}}$). \\textit{Bottom:} Same diagram as above, but here the galaxies are color coded according to their total dust attenuation A$_\\text{V,tot}$. More obscured starbursts preferentially display AGN\/LINER properties.}\\label{BPT1}\n\\end{figure}\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[angle=0,width=\\linewidth,trim={0.1cm 0.2cm 1.7cm 2.3cm},clip]{Mdyn_Mall.pdf}\n    \\includegraphics[angle=0,width=\\linewidth,trim={0.1cm 0.2cm 1.7cm 2.3cm},clip]{Virial_theorem_total.pdf}\n    \\caption{\\small \\textit{Top:} Comparison between the dynamical mass M$_{\\text{dyn}}$ and the total mass content M$_{\\text{tot}}$ ($=$ M$_{gas}$+M$_{\\ast}$+M$_{\\text{dark matter}}$) for our SBs sample, color coded by their total attenuation A$_\\text{V,tot}$. \\textit{Bottom:} Diagram showing the square of the total FWHM velocity width as a function of M$_{\\text{tot}}$\/FWHM$_{\\text{radio}}$, using the same color coding based on A$_\\text{V,tot}$. On the y-axis, const$=1.3 \\times G$\/($4$<sin$^2$(i)>) groups the coefficients in Equation~3 so as to facilitate comparison with the virialized case (1:1 relation, shown as a grey dotted line). The blue continuous line represents a linear fit to our sample, excluding galaxies with an upper limit on their radio size, while the blue shaded area shows the $\\pm1\\sigma$ limits of this best-fit relation. Both panels of the figure suggest that our galaxies may be approaching virialization, and more obscured starbursts are closer to the equilibrium.}\\label{virial}\n\\end{figure}\n\n\\begin{figure}[t!]\n    \\centering\n   \n    \\includegraphics[angle=0,width=8.4cm,trim={0.1cm 0.cm 1.3cm 2.cm},clip]{Sigma_N2index_single.pdf}\n    \\caption{\\small Correlation between the line velocity width of single Gaussian components (FWHM$_{\\text{line}}$) and the N2 index (both measured from our Magellan-FIRE spectra), indicating that the two quantities are tightly (scatter $=$ 0.26 dex) physically related.}\\label{sigmaline}\n\\end{figure}\n\n\\subsubsection{The dynamical masses of our sample}\\label{dynamical_masses}\n\nIn order to better understand the dynamical status of our starbursts and how far they are from relaxation, we compared their total masses M$_{\\text{tot}}$ to the dynamical masses M$_{\\text{dyn}}$ estimated from the line velocity widths and radio sizes. For the latter, we used the formulation of \\citet{daddi10} as :\n\\begin{equation}\n    M_{\\text{dyn}}=1.3\\times\\frac{ \\text{FWHM}_\\text{radio} \\times (\\text{FWHM}_\\text{line,total}\/2)^2}{\\text{G sin}^2(i)}\n\\end{equation}\nwhere FWHM$_\\text{line,total}$ is the one-dimensional total line velocity width (accounting for both rotation and dispersion), while sin$^2$(i) is the correction for inclination that we take as the average value for randomly oriented galaxies ($57^\\circ$). In order to determine the total uncertainty on M$_{\\text{dyn}}$, we considered an additional error on the inclination factor of $0.3$ dex, as in \\citet{coogan18}. This represents the main contribution to the error ($\\sim 90\\%$ in median), since the line width and radio sizes are always well measured with high S\/N. \n\nThen we compared this quantity to the total mass content (baryonic + dark matter) of the systems, estimated as :\n\\begin{equation}\n    \\text{M}_{\\text{tot}}=\\text{M}_\\ast+\\text{M}_{\\text{gas}}+\\text{M}_\\text{dark matter}\n\\end{equation}\nin which M$_\\text{gas}$ was determined, as described in Paper I, as M$_\\text{gas} = 8.05+0.81 \\times$ log(SFR$_\\text{IR}$) \\citep{sargent14}, valid for a starburst regime, and we assumed M$_\\text{dark matter} = 10\\% \\pm 10 \\%$ of M$_\\ast$. Since this contribution is highly uncertain, it was set nearly uncontrained. However, this range is consistent with studies of high-$z$ ($>0.5$) massive star-forming galaxies, which found a modest to negligible dark matter fraction inside the half-light radius \\citep[e.g.,][]{daddi10,genzel17}. In any case, given the small contribution, its exact value does not affect the results of this paper. \nFor the error determination, we considered the above uncertainty on M$_\\text{dark matter}$, a $0.1$ dex error on M$_\\ast$ \\citep{laigle16}, and $20\\%$ incertitude on the gas mass (even though its contribution is negligible given that M$_{\\text{gas}} \\simeq 0.1$ M$_{\\ast} $ on average for our sample).\n\nThe comparison between M$_\\text{dyn}$ and M$_\\text{tot}$ in Fig. \\ref{virial} shows that, on average, our galaxies are not completely virialized: while $\\sim$ half of the sample is consistent within $2\\sigma$ with the 1:1 relation, the remaining part is located above at higher M$_\\text{dyn}$. The largest departures from virialization are observed for the pre-coalescence and less obscured systems, i.e. supposedly earlier stage mergers. On the contrary, the systems with better agreement may be fully coalesced starburst cores with higher A$_\\text{V,tot}$. \n\nThe tight connection between velocity and gravitational potential is clarified in the bottom panel of Fig. \\ref{virial}, as the FWHM$_\\text{line}^2$ and M$_\\text{tot}\/$FWHM$_\\text{radio}$ correlate at 5$\\sigma$ significance (with R=0.74 and p-value=0.0002). Also here, while pre-coalescence mergers have larger