diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhiyd" "b/data_all_eng_slimpj/shuffled/split2/finalzzhiyd" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhiyd" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\subsection{Summary of Results}\n \nThis paper proposes an approach to the Crepant Resolution Conjecture for open Gromov-Witten invariants, and supports it with a series of results and verifications about threefold $A_n$-singularities and their resolutions.\\\\\n\nLet $\\mathcal{Z}$ be a smooth toric Calabi--Yau Deligne--Mumford stack with\ngenerically trivial stabilizers and let $L$ be an Aganagic-Vafa brane\n(Sec. \\ref{sec:ogw}). Fix a Calabi--Yau torus action $T$ on $\\mathcal{Z}$ and\ndenote by $\\Delta_\\mathcal{Z}$ the free module over $H^\\bullet(BT)$ spanned by\nthe $T$-equivariant lifts of orbifold cohomology classes\nof Chen--Ruan degree at most two. We define\n(Sec. \\ref{ssec:dcrc}) a family of elements of Givental space,\n\\beq\n\\widehat{\\mathcal{F}}_{L,\\mathcal{Z}}^{\\rm disk}: H_T^\\bullet(\\mathcal{Z}) \\to \\mathcal{H}_\\mathcal{Z} = H_T^\\bullet(\\mathcal{Z})((z^{-1})),\n\\eeq\nwhich we call the \\textit{winding neutral disk potential}. Upon appropriate\nspecializations of the variable $z$, $\\widehat{\\mathcal{F}}^{\\rm disk}_{L,\\mathcal{Z}}$ encodes disk\ninvariants of $(\\mathcal{Z},L)$ at any winding $d$. \\\\\n\nConsider a \\textit{crepant\n resolution diagram} $\\mathcal{X} \\to X \\leftarrow Y$, where $X$ is the coarse moduli\nspace of $\\mathcal{X}$ and $Y$ is a crepant resolution of the singularities of $X$. A\nLagrangian boundary condition $L$ is chosen on $\\mathcal{X}$ and we denote by $L'$\nits transform in $Y$. \nOur\nversion of the open crepant resolution conjecture is a comparison of the (restricted) winding neutral disk potentials.\n\n\\begin{proposal}[The OCRC]\nThere exists a $\\mathbb{C}((z^{-1}))$-linear map of Givental spaces\n$\\mathbb{O}: \\mathcal{H}_\\mathcal{X} \\to \\mathcal{H}_Y$ and analytic functions $\\mathfrak{h}_\\mathcal{X}: \\Delta_\\mathcal{X}\n\\to \\mathbb{C}$, $\\mathfrak{h}_Y: \\Delta_Y \\to \\mathbb{C}$ such that\n\\beq\n\\mathfrak{h}_Y^{1\/z}{\\widehat \\mathcal{F}_{L,Y}^{\\rm disk}}\\big|_{\\Delta_Y}= \\mathfrak{h}_\\mathcal{X}^{1\/z} \\mathbb{O}\\circ \\widehat \\mathcal{F}_{L,\\mathcal{X}}^{\\rm disk}\\big|_{\\Delta_\\mathcal{X}}\n\\eeq\nupon analytic continuation of quantum cohomology parameters.\n\\end{proposal}\n\nFurther, we conjecture (Conjecture \\ref{conj:iri}) that both $\\mathbb{O}$ and\n$\\mathfrak{h}_\\bullet$ are completely determined\nby the classical toric geometry of $\\mathcal{X}$ and $Y$. In particular, we give a \nprediction for the transformation $\\mathbb{O}$ depending on a choice of identification of the\n$K$-theory lattices of $\\mathcal{X}$ and $Y$. \\\\\n\nWhen $\\mathcal{X}$ is a Hard Lefschetz\nCalabi--Yau orbifold, the OCRC extends to functions on all of $H_T^\\bullet(\\mathcal{Z})$.\nTogether with WDVV, this gives a Bryan--Graber-type statement for potentials encoding invariants from genus $0$ maps with an arbitrary number of boundary components:\n\n\\begin{prop1}\nLet $\\mathcal{X} \\rightarrow X \\leftarrow Y$ be a Hard Lefschetz diagram for which the OCRC holds. Defining $\\mathbb{O}^{\\otimes n}= \\mathbb{O}(z_1)\\otimes\\ldots\\otimes \\mathbb{O}(z_n)$, we have:\n\\beq\n{\\widehat \\mathcal{F}_{L',Y}^{n}}= \\mathbb{O}^{\\otimes n}\\circ \\widehat \\mathcal{F}_{L,\\mathcal{X}}^{n},\n\\eeq\nwhere $\\widehat \\mathcal{F}^{n}$ is the $n$-boundary components analog of $\\widehat\n\\mathcal{F}^{\\rm disk}$ defined in \\eqref{mholepot}. \\\\\n\\end{prop1}\n\n\nConsider now the family of threefold $A_n$ singularities, where $\\mathcal{X}=[\\mathbb{C}^2\/\\mathbb{Z}_{n+1}]\\times \\mathbb{C}$ and $Y$ is its canonical minimal\nresolution. \n\\begin{mt1}\nThe OCRC and Conjecture \\ref{conj:iri} hold for the $A_n$-singularities for any choice of Aganagic-Vafa brane on $\\mathcal{X}$.\n\\end{mt1}\nThe main theorem is an immediate consequence of Proposition \\ref{prop:wncrc}\nand Theorem \\ref{thm:sympl}. From it we deduce a series of comparisons of\ngenerating functions in the spirit of Bryan-Graber's formulation of the CRC. \\\\\n\nIn \\eqref{cohdp} we define the \\textit{cohomological disk potential}\n$\\mathcal{F}_{L}^{\\rm disk}$ - a cohomology valued generating function for disk\ninvariants that ``remembers\" the twisting and the attaching fixed point of an\norbi-disk map. We also consider the coarser \\textit{scalar disk potential}\n(see \\eqref{sdp}), which keeps track of the winding of the orbimaps but\nforgets the twisting and attaching point.\nThere are essentially two different choices for the Lagrangian boundary condition on $\\mathcal{X}$; the simpler case occurs when $L$ intersects one of the effective legs of the orbifold. In this case we have the following result.\n\\begin{cocrceff}\nIdentifying identically the winding parameters and setting $\\mathbb{O}_\\mathbb{Z}(\\mathbf{1_k})=P_{n+1}$ for every $k$, we have:\n\\beq\n\\mathcal{F}_{L',Y}^{\\rm disk}(t,y,\\vec{w}) = \\mathbb{O}_\\mathbb{Z} \\circ \\mathcal{F}_{L,\\mathcal{X}}^{\\rm\n disk}(t,y,\\vec{w}). \n \\eeq\n\\end{cocrceff}\nIt is immediate to observe that the scalar disk potentials coincide (Corollary \\ref{cor:esc}).\\\\\n\nThe case when $L$ intersects the ineffective leg of the orbifold is more subtle.\n\\begin{cocrcgerby}\nWe exhibit a matrix $\\mathbb{O}_\\mathbb{Z}$ of roots of unity and a specialization of the winding parameters depending on the equivariant weights such that\n\\beq\n\\mathcal{F}_{L',Y}^{\\rm disk}(t,y,\\vec{w}) = \\mathbb{O}_\\mathbb{Z} \\circ \\mathcal{F}_{L,\\mathcal{X}}^{\\rm disk}(t,y,\\vec{w}) .\n\\eeq\n\\end{cocrcgerby}\nThe comparison of scalar potentials in this case does not hold\nanymore. Because of the special form of the matrix $\\mathbb{O}_\\mathbb{Z}$ we deduce in\nCorollary \\ref{cor:sc} that the scalar disk potential for $Y$ corresponds to\nthe contribution to the potential for $\\mathcal{X}$ by the untwisted disk maps. \nAs the $A_n$-singularities satisfy the Hard Lefschetz condition, it is\nan exercise in book-keeping to extend the statements of Theorems \\ref{thm:dcrccoh}\nand \\ref{thm:dcrccoheff} to compare generating functions for arbitrary genus\nzero open invariants, even treating all boundary Lagrangian conditions at the\nsame time. \\\\\n\nIn order to prove our main theorem, we must establish a fully equivariant\nversion of the symplectomorphism of Givental spaces which verifies the closed\nCRC for the $A_n$ geometries. Our analysis is centered on a new global\ndescription of the gravitational quantum cohomology of these targets which enjoys a number of remarkable features, and\nmay have an independent interest {\\it per se}.\n\n\n\\begin{thmmir}\nBy identifying the $A$-model moduli space with a genus zero double Hurwitz space, we construct a global quantum $D$-module $(\\mathcal{F}_{\\lambda,\\phi}, T\\mathcal{F}_{\\lambda,\\phi}, \\nabla^{(g,z)},H(,)_{g})$ which is locally isomorphic to $\\mathrm{QDM}(\\mathcal{X})$ and $\\mathrm{QDM}(Y)$ in appropriate neighborhoods of the orbifold and large complex structure points.\n\\end{thmmir} \n \n\n\\subsection{Context, Motivation and Further Discussion}\n\nOpen Gromov-Witten (GW) theory \nintends to study holomorphic maps from bordered Riemann surfaces, where the image of the boundary is constrained\n to lie in a Lagrangian submanifold of the target. While some general foundational\n work has been done \\cite{Solomon:2006dx, MR2425184}, at this point most\n of the results in the theory rely on additional structure. In \\cite{hht1, hht2} Lagrangian Floer theory is employed to study the case when the boundary condition is a fiber of the moment map.