displacements from the 1:1 relation, they are confined in a region at lower velocity widths and shallower potential wells. This suggests that also other starbursts (ID 249989, 466112, 326384) in this region may be pre-coalescence mergers that we were not able to securely identify, due to their lower S\/N 2D spectra, and indeed their optical morphology strengthens this suspicion. In the upper-right part of Fig. \\ref{virial}-\\textit{bottom}, separated from the previous sample, are clustered the more obscured starbursts, i.e., supposedly coalesced mergers. We notice also that all X-ray detected AGNs are localized in this region of the diagram, indicating a possible link between evolutionary phase and AGN properties, that we will further investigate in the following Section. \n\n\nOverall, the above results suggest a time-evolutionary scenario, in which more advanced, already coalesced mergers are close to virialization, and the increased central potential wells (due to the contribution of both merger components) are responsible for the enhancement of both the kinetic energy content and shocks towards the later stages of the interaction. The tight relation between the line velocity width of single Gaussian components (a proxy for the velocity dispersion in the system) and shock production (traced by the N2 index) is further indicated by the color coding of the BPT diagram in Fig. \\ref{BPT1}-\\textit{top}, and by the correlation between FWHM$_{\\text{line}}$ and \\ion{N}{II}\/H$\\alpha$ in Fig. \\ref{sigmaline}, which has a significance higher than $5\\sigma$ (R=0.67, p-value=$3\\times10^{-4}$) and a dispersion of $0.26$ dex. \n\n\n\\subsection{Lower line equivalent widths toward late merger stages}\\label{lower_line_equivalent_widths}\n\n\\begin{figure*}[t!]\n    \\centering\n    \\includegraphics[angle=0,width=17.5cm,trim={4.8cm 0.1cm 3.cm 0.3cm},clip]{EW3_vs_APaBirx_triple_last.pdf}\n    \\vspace{-0.25cm}\n    \\includegraphics[angle=0,width=17.5cm,trim={4.8cm 0.cm 3.cm 1.4cm},clip]{EW3_vs_N2new_2_triple.pdf}\n    \\caption{\\small \\textit{Top:} Comparison between the EW of H$\\delta$, H$\\beta$ and Pa$\\beta$ lines to the dust attenuation parameter A$_{Pa\\beta,\\text{IRX}}$, defined as $2.5 \\times log_{10}(1+\\text{SFR}_{\\text{IR}}\/\\text{SFR}_{Pa\\beta,\\text{obs}}$ (Paper I). We remark that the last panel is equivalent to the bottom-right plot in Fig. \\ref{correlations}, even though a different scale has been used (the relation between A$_{Pa\\beta,\\text{IRX}}$ and A$_\\text{V,tot}$ is given in Eq. \\ref{eqAV}). \\textit{Bottom:} Correlations between the EW of H$\\delta$, H$\\beta$, Pa$\\beta$ lines and the N2 index ($=log_{10}([NII]\/H\\alpha)$). The blue continuous lines are the best-fit linear relations, determined as explained in Section \\ref{results}, while the blue shaded area show the $\\pm1\\sigma$ scatter of our data around the best-fit relations.}\\label{EW_ALL}\n\\end{figure*}\n\nThe equivalent widths (EW) of hydrogen recombination lines give a relatively dust-unbiased picture (assuming that stars and emission lines are equally extincted) of the contribution of the SFR to the stellar mass content\n, and they are sensitive to the luminosity weighted age of the stellar populations, so that they could provide useful information about the evolutionary stage of the merger. However, these EWs would only probe what is happening in the outer parts of the system, since the core is completely obscured in optical\/near-IR.\n\nWe show in the last diagram in Fig. \\ref{correlations} and the upper part of Fig. \\ref{EW_ALL} that, when the starbursts become more obscured, the EWs of Pa$\\beta$, H$\\delta$ and H$\\beta$ decrease, indicating a gradual SFR decline in the outer skin of more obscured and compact starbursts. Additional correlations are found also independently between those EWs and the other quantities, such as the N2 index (Fig. \\ref{EW_ALL}-\\textit{bottom}). We additionally remark that the different Paschen and Balmer lines correlate each other (see Fig. \\ref{EW_EW} in the Appendix), for which reason our results, derived adopting the Pa$\\beta$ line as a reference (because it is the least attenuated), are also valid when considering the H$\\delta$, H$\\beta$ and H$\\alpha$ lines.\n\nIn our sample, we also found that the Balmer EWs, while having a large dynamical range, can reach very low values: in five galaxies (ID 303305, 685067, 777034, 862072 and 345018) we measure an EW(H$\\delta$) < $-4\\AA$ (i.e. in absorption), which are typically found in E+A dusty galaxies \\citep{poggianti00}. \nLow EW hydrogen recombination lines (in strong absorption) are clear signatures of the prevalence of A-type stars, indicating that a recent ($<$ 1 Gyr ago) massive star-formation episode has taken place during the past $10^8$-$10^9$ yr, while the youngest stellar  populations (mainly OB stars) are nearly all obscured by dust in the inner starburst core. \nOur dusty starburst systems should also not be confused with post starburst (PSB) galaxies, which have similar absorption EWs (e.g., EW(H$\\delta$) lower than $\\sim -5 \\AA$ as in \\citet{goto07} and \\citet{maltby16}), but are thought to be nearly (or already) quenched systems, with much lower SFR levels and lower dust content compared to our sample (see \\citet{pawlik18} for a full discussion of the different types of PSB galaxies). We caution that the quenched PSB selection from only the Balmer EW can actually return real starbursts and not post starburst systems.