\nIn the toric context, a mathematical approach\n\\cite{Katz:2001vm, Diaconescu:2003qa, MR2861610,r:lgoa} to construct operatively\na virtual counting theory of open maps is via the use of localization. \nA variety of striking relations have been verified connecting open GW theory and several other types of invariants,\nincluding open $B$-model invariants and matrix models \\cite{Aganagic:2000gs,\n Aganagic:2001nx, Lerche:2001cw, Bouchard:2007ys, fang2012open}, quantum knot invariants\n\\cite{Gopakumar:1998ki, Marino:2001re}, and ordinary\nGromov--Witten and Donaldson--Thomas theory via ``gluing along the boundary'' \\cite{Aganagic:2003db,\n Li:2004uf, moop}.\\\\\n\nSince Ruan's influential conjecture \\cite{MR2234886}, an intensely studied\nproblem in Gromov--Witten theory has been to determine the relation between GW invariants of target spaces\nrelated by a crepant birational transformation (CRC). The most general\nformulation of the CRC is framed in terms of Givental formalism\n(\\cite{MR2529944}, \\cite[Conj 4.1]{coates2007quantum}); the conjecture has been proved in\na number of examples \\cite{MR2510741, MR2529944, MR2486673} and has by now gained folklore status, with\na general proof in the toric setting announced for some time \\cite{ccit2}. A natural question one can ask is whether\nsimilar relations exist in the context of open Gromov--Witten theory. Within\nthe toric realm, physics arguments based on open mirror symmetry\n\\cite{Bouchard:2007ys, Bouchard:2008gu, Brini:2008rh} have given strong indications that\nsome version of the Bryan--Graber \\cite{MR2483931} statement of the crepant\nresolution conjecture should hold at the level of disk invariants. This was\nproven explicitly for the crepant resolution of the Calabi--Yau orbifold $[\\mathbb{C}^3\/\\mathbb{Z}_2]$\nin \\cite{cavalieri2011open}. \nAround the same time, it was suggested \n\\cite{Brini:2011ij, talk-banff} that a general statement of a Crepant Resolution\nConjecture for open invariants should have a natural formulation within\nGivental's formalism, as in \\cite{MR2510741, coates2007quantum}. Some implications of this\nphilosophy were verified in\n\\cite{Brini:2011ij} for the crepant resolution $\\mathcal{O}_{\\mathbb{P}^2}(-3)$ of the orbifold\n$[\\mathbb{C}^3\/\\mathbb{Z}_3]$. \\\\\n\nThe OCRC we propose here is a natural\nextension to open Gromov--Witten theory of the Coates--Corti--Iritani--Tseng\napproach \\cite{MR2529944} to Ruan's conjecture.\nThe observation that the disk function of \\cite{MR2861610,r:lgoa} can be interpreted as an endomorphism of Givental space makes the OCRC statement follow almost tautologically from the Coates--Corti--Iritani--Tseng\/Ruan picture of the ordinary \nCRC via toric mirror symmetry \\cite{MR2510741}. \nThe more striking aspect of our conjecture is then that the linear function $\\mathbb{O}$ comparing the winding neutral disk potentials is considerably simpler than the symplectomorphism $\\mathbb{U}_{\\rho}^{\\X,Y}$ in the closed CRC and it is characterized in terms of {\\it purely classical data}: essentially, the equivariant Chern characters of $\\mathcal{X}$ and $Y$. This is closely related to Iritani's proposal \\cite{MR2553377} that the analytic continuation for the flat sections of the global quantum $D$-module is realized via the composition of $K$-theoretic central charges; our disk endomorphisms are very close to just being inverses to the $\\Gamma$ factors appearing in Iritani's central charges and therefore ``undo\" most of the transcendentality of $\\mathbb{U}_{\\rho}^{\\X,Y}$. \\\\\n\nIritani's proposal is inspired and consistent with the idea of global mirror\nsymmetry, i.e. that there should be a global quantum $D$-module on the\n$A$-model moduli space which locally agrees with the Frobenius structure given\nby quantum cohomology. In order to verify Iritani's proposal in the fully\nequivariant setting, we construct explicitly such a global structure. Motivated by the connection of the Gromov--Witten theory of $A_n$ to certain integrable systems \\cite{agps}, we realize the Dubrovin\nlocal system as a system of one-dimensional hypergeometric periods. As a\nspecial feature of this case, structure constants of quantum cohomology\nare rational in exponentiated flat coordinates (or, equivalently, the inverse\nmirror map is a rational function of the $B$-model variables). Moreover, the $n$-dimensional\noscillating integrals describing the periods of the system reduce to Euler--Pochhammer line integrals in the\ncomplex plane. As a consequence, the computation of the analytic\ncontinuation of flat sections is drastically\nsimplified with respect to the standard toric mirror symmetry methods. Furthermore, in this context integral structures in\nequivariant cohomology emerge naturally from the interpretation of flat\nsections of the Dubrovin connection as twisted period maps. The\nDeligne--Mostow monodromy of hypergeometric periods translates then to an\naction of the colored braid group \nin equivariant\n$K$-theory. \nAn enticing speculation is that, upon mirror symmetry, this may correspond to\nautoequivalences of $D_T^b(Y)$ and surject to the\nSeidel--Thomas braid group action \\cite{MR1831820} in the\nnon-equivariant limit.\n\n\\begin{comment}\n Indeed, in\n\\cite{MR1831820} the authors famously constructed a faithful\nrepresentation of the braid group $B_{n+1}$ in terms of derived equivalences of $D^b(Y)$\ninduced by spherical twists. It is natural to speculate that our monodromy\ndescription of the global quantum $D$-module in Section~\\ref{sec:monodromy}\nsurjects to the Seidel--Thomas braid group, and recovers it in the non-equivariant limit.\n\\end{comment}\n\n\n\\subsection*{Acknowledgements} We are particularly grateful to Tom Coates for\nhis collaboration at the initial stages of this project, and the many\nenlightening conversations that followed. We would also like to thank Hiroshi\nIritani, Yunfeng Jiang, \\'Etienne Mann, Stefano Romano and Ed Segal for useful\ndiscussions and\/or\ncorrespondence. This project originated from discussions at the Banff Workshop on ``New recursion\nformulae and integrability for Calabi--Yau manifolds'', October 2011; we are\ngrateful to the organizers for the kind invitation and the great scientific\natmosphere at BIRS. A.~B.~has been supported by a Marie Curie Intra-European Fellowship\nunder Project n$^\\circ$ 274345 (GROWINT). R.~C.~ has been supported by NSF grant DMS-1101549. Partial support from the GNFM-INdAM under the\nProject ``Geometria e fisica dei sistemi integrabili'' is also acknowledged. \\\\\n\n\n\\section{Background}\n\nThis section gathers background for the formulation of\nthe open string Crepant Resolution Conjecture of\nSection~\\ref{sec:ocrc} and its proof in Section~\\ref{sec:j}. We give a self-contained account of the\nquantum $D$-module\/Givental space approach to the study of the closed string\nCrepant Resolution Conjecture in genus zero along the lines of\nCoates--Corti--Iritani--Tseng \\cite{MR2510741} and Iritani\n\\cite{MR2553377} (Section~\\ref{sec:qdm}). Section~\\ref{sec:ogw} provides an overview of open Gromov--Witten theory for\ntoric Calabi--Yau threefolds \\`a la Katz--Liu as well as its extension to toric\norbifolds. Section~\\ref{sec:an} collects relevant material on\nthe classical and quantum geometry of $A_n$-resolutions. \\\\\n\nThe content of Section~\\ref{sec:qdm} is surveyed in Iritani's\nexcellent review article \\cite{MR2683208}, to which the reader is referred for\nfurther details. For a more comprehensive introduction to the open\nGromov--Witten theory for toric orbifolds, see e.g. \\cite{MR2861610, r:lgoa}.\n\n\n\\subsection[Quantum $D$-modules and the CRC]{Quantum $D$-modules and the Crepant Resolution Conjecture}\n\\label{sec:qdm}\n\n\n\n\nLet $\\mathcal{Z}$ be a smooth Deligne--Mumford stack with coarse moduli\nspace $Z$ and suppose that $\\mathcal{Z}$ carries an algebraic $T\\simeq\\mathbb{C}^*$ action with\nzero-dimensional \nfixed loci. Write $I\\mathcal{Z}$ for the inertia stack of $\\mathcal{Z}$, \n$\\mathrm{inv}:I\\mathcal{Z}\\to I\\mathcal{Z}$ for its canonical involution and\n$i:I\\mathcal{Z}^T\\hookrightarrow I\\mathcal{Z}$ for\nthe inclusion of the $T$-fixed loci into $I\\mathcal{Z}$. \nThe equivariant Chen--Ruan cohomology ring $H(\\mathcal{Z}) \\triangleq H^{\\bullet}_{T,CR}(\\mathcal{Z})$ of $\\mathcal{Z}$ is a finite rank free module over\nthe $T$-equivariant cohomology of a point $H_T(\\mathrm{pt})\\simeq\n\\mathbb{C}[\\nu]$, where $\\nu=c_1(\\mathcal{O}_{BT}(1))$; we define\n$N_\\mathcal{Z} \\triangleq\\operatorname{rank}_{\\mathbb{C}[\\nu]} H(\\mathcal{Z})$. We furthermore suppose\nthat odd cohomology groups vanish in all degrees. \\\\\n\n\nThe $T$-action on $\\mathcal{Z}$ gives a non-degenerate inner product on\n$H(\\mathcal{Z})$ via the equivariant orbifold Poincar\\'e pairing\n\\beq\n\\eta(\\theta_1,\\theta_2)_{\\mathcal{Z}} \\triangleq \\int_{I\\mathcal{Z}^T}\\frac{i^*(\\theta_1 \\cup \\mathrm{inv}^*\n \\theta_2)}{e(N_{I\\mathcal{Z}^T\/I\\mathcal{Z}})},\n\\label{eq:pair}\n\\eeq\nand it induces a torus action on the moduli\nspace $\\overline{\\mathcal M_{g,n}}(\\mathcal{Z}, \\beta)$ of degree $\\beta$ twisted stable maps\n\\cite{MR2450211, MR1950941} from genus $g$ orbicurves to $\\mathcal{Z}$. For classes $\\theta_1, \\dots, \\theta_n\\in H(\\mathcal{Z})$ and\nintegers $r_1, \\dots, r_n \\in \\mathbb{N}$, the Gromov--Witten\ninvariants of $\\mathcal{Z}$\n\\bea\n\\left\\langle \\sigma_{r_1}(\\theta_1) \\dots \\sigma_{r_n}(\\theta_n) \\right\\rangle_{g,n,\\beta}^\\mathcal{Z}\n& \\triangleq & \\int_{[\\overline{\\mathcal M_{g,n}}(\\mathcal{Z},\n \\beta)]_T^{\\rm vir}} \\prod_{i=1}^n \\mathrm{\\operatorname{ev}}^*_i \\theta_i\n\\psi_i^{r_i}, \\label{eq:gwdesc} \\\\\n\\left\\langle \\theta_1 \\dots \\theta_n \\right\\rangle_{g,n,\\beta}^\\mathcal{Z} & \\triangleq & \\left\\langle \\sigma_{0}(\\theta_1) \\dots\n\\sigma_{0}(\\theta_n) \\right\\rangle_{g,n,\\beta}^\\mathcal{Z}, \n\\label{eq:gwprim}\n\\end{eqnarray}\ndefine a sequence of multi-linear functions on $H(\\mathcal{Z})$ with values in the\nfield of fractions $\\mathbb{C}(\\nu)$ of $H_T({\\rm pt})$. The correlators \\eqref{eq:gwprim}\n(respectively, \\eqref{eq:gwdesc} with $r_i>0$) are the {\\it\n primary} (respectively, {\\it descendent}) Gromov--Witten invariants of\n$\\mathcal{Z}$. \\\\\n\nFix a basis\n$\\{\\phi_i\\}_{i=0}^{N_\\mathcal{Z}-1}$ of $H(\\mathcal{Z})$ such that $\\phi_0=\\mathbf{1}_\\mathcal{Z}$\nand $\\phi_j$, $1\\leq j \\leq b_2(Z)$ are untwisted Poincar\\'e duals of $T$-equivariant divisors\nin $Z$. Denote by $\\{\\phi^i\\}_{i=0}^{N_\\mathcal{Z}-1}$ the dual basis with respect to the pairing \\eqref{eq:pair}. Let $\\tau=\\sum\\tau_i\\phi_i$ denote a general point of $H(\\mathcal{Z})$. The WDVV equation for primary Gromov--Witten invariants \\eqref{eq:gwprim} defines a family of associative\ndeformations $\\circ_\\tau$ of the $T$-equivariant Chen--Ruan cohomology ring of $\\mathcal{Z}$ via\n\\beq\n\\eta\\l(\\theta_1 \\circ_\\tau \\theta_2, \\theta_3\\r)_{\\mathcal{Z}} \\triangleq \\left\\langle\\bra \\theta_1, \\theta_2, \\theta_3 \\right\\rangle\\ket_{0,3}^\\mathcal{Z}(\\tau)\n\\eeq\nwhere\n\\beq\n\\left\\langle\\bra \\theta_1, \\dots, \\theta_k \\right\\rangle\\ket_{0,k}^\\mathcal{Z}(\\tau) \\triangleq \\sum_{\\beta}\\sum_{n\\geq 0} \\frac{\\big\\langle \\theta_1,\\dots,\\theta_k,\n \\overbrace{\\tau,\\tau,\\ldots,\\tau}^{\\text{$n$\n times}} \\big\\rangle_{0,n+k,\\beta}^\\mathcal{Z}}{n!} \\in \\mathbb{C}((\\nu)) ,\n\\eeq\nand the index $\\beta$ ranges over the cone of effective curve classes\n$\\mathrm{Eff}(\\mathcal{Z}) \\subset H_2(Z, \\mathbb{Q})$; we denote by $l_\\mathcal{Z} \\triangleq b_2(Z)$\nits dimension. \\\\\n\n\nBy the Divisor Axiom \\cite{MR2450211} this can be rewritten as\n\\beq\n\\eta\\l(\\theta_1 \\circ_\\tau \\theta_2, \\theta_3\\r)_{\\mathcal{Z}}= \\sum_{\\beta\\in \\mathrm{Eff}(\\mathcal{Z}), n\\geq 0} \\frac{\\big\\langle \\theta_1,\\theta_2,\\theta_3,\n \\overbrace{\\tau',\\tau',\\ldots,\\tau'}^{\\text{$n$\n times}} \\big\\rangle_{0,n+3,\\beta}^\\mathcal{Z}}{n!}\\mathrm{e}^{\\tau_{0,2} \\cdot \\beta}\n\\label{eq:qprod2}\n\\eeq\nwhere we have decomposed $\\tau=\\sum_{i=0}^{N_\\mathcal{Z}-1} \\tau_i \\phi_i = \\tau_{0,2}+\\tau'$ as\n\\bea\n\\tau_{0,2} &=& \\sum_{i=1}^{l_\\mathcal{Z}} \\tau_{i} \\phi_{i}, \\\\\n\\tau' &=& \\tau_0 \\mathbf{1}_\\mathcal{Z} + \\sum_{i=l_\\mathcal{Z}+1}^{N_\\mathcal{Z}-1} \\tau_i \\phi_i.\n\\label{eq:Tprime}\n\\end{eqnarray}\n\nThe quantum product \\eqref{eq:qprod2} is a formal Taylor series in $(\\tau',\n\\mathrm{e}^{\\tau_{0,2}})$. Suppose that it is actually {\\it convergent} in a contractible\nopen set $U \\ni (0,0)$; this is the case for many toric orbifolds\n\\cite{MR1653024, Coates:2012vs} and,\nas we see explicitly, for\nall the examples of Section~\\ref{sec:an}. Then the quantum product $\\circ_\\tau$ is an\nanalytic deformation of the Chen--Ruan cup product $\\cup_{\\rm CR}$, to which\nit reduces in the limit $\\tau' \\to 0$, $\\mathfrak{Re}(\\tau_{0,2}) \\to -\\infty$. Thus, the holomorphic\nfamily of rings $H(\\mathcal{Z}) \\times U \\to U$, together with the inner pairing \\eqref{eq:pair} and the\nassociative product \\eqref{eq:qprod2}, gives $U$ the structure of a\n(non-conformal) Frobenius manifold $QH(\\mathcal{Z})\\triangleq(U, \\eta, \\circ_\\tau)$\n\\cite{Dubrovin:1994hc}; this is the {\\it quantum cohomology ring} of $\\mathcal{Z}$. We refer to the Chen--Ruan limit $\\tau' \\to 0$, $\\mathfrak{Re}(\\tau_{0,2})\n\\to -\\infty$ as the {\\it large radius limit point} of $\\mathcal{Z}$. \\\\\n\n\nAssigning a Frobenius structure on $U$ is tantamount to endowing the trivial\ncohomology bundle $TU \\simeq H(\\mathcal{Z}) \\times U \\to U$ with a flat\npencil of affine connections \\cite[Lecture\n 6]{Dubrovin:1994hc}. Denote by $\\nabla^{(\\eta)}$ the Levi--Civita connection\nassociated to the Poincar\\'e pairing on $H(\\mathcal{Z})$; in Cartesian coordinates\nfor $U\\subset H(\\mathcal{Z})$ this reduces to the ordinary de Rham differential\n$\\nabla^{(\\eta)}=d$. Consider then the one parameter family of covariant\nderivatives on $TU$\n\\beq\n\\nabla^{(\\eta,z)}_X \\triangleq \\nabla^{(\\eta)}_X+z^{-1} X \\circ_\\tau.\n\\label{eq:defconn1}\n\\eeq\nThe fact that the quantum product is commutative, associative and integrable implies that\n$R_{\\nabla^{(\\eta,z)}}=T_{\\nabla^{(\\eta,z)}}=0$ identically in $z$; this is equivalent to the WDVV\nequations for the genus zero Gromov--Witten potential. The equation for the horizontal\nsections of $\\nabla^{(\\eta,z)}$,\n\\beq\n\\nabla^{(\\eta,z)} \\omega =0,\n\\label{eq:QDE}\n\\eeq\nis a rank-$N_\\mathcal{Z}$ holonomic system of\ncoupled linear PDEs. We denote by $\\mathcal{S}_\\mathcal{Z}$ the vector space of solutions\nof \\eqref{eq:QDE}: a $\\mathbb{C}((z))$-basis of $\\mathcal{S}_\\mathcal{Z}$ is by definition given by the gradient of\na flat frame $\\tilde \\tau (\\tau,z)$ for the deformed connection\n$\\nabla^{(\\eta,z)}$. The Poincar\\'e\npairing induces a non-degenerate inner product $H(s_1,s_2)_{\\mathcal{Z}}$ on $\\mathcal{S}_\\mathcal{Z}$ via\n\\beq\nH(s_1, s_2)_\\mathcal{Z} \\triangleq \\eta(s_1(\\tau, -z),s_2(\\tau,z))_\\mathcal{Z}.