\n\nPutting all together, the time-evolutionary scenario that we have suggested has the advantage of explaining in a simple way these new results. If we follow the merger evolution towards the coalescence, the outer starburst skin becomes increasingly dominated by A-type stars, recognizable through the deep absorption lines in the optical\/near-IR and which were formed at earlier times when the separation between the merging nuclei was larger. At the same time, the star-formation in the skin is being suppressed, possibly driven by supernova feedback. \n\n\n\\subsection{Outliers}\\label{outliers}\n\nWe found in Section \\ref{results} (Fig. \\ref{correlations}) that $4$ galaxies are outside the $1\\sigma$ dispersion of the best-fit relations between the dust attenuation A$_\\text{V,tot}$ and, simultaneously, the [NII]\/H$\\alpha$ ratio (N2), the line velocity width (FWHM$_\\text{line}$) and the EW(Pa$\\beta$). \nIn particular, they have lower N2, FWHM$_\\text{line}$ and EW(Pa$\\beta$) than expected from their A$_\\text{V,tot}$, suggesting that, compared to other highly obscured galaxies, there is a minor impact from shocks or a dominant contribution of star-formation to the emission lines.\n\nWithin our SB sample, we recognize that these $4$ outliers have the largest dust obscurations A$_\\text{V,tot}\\geq 18$ mag, and are among the most compact, with radio FWHM sizes below $1$ kpc. These extreme and peculiar features suggest they may represent the very end stages of the merger evolution, and that the correlations with A$_\\text{V,tot}$ may saturate toward these late phases. We also notice that the same objects are not systematically outliers when we consider their N2, FWHM$_\\text{line}$ and EW(Pa$\\beta$) values, confirming the close physical connection among these quantities, as shown in the two previous Sections \\ref{velocity_enhancement} and \\ref{lower_line_equivalent_widths}.\n\n\\subsection{The complete sequence of merger stages at intermediate redshift}\\label{cartoon_section}\n\n\\begin{figure*}[t!]\n  \\centering\n  \\includegraphics[angle=0,width=0.9\\linewidth,trim={0cm 0.0cm 0.cm 0cm},clip]{Cartoon_revised_compressed2.pdf}\n  \\caption{\\small Schematic illustration of the time-evolutionary behavior of the physical parameters studied in the text: dust attenuation, characteristic size of the system, line velocity width (or, equivalently, the N2 index) and the EW of Balmer and Paschen lines. The time sequence is divided into 5 fundamental merger stages, with the QSO and passive spheroidal system representing the final stages according to the classical merger paradigm \\citep{sanders96,hopkins08a,hopkins08c}. Solid lines are qualitative trends during the SB phase inferred from our results, while dashed lines are predictions about the future evolution of the $4$ parameters shown on the y-axis (line width and N2 index behave similarly). In the upper part of the figure, we show a qualitative merger timescale following Fig.~1 of \\citet{hopkins08c}, assuming for the merger a total starburst duration of $200$ Myr. For each phase, we show in the bottom part the ACS-F814W cutout of a representative case. The first three images are SB galaxies from our sample: ID 223715, ID 777034 and ID 472775. They were chosen as having increasing dust attenuations and radio compactness, suggestive of more advanced merger phases: the first was identified as a pre-coalescence merger in Section \\ref{pre-coalescence}, while the latter is unresolved in radio and is highly obscured (A$_\\text{V,tot}=18$ mag). The last $2$ cutouts show a quasar at $z=0.73$ and an ETG at $z=0.66$, selected in COSMOS field from the catalogs of \\citet{prescott16} and \\citet{tasca09}, respectively.\n  }\\label{cartoon}\n\\end{figure*}\n\nOur observations and results, presented in previous sections, suggest we are starting to see an evolutionary sequence in high-redshift mergers. This can be traced through a variety of physical measurable quantities of our galaxies, including the total attenuation towards the center, the characteristic size of the starburst region, the EW of hydrogen absorption lines, and finally the [\\ion{N}{II}]\/H$\\alpha$ ratios and line velocity widths, which behave similarly.\nIn Fig. \\ref{cartoon} we schematize with a cartoon all the results that we have found so far, showing with a red continuous line the qualitative trend of the different physical quantities as a function of time. We divided the time axis into five merger evolutionary stages, which are arranged in relation to the two most crucial transformation events during the merger: the coalescence and the blow-out\/QSO appearance.