\n\\label{eq:pairDmod}\n\\eeq\nThe triple $\\mathrm{QDM}(\\mathcal{Z})\\triangleq(U,\\nabla^{(\\eta,z)}, H(,)_\\mathcal{Z})$ defines a {\\it\n quantum D-module} structure on $U$, and the system \\eqref{eq:QDE} is the {\\it quantum differential\n equation} (in short, QDE) of $\\mathcal{Z}$. \n\\begin{rmk}\n\\label{rmk:fuchsLR}\nNotice that the assumption that the quantum product\n \\eqref{eq:qprod2} is analytic in $(\\tau',\\mathrm{e}^{\\tau_{0,2}})$ around the large radius\n limit point translates into the statement that the QDE \\eqref{eq:QDE} has a\n Fuchsian singularity along $\\cup_{i=1}^{l_\\mathcal{Z}} \\{q_i\\triangleq\\mathrm{e}^{\\tau_i}=0\\}$. \\\\\n\\end{rmk}\nIn the same way in which the genus zero primary theory of $\\mathcal{Z}$ defines a quantum\n$D$-module structure on $H(\\mathcal{Z}) \\times U$, the genus zero gravitational\ninvariants \\eqref{eq:gwdesc} furnish a basis of horizontal sections\nof $\\nabla^{(\\eta,z)}$ \\cite{MR1408320}. For every $\\theta\\in\nH(\\mathcal{Z})$, a flat section of the $D$-module is given by an\n$\\mathrm{End}(H(\\mathcal{Z}))$-valued function $S_\\mathcal{Z}(\\tau,z):H(\\mathcal{Z})\\to \\mathcal{S}_\\mathcal{Z}$ defined as\n\\beq\nS_\\mathcal{Z}(\\tau,z)\\theta \\triangleq \\theta-\\sum_{k=1}^{N_\\mathcal{Z}}\\phi^k\\left\\langle\\bra\\phi_k,\\frac{\\theta}{z+\\psi}\\right\\rangle\\ket_{0,2}^\\mathcal{Z}(\\tau)\n\\label{eq:fundsol}\n\\eeq\nwhere $\\psi$ is a cotangent line class and we expand the denominator as a\ngeometric series\n$\\frac{1}{z+\\psi}=\\frac{1}{z}\\sum\\left(-\\frac{\\psi}{z}\\right)^k$. We call the\npair $(\\mathrm{QDM}(\\mathcal{Z}), S_\\mathcal{Z})$ a {\\it calibration} of the Frobenius structure\n$(H(\\mathcal{Z}), \\circ_\\tau, \\eta)$. \\\\\n\nThe flows of coordinate vectors for the flat frame of $TH(\\mathcal{Z})$ induced by\n$S_\\mathcal{Z}(\\tau,z)$ give a basis of deformed flat coordinates\n of\n$\\nabla^{(\\eta,z)}$, which is defined uniquely up to an additive $z$-dependent\n constant. A canonical basis is obtained upon applying the String Axiom:\ndefine the {\\it $J$-function} $J^\\mathcal{Z}(\\tau,z):U \\times \\mathbb{C} \\to H(\\mathcal{Z})$ by\n\\beq\nJ^\\mathcal{Z}(\\tau,z) \\triangleq zS_\\mathcal{Z}(\\tau,-z)^*\\mathbf{1}_\\mathcal{Z}\n\\label{eq:Jfun1}\n\\eeq\nwhere $S_\\mathcal{Z}(\\tau,z)^*$ denotes the adjoint to $S_\\mathcal{Z}(\\tau,z)$ under $H(-,-)_\\mathcal{Z}$. Explicitly, \n\\beq\n\\label{eq:resj}\nJ^\\mathcal{Z}(\\tau,z) = (z+\\tau_0)\\mathbf{1}_\\mathcal{Z}+\\tau_1\\phi_1+...+\\tau_{N_\\mathcal{Z}} \\phi_{N_\\mathcal{Z}}+\\sum_{k=1}^{N_\\mathcal{Z}} \\phi^k\\left\\langle\\bra\n\\frac{\\phi_k}{z-\\psi_{n+1}}\\right\\rangle\\ket_{0,1}^\\mathcal{Z}(\\tau).\n\\eeq\nComponents of $J^\\mathcal{Z}(\\tau,z)$ in the $\\phi$-basis give flat coordinates of\n\\eqref{eq:defconn1}; this is a consequence of \\eqref{eq:Jfun1} combined with\nthe String Equation. From \\eqref{eq:resj}, the undeformed flat coordinate system is obtained in the\nlimit $z\\to\\infty$ as\n\\beq\n\\lim_{z\\to \\infty} \\l(J^\\mathcal{Z}(\\tau,z)-z \\mathbf{1}_\\mathcal{Z}\\r) = \\tau.\n\\eeq\n\\\\\n\nBy Remark~\\ref{rmk:fuchsLR}, a loop around the origin in the variables $q_i=\\mathrm{e}^{\\tau_i}$\ngives a non-trivial monodromy action on the $J$-function. Setting $\\tau'=0$ in \\eqref{eq:resj} and applying the Divisor\nAxiom then gives \\cite[Proposition~10.2.3]{MR1677117}\n\\bea\n& J^{\\mathcal{Z}, \\rm small}(\\tau_{0,2},z) \\triangleq J^\\mathcal{Z}(\\tau,z)\\Big|_{\\tau'=0} \\nn \\\\\n=& z \\mathrm{e}^{\\tau_1 \\phi_1\/z}\\dots\\mathrm{e}^{\\tau_{l_\\mathcal{Z}} \\phi_{l_\\mathcal{Z}}\/z}\n\\l(\\mathbf{1}_\\mathcal{Z}+ \\sum_{\\beta,k}\\mathrm{e}^{\\tau_1 \\beta_1}\\dots\\mathrm{e}^{\\tau_{l_\\mathcal{Z}}\\beta_{l_\\mathcal{Z}}}\\phi^k\\left\\langle\n\\frac{\\phi_k}{z(z-\\psi_{1})}\\right\\rangle_{0,1,\\beta}^\\mathcal{Z}\\r).\n\\label{eq:Jred}\n\\end{eqnarray}\nIn our situation\nwhere the $T$-action has only zero-dimensional fixed loci $\\{P_i\\}_{i=1}^{N_\\mathcal{Z}}$, write \n\\beq\n\\phi_i \\to \\sum_{j=1}^{N_\\mathcal{Z}} c_{ij}(\\nu) P_j, \\quad i=1, \\dots, l_\\mathcal{Z},\n\\eeq\nfor the image of $\\{\\phi_i \\in H^2(\\mathcal{Z}, \\mathbb{C})\\}_{i=1}^{l_\\mathcal{Z}}$ under the\nAtiyah--Bott isomorphism.\n \n The image of each $\\phi_i$ is concentrated on the fixed point cohomology classes with trivial isotropy which \n are idempotents of the\nclassical Chen-Ruan cup\n product on $H(\\mathcal{Z})$. Therefore, the components of the $J$-function in the fixed points basis\n\\beq\nJ^{\\mathcal{Z}, \\rm small}(\\tau_{0,2},z) =: \\sum_{j=1}^{N_\\mathcal{Z}} J_j^{\\mathcal{Z}, \\rm small}(\\tau_{0,2},z) P_j\n\\eeq\nsatisfy\n\\beq\nJ_j^{\\mathcal{Z}, \\rm small}(\\tau_{0,2},z) = z \\mathrm{e}^{\\sum_{i=1}^{l_\\mathcal{Z}} \\tau_i\n c_{ij}\/z}\\l(1+\\mathcal{O}\\l(\\mathrm{e}^{\\tau_{0,2}}\\r)\\r)\n\\label{eq:Jloc}\n\\eeq\nwhere the $\\mathcal{O}\\l(\\mathrm{e}^{\\tau_{0,2}}\\r)$ term on the right hand side is an analytic power\nseries around $\\mathrm{e}^{\\tau_{0,2}}=0$ by \\eqref{eq:Jred} and the assumption of convergence\nof the quantum product. The localized basis $\\{P_j\\}_{j=1}^{N_\\mathcal{Z}}$ therefore\ndiagonalizes the monodromy around large radius: by \\eqref{eq:Jloc}, each\n$J_j^{\\mathcal{Z}, \\rm small}(\\tau_{0,2},z)$ is an eigenvector of the monodromy around a loop in the\n$q_i$-plane encircling the large radius\nlimit of $\\mathcal{Z}$ with eigenvalue $\\mathrm{e}^{2\\pi\\mathrm{i} c_{ij}\/z}$.\n\n\n\\subsubsection{Global mirror symmetry and the closed CRC}\n\n\n\n\n \nConsider a toric Gorenstein orbifold $\\mathcal{X}$, and let $X \\leftarrow Y$ be a crepant resolution of its coarse moduli space.\nRuan's Crepant Resolution Conjecture can be phrased as the existence of a {\\it global quantum $D$-module}\nunderlying the quantum differential systems of $\\mathcal{X}$ and $Y$. This is a 4-tuple\n$(\\mathcal M_A, F, \\nabla, H(,)_F)$ with\n\\bit\n\\item $\\mathcal M_A$ a complex quasi-projective variety\n \n \n \n \n\\item $F\\to \\mathcal M_A$ a rank-$N_\\mathcal{Z}$ holomorphic vector bundle on $\\mathcal M_A$; \n\\item $\\nabla$ a flat $\\mathcal{O}_{\\mathcal M_A}$-connection on $F$;\n\\item $H(,)_F \\in \\mathrm{End}(F)$ a non-degenerate $\\nabla$-flat inner product.\n\\end{itemize}\nIn the quantum $D$-module picture, the Crepant Resolution\nConjecture states that there exist open subsets $V_\\mathcal{X}$, $V_Y \\subset \\mathcal M_A$\nand functions $\\mathfrak{h}_\\mathcal{X}, \\mathfrak{h}_Y \\in \\mathcal{O}_{\\mathcal M_A}$\nsuch that \nthe global\n$D$-module $(\\mathcal M_A, F, \\nabla, H(,)_F)$ is locally isomorphic to $\\mathrm{QDM}(\\mathcal{X})$ and\n$\\mathrm{QDM}(Y)$:\n\\bea\n(\\mathcal M_A, F, \\nabla \\circ \\mathfrak{h}_\\mathcal{X}^{1\/z} , H(,)_F)\\big|_{V_\\mathcal{X}} &\\simeq &\\mathrm{QDM}(\\mathcal{X}), \\\\\n(\\mathcal M_A, F, \\nabla \\circ \\mathfrak{h}_Y^{1\/z} , H(,)_F)\\big|_{V_Y} &\\simeq &\\mathrm{QDM}(Y).\n\\end{eqnarray}\nNotice that the Dubrovin connections on $TH(\\mathcal{X})$ and $TH(Y)$ correspond to\ndifferent trivialization of the global flat system $\\nabla$ when $\\mathfrak{h}_\\mathcal{X}\\neq\n\\mathfrak{h}_Y$. \nAny 1-chain $\\rho$ in $\\mathcal M_A$ gives an analytic continuation map\nof $\\nabla$-flat sections \n$\\mathbb{U}^{\\mathcal{X}, Y}_{\\mathcal{S},\\rho}:\\Gamma(V_Y, \\mathcal{O}(F)) \\to \\Gamma(V_\\mathcal{X}, \\mathcal{O}(F))$,\nwhich is an isometry of $H(,)_F$ \nand identifies the quantum $D$-modules of\n$\\mathcal{X}$ and $Y$.\n\\begin{rmk}\nWhen $\\mathfrak{h}_{\\mathcal{X}}\\neq \\mathfrak{h}_{Y}$, the induced Frobenius structures on $H(\\mathcal{X})$ and\n$H(Y)$ are inequivalent. A sufficient condition \\cite{MR2529944} for the two Frobenius\nstructures to coincide is given by the Hard Lefschetz criterion for $\\mathcal{X} \\to X$:\n\\beq\n\\mathrm{age}(\\theta) - \\mathrm{age}(\\mathrm{inv}^*\\theta) = 0\n\\eeq\nfor any class $\\theta\\in H(\\mathcal{X})$.\n\\end{rmk}\n\n\\begin{rmk}\nSuppose that\n$c_1(\\mathcal{Z})\\geq 0$ and that the coarse moduli space $Z$ is a\nsemi-projective toric variety given by a GIT quotient of $\\mathbb{C}^{\\dim\\mathcal{Z}+l_\\mathcal{Z}}$ by $(\\mathbb{C}^*)^{l_\\mathcal{Z}}$.\nIn this setting, the global quantum $D$-module arises naturally in the\nform of the GKZ system associated to $\\mathcal{Z}$ \\cite{MR1653024,\n MR2271990, ccit2}. The scaling factor $\\mathfrak{h}_\\mathcal{Z}^{1\/z}$ then measures the\ndiscrepancy between the small $J$-function and the canonical basis-vector of\n solutions of the GKZ system (the {\\it $I$-function}), restriced to zero\n twisted insertions:\n\\beq\n\\mathfrak{h}_\\mathcal{Z}^{1\/z}(\\tau_{0,2}) J^{\\mathcal{Z}, \\rm small}(\\tau_{0,2}, z) =\nI^\\mathcal{Z}({\\frak a}(\\tau_{0,2}),z),\n\\label{eq:scalingIJ}\n\\eeq\nwhere ${\\frak a}(\\tau_{0,2})$ is the inverse mirror map. As a consequence of \\eqref{eq:scalingIJ}, the\nscaling factor $\\mathfrak{h}_{\\mathcal{Z}}$ is \ndetermined by the toric data defining $\\mathcal{Z}$ \\cite{MR1653024,\n MR2529944, ccit2}. Let $\\Xi_i\\in H^2(Z)$ be the $T$-equivariant Poincar\\'e dual of the reduction to the quotient of the $i^{\\rm th}$\ncoordinate hyperplane in $\\mathbb{C}^{\\dim\\mathcal{Z}+l_\\mathcal{Z}}$ and write\n$\\zeta^{(j)}_i=\\mathrm{Coeff}_{\\phi_j}\\Xi_i \\in \\mathbb{C}[\\nu]$ for the coefficient of the projection of\n$\\Xi_i$ along $\\phi_j\\in H(\\mathcal{Z})$ for $j=0, \\dots, l_\\mathcal{Z}$. Defining, for every\n$\\beta$, $D_i(\\beta) \\triangleq \\int_\\beta \\Xi_i$ and $J^\\pm_\\beta\\triangleq\\l\\{j \\in\n\\{1,\\dots,\\dim\\mathcal{Z}+l_\\mathcal{Z} \\} | \\pm D_j(\\beta)>0\\r\\}$, we have\n\\bea\n\\tau_l &=& \\log{{\\frak a}_l} + \\sum_{\\beta\\in \\mathrm{Eff}(\\mathcal{Z})}{\\frak a}^\\beta \\frac{\\prod_{j_{-}\\in\n J^-_\\beta}(-1)^{D_{j_{-}}(\\beta)} |D_{j_{-}}(\\beta)|!}{\\prod_{j_{+}\\in\n J^+_\\beta}D_{j_{+}}(\\beta)!}\\sum_{k_{-}\\in\n J^-_\\beta}\\frac{-\\zeta^{(l)}_{k_{-}}}{D_{k_{-}}(\\beta)}, \\quad l=1,\\dots,\nl_\\mathcal{Z}, \n\\end{eqnarray}\n\\bea\n\\mathfrak{h}_\\mathcal{Z} &=& \\exp\\l[\\sum_{\\beta\\in \\mathrm{Eff}(\\mathcal{Z})}{\\frak a}^\\beta \\frac{\\prod_{j_{-}\\in\n J^-_\\beta}(-1)^{D_{j_{-}}(\\beta)} |D_{j_{-}}(\\beta)|!}{\\prod_{j_{+}\\in\n J^+_\\beta}D_{j_{+}}(\\beta)!}\\sum_{k_{-}\\in J^-_\\beta}\\frac{-\\zeta^{(0)}_{k_{-}}}{D_{k_{-}}(\\beta)}\\r].\n\\end{eqnarray}\n\\end{rmk}\n\n\n\\subsubsection{Givental's symplectic formalism}\n\\label{sec:givental}\n\nThe global quantum D-module picture is intimately connected to\n the CRC statement of \\cite{MR2510741, coates2007quantum}. In view of our\n statement of the OCRC in Section \\ref{sec:ocrc}, we find it useful to spell\n it out here. Givental's symplectic space $(H_\\mathcal{Z},\\Omega_\\mathcal{Z})$ is the infinite dimensional vector space\n\\beq\n\\mathcal{H}_\\mathcal{Z}\\triangleq H(\\mathcal{Z})\\otimes\\mathcal{O}(\\mathbb{C}^*)\n\\eeq\nalong with the symplectic form\n\\beq\n\\Omega_\\mathcal{Z}(f,g)\\triangleq \\Res_{z=0} \\eta(f(-z),g(z))_\\mathcal{Z}.\n\\label{eq:sympform}\n\\eeq\nA general point of $\\mathcal{H}_\\mathcal{Z}$ can be written as\n\\beq\n\\sum_{k\\geq 0}\\sum_{\\alpha=0}^{N_\\mathcal{Z}-1} q_{k,\\alpha} \\phi_\\alpha z^k+\\sum_{l\\geq 0}\\sum_{\\beta=0}^{N_\\mathcal{Z}-1} p_{l,\\beta} \\phi_\\beta z^{-k-1}.\n\\eeq\nNotice that $\\{q_{k,\\alpha}, p_{l,\\beta}\\}$ are Darboux coordinates for\n\\eqref{eq:sympform}; call $\\mathcal{H}_\\mathcal{Z}^+$ the Lagrangian subpace spanned by\n$q_{k,\\alpha}$. The generating function of genus zero descendent Gromov--Witten invariants of\n$\\mathcal{Z}$,\n\\beq\n\\mathcal{F}_0^\\mathcal{Z} \\triangleq \\sum_{n=0}^\\infty \\sum_{\\beta \\in \\mathrm{Eff}(\\mathcal{Z})}\\sum_{\\substack{a_1, \\dots a_n \\\\ p_1\n \\dots p_n}} \\frac{\\prod_{i=1}^n \\tau_{a_i,r_i}}{n!}\\left\\langle\n\\sigma_{r_1}(\\phi_{a_1}) \\dots \\sigma_{r_n}(\\phi_{a_n}) \\right\\rangle_{0,n,\\beta}^\\mathcal{Z},\n\\label{eq:descpot}\n\\eeq\nis the germ of an analytic function on $\\mathcal{H}_\\mathcal{Z}^+$ upon identifying\n$\\tau_{0,0}=q_{0,0}+1$, $\\tau_{\\alpha,n}=q_{\\alpha,n}$; under the assumption of convergence\nof the quantum product, coefficients of monomials in $\\tau_{\\alpha,n}$ with\n$\\deg_{\\rm CR} \\phi_\\alpha \\neq 0$, $n > 0$ are analytic functions of $\\mathrm{e}^{\\tau_{0,2}}$\nin a neighbourhood of the origin. The graph of the differential of\n\\eqref{eq:descpot}, \n\\beq\np_{l,\\beta}=\\frac{{\\partial}\\mathcal{F}_0^\\mathcal{Z}}{{\\partial} q^{l,\\beta}},\n\\eeq\nthen yields a formal germ of a Lagrangian submanifold $\\mathcal{L}_\\mathcal{Z}$ (in\nfact, a ruled cone, as a consequence of the genus zero Gromov--Witten axioms),\ndepending analytically on the small quantum cohomology variables\n$\\tau_{0,2}$. By the equations defining the cone, the $J$-function $J^\\mathcal{Z}(\\tau,-z)$ yields a family of\nelements of $\\mathcal{L}_\\mathbb{Z}$ parameterized by $\\tau \\in H(\\mathcal{Z})$, which is uniquely\ndetermined by its large $z$ asymptotics $J(\\tau, -z)=-z+\\tau + \\mathcal{O}(z^{-1})$. Conversely, the genus zero topological recursion relations imply that $\\mathcal{L}_\\mathcal{Z}$ can be reconstructed entirely from $J^\\mathcal{Z}(\\tau, z)$.\n\\\\\n\nThe Crepant Resolution Conjecture has a natural formulation in terms of\nmorphisms of Givental spaces, as pointed out by\nCoates--Corti--Iritani--Tseng (CCIT) \\cite{MR2510741} and further explored by Coates--Ruan\n\\cite{coates2007quantum}. \n\\begin{conj}[\\cite{MR2510741}, \\cite{coates2007quantum}]\nThere exists $\\mathbb{C}((z^{-1}))$-linear symplectic\nisomorphism of Givental spaces $\\mathbb{U}_\\rho^{\\mathcal{X},Y}:\\mathcal{H}_\\mathcal{X}\\rightarrow \\mathcal{H}_Y,$\nmatching the Lagrangian cones of $\\mathcal{X}$ and $Y$ upon a suitable analytic\ncontinuation of small quantum cohomology parameters:\n\\beq\n\\mathbb{U}_{\\rho}^{\\X,Y}(\\mathcal{L}_\\mathcal{X})=\\mathcal{L}_Y.\n\\eeq\n\\end{conj}\nThis version of the CRC is equivalent to the quantum $D$-module approach via the\nfundamental solutions, which give a canonical $z$-linear identification\n\\beq\\label{eq:givetosect}\nS_\\mathcal{Z}(\\tau,z):\\mathcal{H}_\\mathcal{Z}\\stackrel{\\cong}{\\longrightarrow}\\mathcal{S}_\\mathcal{Z}.