\n\nWe notice that the first phase may not necessarily represent the beginning of the interaction, i.e., when the two galaxies approach for the first time. Even though the whole merger episode may last 1-1.5 Gyr in total, from the first encounter to the formation of a passive spheroidal system, the starburst activity is typically shorter, ranging 200-300 Myr \\citep{dimatteo08} and may be triggered intermittently at various stages of the evolution. Furthermore, whether or not a strong burst is already activated at the first approach depends on many factors, including the impact geometry, the morphology, the stellar mass ratio and the gas content of the colliding galaxies \\citep{dimatteo08}. \n\n\nBesides the observable starburst phases studied in this work, can we also make some predictions on the future evolution of these systems ? In general, it is very hard to demonstrate visually a connection between mergers and their descendants. Indeed, not all merger-induced starbursts exhibit morphological disturbances \\citep{lotz08}, and when merger residual signatures are present, they fade rapidly, becoming almost invisible beyond the local Universe even in the deepest optical images \\citep[e.g.][]{hibbard97}. We can in principle rely on hydrodynamical simulations, which allow to trace the full time-sequence of mergers, even though they also present limitations due to the many assumptions, initial conditions and physical complexity involved in such events. \n\nIn the classical theoretical merger paradigm, the infalling gas triggers obscured AGN accretion \\citep{bennert08a}, whose peak of activity typically occurs $\\sim 250$ Myr after the onset of the starburst \\citep{wild10}, and $\\sim 100$ Myr after the peak of SFR \\citep{davies07,hopkins12}. It is during these later starburst phases that the AGN feedback can blow out with strong feedback winds the surrounding dust and gas cocoon, eventually revealing itself as a bright QSO \\citep{hopkins08c}. This phase is generally very short, lasting for $\\lesssim 100$ Myr \\citep{hopkins10c}, and has been claimed since a long time: \\citet{lipari03} suggested that QSOs could be indeed young IR active galaxies at the end phase of a strong starburst. \n\n\nSince the QSO dominates the luminosity of the system at all wavelengths, it would be extremely hard to analyze the physical properties of the host galaxies during this phase. Indeed, \\citet{zakamska16} show that even in radio-quiet QSOs both the infrared and the radio emission are dominated by the quasar activity, not by the host galaxy. \nAn alternative possibility is to look far from the central bright source. Recent works are revealing Ly-$\\alpha$ nebulae surrounding high-redshift quasars, with extension that can reach tens of kpc ($\\lesssim$ 50 kpc) from the center \\citep{arrigonibattaia18}.\nOn the other hand, one may focus on local samples, increasing simultaneously the images resolution. For example, \\citet{lipari03} and \\citet{bennert08b} discovered with HST the presence of outflows, arcs, bubbles and tidal tales in optical band in a sample of local QSOs, possibly formed through strong galactic winds or merger processes. Again in nearby (z $<0.3$) QSOs, near-IR H band adaptive optics observations \\citep{guyon06} revealed that $\\sim 30\\%$ of their hosts show signs of disturbances, and the most luminous QSOs are harbored exclusively in ellipticals or in mergers (which may become ellipticals soon). Furthermore, while the SFRs of the hosts are similar to those of normal star-forming galaxies, their mid- and far-IR colors resemble those of warm ULIRGs, strengthening a connection between these two objects.\n\n\nIn the following two Sections \\ref{mass-size-section} and \\ref{QSO_in_formation}, we discuss separately the two ending stages of the merger sequence, and investigate how our work can provide some clues to understand what are the physical properties of the systems into which our starbursts will evolve. In the cartoon of Fig. \\ref{cartoon}, the predicted evolution for all the quantities studied in this paper (see Section \\ref{results}) is shown with a dashed line. These qualitative trends are motivated mainly from simulations, and are not confirmed observationally.\n\n\\subsection{Mass-size relation and comparison with higher and lower-z starbursts}\\label{mass-size-section}\n\nThe merger-induced starbursts are supposed to end up in a passive system, but we do not know the exact physical properties (e.g., size, stellar mass, morphology) of these merger remnants. Sub-millimeter galaxies (SMGs) at high redshift ($>2$), which are commonly viewed as higher luminous counterparts of lower redshift ULIRGs, have been suggested to be direct progenitors of massive ETGs \\citep{tacconi08,toft14}.\nWe can investigate this connection by comparing in Fig. \\ref{Mass_Size} the stellar masses and the characteristic sizes of our starbursts with those of disk galaxies and spheroids at z $\\sim0.7$ \\citep{vanderwel14}. To be conservative, we are adopting here the M$_\\ast$-size relations for circularized radii. If we consider instead the non-circularized cases, the same relations would slightly shift upwards by $\\sim 0.1$ dex.\nIn addition, we remark that we are comparing our radio (starburst) extensions to optical rest-frame sizes tracing the stellar mass distribution of disks and elliptical galaxies. Indeed, we implicitly assume that, after the gas in our starburst cores is converted into stars, the extensions of these cores will represent also the stellar component sizes of their passive remnants. On the other hand, they may still represent the dense star-forming gas components of post-starburst systems if some residual is left after the merger, as they may remain compact for at least 1 Gyr \\citep{davis18}.