\n\\eeq\ntranslating the analytic continuation map $\\mathbb{U}_{\\mathcal{S},\\rho}^{\\mathcal{X},Y}$ to a\nlinear isomorphism of Givental spaces which is symplectic, as\n$\\mathbb{U}_{\\mathcal{S},\\rho}^{\\mathcal{X},Y}$ preserves the pairing \\eqref{eq:pairDmod}. \n\\\\\n\nSuppose now that $c_1(\\mathcal{X})=0$, $\\mathrm{dim}_\\mathbb{C}\\mathcal{X}=3$ and assume further that\nthe $J$-functions $J^\\mathcal{Z}$, for $\\mathcal{Z}$ either $\\mathcal{X}$ or $Y$, and $\\mathbb{U}_{\\rho}^{\\X,Y}$ admit well-defined non-equivariant limits,\n\\beq\nJ_{\\rm n-eq}^\\mathcal{Z}(\\tau,z) \\triangleq \\lim_{\\nu\\to 0}J^\\mathcal{Z}(\\tau,z), \\qquad \\mathbb{U}^{\\mathcal{X},Y}_{\\rho,0} \\triangleq \\lim_{\\nu\\to 0} \\mathbb{U}_{\\rho}^{\\X,Y}. \n\\eeq\nBy homogeneity, $\\mathrm{e}^{-\\tau_0\/z} J_{\\rm n-eq}^\\mathcal{Z}(\\tau,z)$ is a Laurent\npolynomial of the form \\cite[\\S10.3.2]{MR1677117}\n\\beq\nJ_{\\rm n-eq}^\\mathcal{Z}(\\tau,z) = \\mathrm{e}^{-\\tau_0\/z}\\l(z + \\sum_{i=1}^{N_\\mathcal{Z}-1}\\l(\\tau_i +\n\\frac{\\mathfrak{f_i}^\\mathcal{Z}(\\tau)}{z}\\r)\\phi_i + \\frac{\\mathfrak{g}^\\mathcal{Z}(\\tau)}{z^2}\\mathbf{1}_\\mathcal{Z}\\r),\n\\eeq\nwhere $\\mathfrak{f}^\\mathcal{Z}(\\tau)$ and $\\mathfrak{g}^\\mathcal{Z}(\\tau)$ are\nanalytic functions around the large radius limit point of $\\mathcal{Z}$. Restricting $J_{\\rm n-eq}^\\mathcal{Z}(\\tau,z)$\nto $\\Delta_\\mathcal{Z}$ and picking up a branch $\\rho$ of analytic continuation of the\nquantum parameters, the vector valued analytic function $\\mathcal{I}_\\rho^{\\mathcal{X},Y}$\ndefined by\n\\beq\n\\begin{xy}\n(0,20)*+{\\Delta_\\mathcal{X}}=\"a\"; (40,20)*+{\\Delta_Y}=\"b\";\n(0,0)*+{\\mathcal{H}_\\mathcal{X}}=\"c\"; (40,0)*+{\\mathcal{H}_Y}=\"d\";\n{\\ar^{\\mathcal{I}_\\rho^{\\mathcal{X},Y}} \"a\";\"b\"};\n{\\ar_{J_{\\rm n-eq}^\\mathcal{X}\\big|_{\\Delta_\\mathcal{X}}} \"a\";\"c\"};{\\ar^{J_{\\rm n-eq}^Y\\big|_{\\Delta_Y}} \"b\";\"d\"};\n{\\ar^{ \\mathfrak{h}_\\mathcal{X}^{1\/z}\\mathbb{U}^{\\mathcal{X},Y}_{\\rho,0} \\mathfrak{h}_Y^{-1\/z}} \"c\";\"d\"};\n\\end{xy}\n\\label{eq:iddelta}\n\\eeq\ngives an analytic\nisomorphism\\footnote{Explicitly, matrix entries $(\\mathbb{U}^{\\mathcal{X},Y}_{\\rho,0})_{ij}$ of\n$\\mathbb{U}^{\\mathcal{X},Y}_{\\rho,0}$ are monomials in $z$; call $\\mathfrak{u}_{ij}$ the\ncoefficient of such monomial. Then \\eqref{eq:iddelta} boils down to the\nstatement that quantum cohomology parameters \n$\\tau^\\bullet_i$ in $\\Delta_\\bullet$ for $i=1, \\dots, l_Y$ are identified as \n\\beq\n\\tau^Y_i = (\\mathcal{I}^{\\mathcal{X},Y}_\\rho \\tau^\\mathcal{X})_i \\triangleq\n\\mathfrak{u}_{i0}+\n\\sum_{j=1}^{l_Y}\\mathfrak{u}_{ij} \\tau_{j}^\\mathcal{X}+\n\\sum_{k=l_Y+1}^{N_Y-1}\\mathfrak{u}_{ik} \\mathfrak{f}^\\mathcal{X}_k(\\tau^\\mathcal{X}).\n\\label{eq:changevargen}\n\\eeq\nSince $\\deg (\\mathbb{U}^{\\mathcal{X},Y}_{\\rho,0})_{ij}>0$ for $j>l_Y$, in the Hard Lefschetz\ncase the condition that the coefficients of $\\mathbb{U}_{\\rho}^{\\X,Y}$ are Taylor series in $1\/z$\nimplies that $\\mathfrak{u}_{ik}=0$ for $k>l_Y$.\n} between neighbourhoods $V_\\mathcal{X}$, $V_Y$ of the\nprojections of the large radius points of $\\mathcal{X}$ and $Y$ to $\\Delta_\\mathcal{X}$ and\n$\\Delta_Y$. \nWhen $\\mathcal{X}$ satisfies the Hard--Lefschetz condition, the coefficients of $\\mathbb{U}_{\\rho}^{\\X,Y}$ contain\nonly non-positive powers of $z$ \\cite{coates2007quantum} and the non-equivariant limit coincides with the\n$z\\to\\infty$ limit; then the isomorphism\n$\\mathcal{I}_\\rho^{\\mathcal{X},Y}$ extends to an affine linear\n change of variables\n$\\widehat{\\mathcal{I}}_\\rho^{\\mathcal{X},Y}:H(\\mathcal{X})\\to H(Y)$ at the level of the full\ncohomology rings of $\\mathcal{X}$ and $Y$, which is \nan isomorphism of Frobenius\nmanifolds.\n\n\n\n\\subsubsection{Integral structures and the CRC}\n\\label{sec:intstr}\n\nIn \\cite{MR2553377}, Iritani uses $K$-groups to define an integral structure\nin the quantum D-module associated to the Gromov--Witten theory of a smooth\nDeligne--Mumford stack $\\mathcal{Z}$; we recall \n the discussion in \\cite{MR2553377, MR2683208}, adapting\nit to the equivariant setting. \\\\\n\nWrite $K(\\mathcal{Z})$ for the Grothendieck group of topological vector bundles\n$V\\to\\mathcal{Z}$ and consider the map $\\Psi:K(\\mathcal{Z})\\to H(\\mathcal{Z})\\otimes\\mathbb{C}((z^{-1}))$ given by \n\\beq\\label{eq:stackymukai}\n\\Psi(V)\\triangleq (2\\pi)^{-\\frac{\\dim\\mathcal{Z}}{2}}z^{-\\mu} \\widehat\\Gamma_\\mathcal{Z}\\cup(2\\pi\\mathrm{i})^{\\deg\/2}\\mathrm{inv}^*\\mathrm{ch}(V),\n\\eeq\nwhere $\\mathrm{ch}(V)$ is the orbifold Chern character, $\\cup$ is the topological cup\nproduct on $I\\mathcal{Z}$, and \n\\bea\n\\label{eq:gammaT}\n\\widehat\\Gamma_\\mathcal{Z} &\\triangleq & \\bigoplus_v\\prod_f\\prod_\\delta\\Gamma(1-f+\\delta), \\\\\n\\mu & \\triangleq &\\left(\\frac{1}{2}\\deg(\\phi)-\\frac{3}{2}\\right)\\phi,\n\\end{eqnarray}\nwhere the sum in \\eqref{eq:gammaT} is over all connected components of the inertia stack, the left\nproduct is over the eigenbundles in a decomposition of the tangent bundle $T\\mathcal{Z}$\nwith respect to the stabilizer action (with $f$ the weight of the action on the eigenspace), and the\nright product is over all of the Chern roots $\\delta$ of the\neigenbundle. Via the fundamental solution \\eqref{eq:fundsol} this induces a map\nto the space of flat sections of $\\mathrm{QDM}(\\mathcal{Z})$; its image is a lattice \\cite{MR2553377}\nin $\\mathcal{S}_\\mathcal{Z}$, which Iritani dubs the {\\it $K$-theory integral structure} of\n $QH(\\mathcal{Z})=(H(\\mathcal{Z}) , \\eta, \\circ_\\tau)$. This implies the existence of an integral\n local system underlying $\\mathrm{QDM}(\\mathcal{Z})$ induced by the $K$-theory of\n $\\mathcal{Z}$. \\\\\n\nIritani's theory has important implications for the Crepant Resolution\nConjecture. At the level of integral structures, the analytic continuation map\n$\\mathbb{U}_{\\mathcal{S},\\rho}^{\\mathcal{X}, Y}$ of flat sections should be induced by an isomorphism\n$\\mathbb{U}_{K,\\rho}^{\\mathcal{X},Y}: K(Y) \\to K(\\mathcal{X})$ at the $K$-group level,\n\\beq\\label{eq:intstructure}\n\\begin{xy}\n(0,20)*+{K(\\mathcal{X})}=\"a\"; (40,20)*+{K(Y)}=\"b\"\n(0,0)*+{\\mathcal{S}_\\mathcal{X}}=\"c\"; (40,0)*+{\\mathcal{S}_Y}=\"d\";%\n{\\ar^{\\mathbb{U}_{K,\\rho}^{\\mathcal{X},Y}} \"a\";\"b\"};\n{\\ar_{S_\\mathcal{X}(x,z)\\Psi_\\mathcal{X}} \"a\";\"c\"};{\\ar^{S_Y(t,z)\\Psi_Y} \"b\";\"d\"};\n{\\ar^{ \\mathfrak{h}_Y^{1\/z}\\mathbb{U}_{\\mathcal{S},\\rho}^{\\mathcal{X},Y} \\mathfrak{h}_\\mathcal{X}^{-1\/z}} \"c\";\"d\"};\n\\end{xy}\n\\eeq\n\n\nThe Crepant Resolution Conjecture can then be phrased in terms of the\nexistence of an identification of the integral structures underlying\nquantum cohomology. In \\cite{MR2553377}, it is conjectured that\n$\\mathbb{U}_{K,\\rho}^{\\mathcal{X},Y}$ should be induced by a natural geometric\ncorrespondence between $K$-groups (see also \\cite{MR2271990} for earlier work\nin this context). In terms of Givental's symplectic formalism, we have \n\\beq\\label{eq:iritanisymp}\n\\mathbb{U}_\\rho^{\\mathcal{X},Y}=\\Psi_Y\\circ\\mathbb{U}_{K,\\rho}^{\\mathcal{X},Y}\\circ\\Psi_\\mathcal{X}^{-1}.