\n\n\\begin{figure*}[t!]\n  \\centering\n  \\includegraphics[angle=0,width=\\linewidth,trim={0.1cm 0.2cm 2cm 1.3cm},clip]{Mass_size_new.pdf}\n  \\caption{\\small Diagram showing the radio size vs stellar mass for our sample. We compare our results to the stellar mass - stellar size relation of LTGs (cyan line with 1$\\sigma$ dispersion) and ETGs (blue line) at z$=$0.7 \\citep{vanderwel14}, and with the bulge properties of low redshift (z$\\sim$0.1) spiral galaxies from \\citet{graham08} (grey squares). The eleven points shown here for the bulges represent the median ($\\pm1\\sigma$) of their distributions of stellar masses and stellar sizes (in K band) as a function of galaxy type, from S0 to Sm spirals. For $3$ galaxies in our sample fitted with a double Gaussian, we also represent the `deblended' radio sizes of each single component with violet crosses, connecting them with a dashed violet line. In these cases, we assigned to each component half of the total stellar mass of the system, even though a precise estimation requires a separate fit on deblended photometric data.}\\label{Mass_Size}\n\\end{figure*}\n\nIn the diagram of Fig. \\ref{Mass_Size}, $6$ SBs are consistent with the late-type galaxy (LTG) relation at z $\\sim0.7$. However all of them are pre-coalescence SBs and, as we have seen before, they should not be considered disk galaxies as their size is primarily reflecting the separation between the merging components. For two of the three galaxies resolved in radio, the single values return below on the early-type galaxy (ETG) relation. The characteristic sizes of this subset (FWHM$_{\\text{size}}$ ranging 3-15 kpc in diameter, with median FWHM$_{\\text{size}}$ of 8 kpc) are similar to those typical observed in SMGs \\citep{casey11,tacconi08,biggs08}, which suggests that SMGs at high-redshift (or at least a fraction of them) could be indeed intermediate-phase mergers composed of unresolved double nuclei, as argued by \\citet{iono09} and \\citet{arribas12}. \n\nIn the bottom part instead, we can immediately notice that a major fraction of our sample (13 galaxies, i.e., $52\\%$ of the total sample) is not consistent with the ETG relation (taking $1$-$\\sigma$ dispersion), and is located well below it by $\\sim0.5$ dex, with an average size of $\\leq 1.2$ kpc, indicating that they are much more compact than their stellar envelopes and than typical ellipticals at z $\\sim0.7$. We underline that such difference would be even higher if we compare this subset to the M$_\\ast$-size relation at redshifts lower than $0.7$, as the ETG sizes at z $=0.25$ are a factor of 1.5 higher than those at z $=0.75$, at our median stellar mass \\citep{vanderwel14}.\nThis sample of very compact starbursts has typical extensions that are similar to those of dense star-forming regions in local ULIRGs \\citep{genzel98,piqueraslopez16}, including Arp 220 \\citep{sakamoto17} and M82 \\citep{barker08}, suggesting they are driven by the same merger mechanisms (as also argued in Paper I).\n\nIf we take for each galaxy its distance from the LTG relation (dist$_{\\text{LTG}}=\\log_{10}$(FWHM$_{\\text{size}}$\/FWHM$_{\\text{LTG}}$)), we can also use this quantity in place of the radius to trace the same sequence found in Section \\ref{results}, taking into account the mild dependence on stellar mass.\nAs the merger proceeds, the system moves from the LTG to the ETG relation and then even below at significantly smaller sizes (by $\\sim 0.5$ dex at least), meaning that the compact starburst cores that form at the coalescence cannot produce directly the ellipticals seen at redshift $0.7$ and below. \n\nThe sizes of our starbursts instead resemble those of typical bulges in lower redshift spirals and lenticular galaxies \\citep{graham08,laurikainen10}, indicating a possible evolutionary link with mergers, as suggested by other works \\citep[e.g.][]{sanders96,lilly99,elichemoral06,querejeta15}.\nThis idea is consistent with the typical observed gas fractions of our starbursts (derived as M$_{gas}$\/(M$_\\ast$+M$_{gas}$), with M$_{gas}$ calculated in Section \\ref{dynamical_masses}), which range between $0.02$ and $0.25$ ($\\sim 0.1$ in median). Assuming that all the remaining gas is consumed before the passivization and that the same amount of gas has been already converted into stars (which depends on the merger phase and dynamics), it means that the current starburst cores can produce approximately $20\\%$, and up to $50\\%$, of the final stellar mass of the galaxies. Higher resolution radio images targeting specific emission lines can further constrain the kinematic properties of the starbursting cores, by looking for rotation, or their luminosity profile, e.g., measuring their Sersic index.\nHow this old stellar component is affected by the merger can depend on many conditions difficult to model in detail, including the geometry of the interaction, the gas content and the mass ratio of the colliding galaxies.  \n\n\n\\begin{figure*}[]\n  \\centering\n  \\includegraphics[angle=0,width=0.338\\linewidth,trim={0cm 0.cm 0cm 0cm},clip]{distanceLTG_LXobserved_SNR3_0.pdf}\n  \\includegraphics[angle=0,width=0.65\\linewidth,trim={1.2cm 0.7cm 3.cm 2cm},clip]{LXobs_LXint_AvTOT_ALL.pdf}\n \n  \\caption{\\small \\textit{Left:} Comparison between the observed X-ray luminosity (L$_\\text{X,obs}$) and the distance from the Mass-size relation of LTGs at $z\\sim0.7$, for our $6$ starbursts detected in X-rays. Upper limit on L$_\\text{X,obs}$ for $6$ mid-IR AGNs undetected in X is shown with a black circle, where the horizontal segment represents the range of dist$_\\text{LTG}$ spanned by this subset. The intrinsic X-ray luminosity due to star-formation is highlighted with a gray line for the median SFR of the sample ($\\pm 0.4$dex scatter from \\citet{mineo14}), and may dominate the total X-ray observed emission for the X-undetected starbursts; \\textit{Center:} X-ray attenuation L$_{X,obs}$\/L$_{X,int}$ as a function of the infrared-based attenuation A$_\\text{V,tot}$ (in a mixed model geometry and towards the center) for our sample of mid-IR detected AGNs. We assumed here a bolometric correction factor L$_{\\text{X,intr,AGN}}=0.04\\times$ L$_{\\text{BOL,AGN}}$ \\citep{vasudevan07}. Stacks on the whole sample and on the X-undetected subset are displayed with hatching circles, while the violet shaded regions indicate the area of no obscuration, which incorporates a factor of 2 uncertainty in the conversion between intrinsic X-ray and bolometric AGN luminosity; \\textit{Right:} Same diagram as before, but assuming an L$_{\\text{BOL,AGN}}$- dependent bolometric correction \\citep{lusso12}, as explained in the text.  \n  }\\label{distltg}\n\\end{figure*}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[angle=0,width=\\linewidth,trim={0.1cm 0cm 0cm 1.3cm},clip]{distanceLTG_Lbol_AGN_Mstar_SNR3_0.pdf}\n  \\caption{\\small L$_{\\text{BOL,AGN}}$\/M$_\\ast$ vs. distance from the Mass-size relation of LTGs (dist$_{LTG}$) for our SBs sample. The Eddington limit is shown with a blue horizontal line, while the shaded area takes into account the spread of M$_\\ast$ among our sample and the uncertainty of the relation between stellar mass and BH mass by \\citet{reines15}. The Eddington ratio is shown on the right y-axis.}\\label{distltg2}\n\\end{figure}\n\n\\subsection{QSOs in formation at $z \\sim 0.7$ ?}\\label{QSO_in_formation}\n\n\nIn the starburst selection phase, we discarded several quasars because of the impossibility to study the properties of their host galaxies (dust attenuation, SFR, stellar mass), as discussed in Section \\ref{cartoon_section}. In order to overcome these limitations, several authors have extensively studied also the transitional moments (just preceding the final blow-out) in the dress of type-I and warm ULIRGs \\citep{kawakatu06,sanders88}. Similarly, we can have some clues of the inner black-hole activity just before the hypothetical QSO by looking at the AGN diagnostics in our starburst sample. \n\nAs mentioned in Section \\ref{AGN_identification}, we detected the mid-IR dusty torus AGN emission in $12$ galaxies, and simultaneous X-ray emission in $6$ of them. Notably, the latter are the only ones (among mid-IR AGNs) whose host galaxies lie below the ETG relation in the Mass-size diagram (see Fig. \\ref{Mass_Size}), at systematically smaller sizes than ellipticals at z $\\leq 0.7$. \nThis suggests that during early merger stages the AGNs are predominantly obscured, while they start to appear in X-rays toward intermediate stages (i.e. when the host starbursts are more compact and obscured), possibly driven by rapid AGN feedback clearing the gas and dust around the black hole.\n\nThis can be seen better in Fig. \\ref{distltg}-\\textit{left}, where all the host galaxies of X-ray detected AGNs are located at larger distances from the Mass-size relation of LTGs compared to X-ray undetected AGNs. Moreover, they have X-ray luminosities at least 1 order of magnitude higher ($\\sim +1.5$ on average) than what inferred from their SFR. For the X-undetected galaxies instead, the upper limit on L$_X$ $=$10$^{41.7}$ erg\/s, determined by average-stacking their fluxes in the $2$-$10$ keV band at the median $z$ of the sample, is consistent with emission produced by star-formation only, suggesting that in this band the AGN is completely obscured.\n\nIn order to assess the level of obscuration, we computed the ratio between the observed and the intrinsic X-ray luminosity L$_{\\text{X,obs}}$\/L$_{\\text{X,intr}}$, comparing this quantity to the total dust attenuation A$_{V,tot}$ inferred in a mixed model from Pa$\\beta$ and the bolometric IR luminosity (Paper I). L$_{\\text{X,intr}}$ comprises the contribution from both star-formation (as explained in Section \\ref{AGN_identification} and from the AGN, assuming L$_{\\text{X,intr,AGN}}=0.04\\times$ L$_{\\text{BOL,AGN}}$ \\citep{vasudevan07} and a bolometric AGN luminosity L$_{\\text{BOL,AGN}}$ $=1.5\\times$ L$_{AGN,IR}$ \\citep{elvis94} (Fig. \\ref{distltg}-\\textit{center}). \nAlternatively, we considered the bolometric correction of \\citet{lusso12} for type-2 AGNs, which depends on L$_{\\text{BOL,AGN}}$ itself through the following, nonlinear equation:\n\\begin{equation}\n    \\text{log}_\\text{10}\\left(\\frac{\\text{L}_\\text{BOL}}{\\text{L}_\\text{2-10keV}}\\right)=0.217x-0.022x^2-0.027x^3+1.289\n\\end{equation}\n, where x $=$ log$_{10}$(L$_\\text{BOL}$) $-12$ and the scatter of the relation is $0.26$ dex.