\n\\eeq\n\n\n\\subsection{Open Gromov--Witten theory}\n\\label{sec:ogw}\n\nFor a three-dimensional toric Calabi--Yau variety, open Gromov-Witten invariants \nare defined ``via\nlocalization\" in \\cite{Katz:2001vm, Diaconescu:2003qa}. This theory\nhas been first introduced for orbifold targets in \\cite{MR2861610} and developed in\nfull generality in \\cite{r:lgoa} (see also \\cite{fang2012open} for recent\nresults in this context). \n Boundary conditions are given by choosing special type of Lagrangian\n submanifolds introduced by Aganagic--Vafa in\n \\cite{Aganagic:2000gs}. These Lagrangians are defined locally in a formal neighborhood of each torus invariant line: in particular if $p$ is a torus fixed point adjacent to the torus fixed line $l$, and the local coordinates at $p$ are $(z,u,v)$, then $L$ is defined to be the fixed points of the anti-holomorphic involution\n \\beq\n (z,u,v)\\rightarrow (1\/\\overline{z}, \\overline{zu}, \\overline{zv})\n \\eeq\n defined away from $z=0$. Boundary conditions can then be thought of as ``formal'' ways\n of decorating the web diagram of the toric target. \\\\\n \n Loci of fixed maps are described in terms of closed\ncurves mapping to the compact edges of the web diagram in the usual way and disks mapping rigidly to\nthe torus invariant lines with Lagrangian conditions. Beside Hodge integrals coming from the contracting\ncurves, the contribution of each fixed locus to the invariants has a factor\nfor each disk, which is constructed as follows. The map from the disk to a neighborhood\nof its image is viewed as the quotient via an involution of a map of a\nrational curve to a canonical target. The obstruction theory in ordinary\nGromov-Witten theory admits a natural $\\mathbb{Z}_2$ action, and the equivariant Euler\nclass of the involution invariant part of the obstruction theory is chosen as\nthe localization contribution from the disk \\cite[Section~2.2]{MR2861610}, \\cite[Section~2.4]{r:lgoa}. This construction is efficiently\nencoded via the introduction of a ``disk function\", which we now review in the\ncontext of cyclic isotropy (see \\cite[Section~3.3]{r:lgoa} for the general\ncase of finite abelian isotropy groups). \\\\\n\nLet $\\mathcal{Z}$ be a three-dimensional CY toric orbifold, $p$ a fixed point such\nthat a neighborhood is isomorphic to $[\\mathbb{C}^3\/\\mathbb{Z}_{n+1}]$, with representation\nweights $(m_1, m_2,m_3)$ and CY torus weights $(w_1,w_2,w_3)$. Define ${n_{e}}=\n(n+1)\/\\gcd(m_1,n+1)$ to be the size of the effective part of the action along\nthe first coordinate axis. \n\n There exist a map from an orbi-disk mapping to the first coordinate axis with winding $d$ and twisting $k$ if the compatibility condition \n\\beq\n\\frac{d}{{n_{e}}}-\\frac{km_1}{n+1}\\in \\mathbb{Z}\n\\label{compat}\n\\eeq\nis satisfied. In this case the positively oriented disk function is\n\\begin{equation}\nD_k^+(d;\\vec{w})=\n\\left( \\frac{ {n_{e}}w_1}{d} \\right)^{\\text{age}(k)-1}\\frac{{n_{e}}}{d(n+1)\\left\\lfloor \\frac{d}{{n_{e}}} \\right\\rfloor !}\\frac{\\Gamma\\left( \\frac{dw_{2}}{{n_{e}}w_1}+\\left\\langle \\frac{k m_{3}}{n+1} \\right\\rangle + \\frac{d}{{n_{e}}} \\right)}{\\Gamma\\left( \\frac{dw_{2}}{{n_{e}}w_1}-\\left\\langle \\frac{k m_{2}}{n+1} \\right\\rangle +1 \\right)}.\n\\end{equation}\nThe negatively oriented disk function is obtained by switching the indices $2$\nand $3$. By renaming the coordinate axes this definition applies to the\ngeneral boundary condition. \\\\\n\nIn \\cite{r:lgoa} the disk function is used to construct the GW orbifold\ntopological vertex, a building block for open and closed GW invariants of\n$\\mathcal{Z}$. The disk potential is efficiently expressed in terms of the\ndisk and of the $J$ function of $\\mathcal{Z}$. Fix a Lagrangian boundary condition $L$\nwhich we assume to be on the first coordinate axis in the local chart ( $\\cong [\\mathbb{C}^3\/\\mathbb{Z}_{n+1}]$) around the\npoint $p$. Denote by $\\{\\mathbf{1_{p,k}}\\}_{k=1,...,n+1}$ the part of the localized basis for $H(\\mathcal{Z})$ supported at $p$.\nRaising indices using the orbifold\nPoincar\\'e pairing, and extending the disk function to be a cohomology valued\nfunction\n\\begin{equation}\n\\mathcal{D}^+(d;\\vec{w})=\\sum_{k=1}^{n+1}D_k^+(d;\\vec{w}) \\mathbf{1_p^k},\n\\end{equation}\nthe (genus zero) \\textit{scalar disk potential} is obtained by contraction with the $J$ function:\n\\bea\nF_{L}^{\\rm disk}(\\tau,y,\\vec{w}) &\\triangleq & \\sum_d\\frac{y^d}{d!}\\sum_n\n\\frac{1}{n!}\\langle \\tau, \\ldots, \\tau \\rangle_{0,n}^{L,d} \\nn \\\\\n&=& \\sum_d \\frac{y^d}{d!}\\left( \\mathcal{D}^+(d;\\vec{w}), J^\\mathcal{Z}\\left(\\tau,\\frac{gw_1}{d}\\right)\\right)_{\\mathcal{Z}},\n\\label{sdp}\n\\end{eqnarray}\nwhere we denoted by $\\langle \\tau, \\ldots, \\tau \\rangle_{0,n}^{L,d}$ the disk\ninvariants with boundary condition $L$, winding $d$\nand $n$ general cohomological insertions.\n\\begin{rmk}\nWe may consider the disk potential relative to multiple Lagrangian boundary conditions. In that case, we define the disk function by adding the disk functions for each Lagrangian, and we introduce a winding variable for each boundary condition. \n\\end{rmk}\n\n\\begin{rmk}\nIt is not conceptually difficult (but book-keeping intensive) to express the general genus zero open potential in terms of appropriate contractions of arbitrary copies of these disk functions with the full descendant Gromov-Witten potential of $\\mathcal{Z}$.\n\\end{rmk}\n\n\n\\subsection{$A_n$ resolutions}\n\\label{sec:an}\n\n\\subsubsection{GIT Quotients}\n\\label{sec:GIT}\n\nHere we review the relevant toric geometry concerning our targets. Let\n$\\mathcal{X}\\triangleq[\\mathbb{C}^3\/\\mathbb{Z}_{n+1}]$ be the 3-fold $A_n$ singularity and $Y$ its resolution.\nThe toric fan for $\\mathcal{X}$ has rays $(0,0,1)$, $(1,0,0)$, and $(1,n+1,0)$, while\nthe fan for $Y$ is obtained by adding the rays $(1,1,0)$, $(1,2,0)$,...,\n$(1,n,0)$. The divisor class group is described by the short exact sequence\n\\beq\n0\\longrightarrow\\mathbb{Z}^{n}\\stackrel{M^T}{\\longrightarrow}\\mathbb{Z}^{n+3}\\stackrel{N}{\\longrightarrow}\\mathbb{Z}^3\\longrightarrow\n0,\n\\label{eq:divclass}\n\\eeq\nwhere\n\\beq\nM=\\left[ \\begin{array}{cccccccc}\n1 & -2 & 1 & 0 & 0 &... &0 & 0\\\\\n0 & 1 & -2 & 1 & 0 &... &0 & 0\\\\\n\\vdots & &\\ddots & &\\ddots & && \\vdots\\\\\n0 &... & 0 & 0 & 1 & -2 & 1 & 0\n\\end{array}\n\\right]\n,\\hspace{.5cm} N=\\left[ \\begin{array}{cccccc}\n1 & 1 & 1 & & 1 & 0\\\\\n0 & 1 & 2 & ... & n+1 & 0\\\\\n0 & 0 & 0 & & 0 & 1\n\\end{array}.\n\\right]\n\\label{eq:MN}\n\\eeq\n\\\\\nBoth $\\mathcal{X}$ and $Y$ are GIT quotients: \n\\bea\\label{orbgit}\n\\mathcal{X} &=& \\left[\\frac{\\mathbb{C}^{n+3}\\setminus V(x_1\\cdot...\\cdot\n x_n)}{(\\mathbb{C}^*)^n}\\right], \\\\\nY &=& \\frac{\\mathbb{C}^{n+3}\\setminus V(I_1, \\dots, I_n),\n}{(\\mathbb{C}^*)^n}\n\\label{resgit}\n\\end{eqnarray}\nwhere \n\\beq\nI_i=\\prod_{j=0, j \\neq i-1, i}^{n+1} x_i,\n\\eeq\nand the torus action is specified by $M$. \nFrom the quotient \\eqref{orbgit}, we can compute pseudo-coordinates on the orbifold\n\\begin{equation}\\label{orbcoords}\n\\left[\\begin{array}{c}\nz_1\\\\\nz_2\\\\\nz_3\n\\end{array}\\right]\n=\n\\left[\\begin{array}{c}\nx_0x_1^{\\frac{n}{n+1}}x_2^{\\frac{n-1}{n+1}}\\cdot...\\cdot x_n^{\\frac{1}{n+1}}\\\\\nx_1^{\\frac{1}{n+1}}x_2^{\\frac{2}{n+1}}\\cdot...\\cdot x_n^{\\frac{n}{n+1}}x_{n+1}\\\\\nx_{n+2}\n\\end{array}\\right].\n\\end{equation}\nThese coordinates are only defined up to a choice of $(n+1)^{\\rm st}$\nroot of unity for each $x_i$. This accounts for a residual $\\mathbb{Z}_{n+1}\\subset\n(\\mathbb{C}^*)^n$ acting with dual representations on the first two coordinates. We\nidentify this residual $\\mathbb{Z}_{n+1}$ as the subgroup generated by \n$\\left(\\omega,\\omega^2, \\dots, \\omega^n\\right)\\in(\\mathbb{C}^*)^n$, where\n $\\omega=\\mathrm{e}^{\\frac{2\\pi \\mathrm{i} }{n+1}}$. This realizes the quotient\n\\eqref{orbgit} as the 3-fold $A_n$ singularity where $\\mathbb{Z}_{n+1}=\\langle \\omega\n\\rangle$ acts by $\\omega \\cdot(z_1,z_2,z_3)=(\\omega z_1,\\omega^{-1} z_2,z_3)$. \n\n\\begin{rmk}\\label{dualrmk}\nThe weights of the $\\mathbb{Z}_{n+1}$ action on the corresponding fibers of $T\\mathcal{X}$ are\ninverse to the weights on the local coordinates because a local trivialization\nof the tangent bundle is given by $\\frac{\\partial}{\\partial z^\\alpha}$ where\n$z^\\alpha$ are the local coordinates. \\\\\n\\end{rmk}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics{webdiagrama3.pdf}\n\\caption{The toric web diagrams for $Y$ and $\\mathcal{X}$ for $n=3$. Fixed points and invariants lines are labelled, together with the relevant torus and representation weights.