\nHowever, the results derived with this second assumption (Fig. \\ref{distltg}-\\textit{left}) do not change significantly compared to the first case.\n\nThe figures presented above confirm that the total attenuation inferred from the L$_\\text{X}$ may be the discriminating parameter between X-ray detected and undetected mid-IR AGNs: while the former are relatively unobscured, the X-ray emission from the second is suppressed at least by a factor of $30$. \nInterestingly, this transition does not seem to be related to an increased bolometric luminosity, since no correlations are observed with this quantity.\nWe also remind that for this analysis we are following the standard procedure, which does not take into account shock contribution to the X-ray luminosity. A possible non negligible shock emission at these high energies (which in any case is difficult to model) would result in an underestimation of the true effective X-ray attenuation towards the AGN.\n\n\n\nThe previous results suggest that the X-ray attenuation decreases as the starburst becomes more dust-obscured (probed by A$_\\text{V,tot}$) during the last merger phases. In a standard framework (i.e., if we exclude dominant contribution from shocks to the X-ray luminosity), this apparent contradiction can be reconciled by considering the different timescales of our diagnostics.\nOn the one hand, Pa$\\beta$ (used to calculate A$_\\text{V,tot}$), yields a luminosity (or, equivalently, a SFR, by applying the \\citet{kennicutt94} conversion) that is averaged over a timescale of $20$-$30$ Myr. Conversely, the AGN luminosity that we measure in X-rays gives an istantaneous information of the AGN activity.\nAs a consequence, with the X-ray analysis we are able to probe the current dust attenuation level, while A$_\\text{V,tot}$ traces the obscuration in the recent past ($\\lesssim 30 Myr$). According to this speculation, the X-ray luminosities measured for a subset of $6$ late stage mergers indicate that the AGN-induced blow-out may have already started since a few Myr ago, clearing the surrounding gas and dust content, and that we might be very close to the final QSO phase.\n\n\nFurthermore, in Fig.\\ref{distltg2} we display the AGN accretion efficiencies of our galaxies as a function of their distance from the LTGs Mass-size relation.\nThe efficiencies were estimated by comparing the observed L$_{\\text{BOL,AGN}}$\/M$_\\ast$ ratios to the maximum value allowed by Eddington (L$_{\\text{BOL,AGN}}$\/M$_\\ast$|$_{EDD}\\simeq$1.5), from which we derived the so-called Eddington ratio (L\/L$_\\text{EDD}$)|$_{AGN}$.\nWe assumed the typical correlation for AGNs between the stellar mass M$_\\ast$ and black hole mass M$_{BH}$ of \\citet{reines15} and a spherically symmetric accreting BH, yielding log(L$_{EDD}$\/M$_\\ast$) $\\simeq$ 0.9685+0.05 log$_{10}$(M$_\\ast$). The M$_\\ast$ in the second term can be approximated with the median value of the sample M$_{\\ast,\\text{median}}$, leaving a small secondary dependence on stellar mass which, for our mass ranges (10$^{10}$-10$^{11}$ M$_\\odot$) produces variations of $<5\\%$. This variation, added to the uncertainty on the relation between M$_\\ast$ and M$_\\text{BH}$ reported by \\citet{reines15} (cfr. their equations 4 and 5), is highlighted in Fig.\\ref{distltg2} with a blue shaded area around the Eddington limit (blue line) calculated above. \n\nFrom this analysis, we found that $2$ AGNs have an Eddington ratio higher than 1 ($1.2<$ (L\/L$_\\text{EDD}$)|$_{AGN}$ $<1.85$), while additional $3$ AGNs are radiating between $57\\%$ and $79\\%$ of their maximum luminosity. However, all these $5$ AGNs are still consistent within $2\\sigma$ errors with (L\/L$_\\text{EDD}$)|$_{AGN}=1$ if we also consider the uncertainty on the conversion factor from the ratio L$_{\\text{BOL,AGN}}$\/M$_\\ast$, as discussed before. We remark that additional uncertainties on the M$_\\ast$- M$_\\text{BH}$ relation can come from the assumptions on the BH accretion geometry, which is not taken into account here.\nThe remaining 7 IR-detected AGNs have instead lower Eddington ratios between $0.35$ and $0.08$.\n\nThe galaxies which are undetected in X, radio and mid-IR may contain low-active AGNs, even though current upper limits on the Eddington ratio are not so stringent and do not allow to discriminate them from the detected subset.\nThe intrinsic variability of AGN accretion may thus explain why we are currently missing many of these sources in our sample, and that only deeper X-ray observations can potentially reveal. The duty cycle above $30\\%$ and $1\\%$ L$_{\\text{EDD}}$ seems to be at least $\\sim 25\\%$ and $\\sim 50\\%$, respectively.\n\n\\section{Summary and conclusions}\n\nUsing our unique sample of 25 starburst galaxies (typically 7 times above the star-forming Main Sequence) at z=0.5-0.9 with near-IR rest frame spectroscopy of Paschen lines, we found in Paper I that they span a large range of attenuations toward the core centers from A$_V=2$ to A$_V=30$, forming a sequence which is consistent with a mixed model geometry of dust and stars. In this paper we have investigated the nature of this attenuation sequence, comparing A$_V$ with other physical properties, such as the radio size (which traces the extension of the starburst), the emission lines velocity widths and [\\ion{N}{II}]\/H$\\alpha$ ratios (which reflect the increasing potential well depth and likely shock contribution towards the final merger stages), and finally the EW of hydrogen absorption lines, which is sensitive to the luminosity-weighted age of the stellar populations surrounding the optically obscured core.\n\n\nWe summarize the main results of this paper as follows:\n\\begin{itemize}\n    \\item We found that the physical quantities introduced above, namely the radio sizes (FWHM$_{\\text{radio}}$), the line velocity widths (FWHM$_{\\text{line}}$), the [\\ion{N}{2}]\/H$\\alpha$ ratios (N2) and the equivalent widths of Paschen\/Balmer lines (EW$_{Balmer,Paschen}$), all correlate each other (Fig.\\ref{correlations}), defining a one-parameter sequence of z$\\sim$0.7 starburst galaxies.\n    \\item These correlations can be interpreted as a time-evolutionary sequence of merger stages. As the merger evolves, the starburst becomes more compact and dust obscured, while the deep potential wells created by merging nuclei produce, according to the Virial Theorem, an increase of the kinetic energy and shocks in the system. At the same time, intermediate aged A-type stars in the outer starburst core regions are primarily responsible for the stronger optical+near-IR absorption lines in later phases.\n   \n    \\item 4 galaxies are outliers simultaneously in 3 of the 10 main correlations, which involve A$_\\text{V,tot}$ and, respectively, N2, FWHM$_\\text{line}$ and EW(Pa$\\beta$). Having the largest dust attenuations and among the smallest radio sizes in our sample, these outliers may represent the very end phases of the merger evolution, where the above 3 relations may reach a saturation level. \n    \\item Using sky-subtracted 2D spectra, we identified a subset of $7$ pre-coalescence mergers by the presence of spatially separated or kinematically detached (i.e., rotation-driven tilted lines with different inclination angles) H$\\alpha$ components, representing earlier, less obscured phases of the interaction. The radio sizes measured for these systems are likely tracing the separation between the merging nuclei rather than the dimensions of single cores. However, our sample may contain additional double nuclei which we are not currently able to resolve.\n    \\item Half of our sample comprises extremely compact starbursts, with average half-light radii of $600$ pc ($6$ galaxies have only upper limits), similar to the sizes of starbursting cores observed in local ULIRGs. These sizes are also $\\sim 0.5$ dex smaller than ETGs at redshift $\\sim$0.7 and below, indicating that our merger-driven starbursts cannot be direct progenitors of the population of massive ellipticals formed in the last $\\sim 7$ Gyr. On the contrary, they are more consistent with typical sizes and masses of bulge structures \\citep{graham08}, suggesting a possible evolutionary connection between our starburst cores and bulges.\n    \\item In our sample, we detected at $>3\\sigma$ the mid-IR dusty torus AGN emission in $12$ starbursts, with Eddington ratios ranging from $1.9$ to less than $0.08$. Among them, only 6 galaxies are simultaneously detected (at $3\\sigma$) in X-rays.\n    Intriguingly, the latter have the largest departures from the Mass-size relation of LTGs (at $z\\leq$ 0.7), suggesting that AGNs start to appear in X-rays during the latest (compact) merger phases, as the blow-out of surrounding dust\/gas may precede a possible final QSO. \n\\end{itemize}\nOverall, the relations among the above physical parameters converge toward a time-evolutionary sequence of merger stages, which represents an observational evidence (translated at higher redshift) of the theoretical merger-induced starbursts framework of \\citet{hopkins08a,hopkins08c,dimatteo05}, and the evolutionary sequence postulated by \\citet{toomre72}. The future advent of JWST will allow to test this scenario up to very high redshift, where the conditions of the Universe and gas content of galaxies were even different compared to the epochs studied here.\n\n\\smallskip\n\n\\begin{acknowledgements}\nWe thank the anonymous referee for useful suggestions that improved the quality of this manuscript, G.Rudie for assistence with Magellan observations, Nicol{\\'a}s Ignacio Godoy for data reduction, and Daniela Calzetti for discussions. M.O. acknowledges support from JSPS KAKENHI Grant Number JP17K14257. N.A. acknowledges support from the Brain Pool Program, funded by the Ministry of Science and ICT through the Korean National Research Foundation (2018H1D3A2000902). M.B. acknowledges FONDECYT regular grant 1170618. R.C. acknowledges financial support from CONICYT Doctorado Nacional No. 21161487 and CONICYT PIA. ACT172033. A.C. acknowledges RadioNet conference funding. This research has made use of the zCosmos database, operated at CeSAM\/LAM, Marseille, France.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}