}\n\\label{fig:web}\n\\end{figure}\n\nThe geometry of the space $Y$ is captured by the toric web diagram in Figure\n\\ref{fig:web}. In particular, $Y$ has $n+1$ torus fixed points (corresponding\nto the $n+1$ 3-dimensional cones in the fan) and a chain of $n$ torus\ninvariant lines connecting these points. We label the points\n$p_1$,...,$p_{n+1}$ where $p_i$ correspondes to the cone spanned by $(0,0,1)$,\n$(1,i-1,0)$, and $(1,i,0)$ and we label the torus invariant lines by\n$L_1$,...,$L_n$ where $L_i$ connects $p_i$ to $p_{i+1}$. We also denote by\n$L_0$ and $L_{n+1}$ the torus invariant (affine) lines corresponding to the\n2-dimensional cones spanned by the rays $(1,0,0), (0,0,1)$ and $(1,n,0),\n(0,0,1)$, respectively. From the quotient \\eqref{resgit} we compute\nhomogeneous coordinates on the line $L_i$\n\\begin{equation}\\label{projcoords}\n\\left[\\begin{array}{c}\nx_0^ix_1^{i-1}\\cdot...\\cdot x_{i-1}\\\\\nx_{n+1}^{n+1-i}x_{n-1}^{n-i}\\cdot...\\cdot x_{i+1}\n\\end{array}\\right]\n\\end{equation}\nwhere $p_i\\leftrightarrow[0:1]$ and $p_{i+1}\\leftrightarrow[1:0]$. \\\\\n\n\nOn the resolution, $H_2(Y)$ is generated by the torus invariant lines $L_i$.\nDefine $\\gamma_i\\in H^2(Y)$ to be dual to $L_i$. The $\\gamma_i$ form a\nbasis of $H^2(Y)$; denote the corresponding line bundles by $\\mathcal{O}(\\gamma_i)$.\nNote that $\\mathcal{O}(\\gamma_i)$ restricts to $\\mathcal{O}(1)$ on $L_i$ and $\\mathcal{O}$ on $L_j$ if\n$j\\neq i$ and this uniquely determines the line bundle $\\mathcal{O}(\\gamma_i)$. On the\norbifold, line bundles correspond to $\\mathbb{Z}_{n+1}$ equivariant line bundles on\n$\\mathbb{C}^3$. We denote $\\mathcal{O}_k$ the line bundle where $\\mathbb{Z}_{n+1}$ acts on fibers\nwith weight $\\omega^k$; then, for example, $T_\\mathcal{X}=\\mathcal{O}_{-1}\\oplus\\mathcal{O}_{1}\\oplus\\mathcal{O}_0$ where the subscripts are computed modulo $n+1$ (c.f. Remark \\ref{dualrmk}).\n\n\\subsubsection{Classical equivariant geometry}\\label{sec:elb}\n\nGiven that we are working with noncompact targets, all of our quantum\ncomputations utilize Atiyah-Bott localization with respect to an additional\n$T=\\mathbb{C}^*$ action on our spaces. Let $T$ act on $\\mathbb{C}^{n+3}$ with weights $(\\alpha_1,0,...,0,\\alpha_2,-\\alpha_1-\\alpha_2)$. Then the induced action on the orbifold and resolution can be read off from the local coordinates in \\eqref{orbcoords} and \\eqref{projcoords}. In particular, the three weights on the fibers of $T_\\mathcal{X}$ are $-\\alpha_1,-\\alpha_2, \\alpha_1+\\alpha_2$. The $T$-equivariant Chen-Ruan cohomology $H(\\mathcal{X})$ is\nby definition the $T$-equivariant cohomology of the inertia stack\n$\\mathcal{I}\\mathcal{X}$. The latter has components $\\mathcal{X}_1, \\dots, \\mathcal{X}_n, \\mathcal{X}_{n+1}$, the last being the untwisted sector\\footnote{While it is more common to index the untwisted sector by $0$, we make this choice of notation for the sake of the computations of Section \\ref{sec:j}, where certain matrices are triangular with this ordering.}: \n\\bea\n\\mathcal{X}_k &=& [\\mathbb{C}\/\\mathbb{Z}_{n+1}], \\quad 1\\leq k \\leq n, \\nonumber \\\\\n\\mathcal{X}_{n+1} &=& [\\mathbb{C}^3\/\\mathbb{Z}_{n+1}]\n\\end{eqnarray}\nWriting $\\mathbf{1}_k$, $k=1, \\dots, n+1$ for the fundamental class of\n$\\mathcal{X}_k$ we obtain a $\\mathbb{C}(\\nu)$ basis of $H(\\mathcal{X})$; the age-shifted grading\nassigns degree $0$ to the fundamental class of the untwisted sector, and\ndegree $1$ to every twisted sector. \nThe Atiyah-Bott localization isomorphism is trivial, i.e. the fundamental class on each twisted sector is identified with the unique $T$-fixed point on that sector. We abuse notation and use $\\mathbf{1}_k$ to also denote the fixed point basis. \nThe equivariant Chen-Ruan pairing in orbifold cohomology is\n\\beq\n\\eta\\l(\\mathbf{1}_i, \\mathbf{1}_j\\r)_\\mathcal{X} = \\frac{\\delta_{i,n+1}\\delta_{j,n+1}+\\alpha_1\\alpha_2 \\delta_{i+j,n+1}}{\\alpha_1 \\alpha_2(\\alpha_1+\\alpha_2)(n+1)}.\n\\eeq\n\\\\\n\nOn the resolution $Y$, the three weights on the tangent bundle at $p_i$ are \\beq(w_i^-,w_i^+,\\alpha_1+\\alpha_2)\\triangleq((i-1)\\alpha_1+(-n+i-2)\\alpha_2,-i\\alpha_1+(n+1-i)\\alpha_2,\\alpha_1+\\alpha_2).\\eeq\nMoreover, $\\mathcal{O}(\\gamma_j)$ is canonically linearized via the homogeneous coordinates in \\eqref{orbcoords}. The weight of $\\mathcal{O}(\\gamma_i)$ at the fixed point $p_i$ is\n\\begin{equation}\\label{canwts}\n\\begin{cases}\n(n+1-j)\\alpha_2 & i\\leq j,\\\\\nj\\alpha_1 & i>j.\n\\end{cases}\n\\end{equation}\nDenote by $\\{P_i\\}_{i=1}^{n+1}$ the equivariant cohomology classes\ncorresponding to the fixed points of $Y$. Choosing the canonical\nlinearization given in \\eqref{canwts}, the Atiyah-Bott localization\nisomorphism on $Y$ is given by\n\\bea\n\\label{eq:ab1}\n\\gamma_j &\\longrightarrow & \\sum_{i\\leq j}(n+1-j)\\alpha_2 P_i +\n\\sum_{i>j}j\\alpha_1P_i, \\\\\n\\gamma_{n+1} & \\longrightarrow &\\sum_{i=1}^{n+1}P_i.\n\\label{eq:ab2}\n\\end{eqnarray}\nwhere $\\gamma_{n+1}$ is the fundamental class on $Y$. \nGenus zero, degree zero GW invariants are given by equivariant triple\nintersections on $Y$,\n\\beq\n\\left\\langle \\gamma_i, \\gamma_j, \\gamma_k \\right\\rangle^Y_{0,3,0} = \\int_Y\\gamma_i\\cup\\gamma_j\\cup\\gamma_k.\n\\eeq\nWith $i\\leq j \\leq k}[r]& \\mathcal{U}\\ar[d]^\\pi \\ar[r]^{\\lambda} & \\mathbb{P}^1 \\\\\n [\\lambda] \\ar@{^{(}->}[r]^{pt.} \\ar@\/^1pc\/[u]^{P_i}& \\mathcal{H}_\\lambda \\ar@\/^1pc\/[u]^{\\Sigma_i}& \n}\n\\eeq\n\\end{defn}\n\n\\begin{rem}\nA genus zero double Hurwitz space is naturally isomorphic to $M_{0,n+3}$, and is therefore an open set in affine space $\\mathbb{A}^n$. This is the only case that we utilize and it may seem overly sophisticated to use the language of moduli spaces to then work on such a simple object. We choose to do so to connect to the work of Dubrovin \\cite{Dubrovin:1992eu, Dubrovin:1994hc} and Romano \\cite{2012arXiv1210.2312R} (after Saito \\cite{MR723468}; see also \\cite{Krichever:1992qe}), who studied\nexistence and construction of Frobenius structures on arbitrary double Hurwitz spaces. \n\\end{rem}\nLet \n$\\phi\\in \\Omega^1_{C}(\\log (\\lambda))$ be a meromorphic one form having simple poles at the support of $(\\lambda)$ with\nconstant residues; we call $(\\lambda, \\phi)$ respectively the {\\it\n superpotential} and the {\\it quasi-momentum differential} of $\\mathcal{H}_\\lambda$.\nBorrowing the terminology from \\cite{2012arXiv1210.2312R, phdthesis-romano},\nwe say that an analytic Frobenius manifold structure $(\\mathcal{F}, \\circ, \\eta)$ on\na complex manifold $\\mathcal{F}$ is\n{\\it weak} if\n\\ben\n\\item the $\\circ$-multiplication gives a commutative and associative\nunital $\\mathcal{O}$-algebra structure\non the space of holomorphic vector fields on $\\mathcal{F}$;\n\\item the metric $\\eta$ provides a flat\n pairing which is Frobenius w.r.t. to $\\circ$;\n\\item the algebra structure\nadmits a {\\it potential}, meaning that the 3-tensor\n\\beq\nR(X,Y,Z) \\triangleq \\eta(X,Y \\circ Z)\n\\eeq\nsatisfies the integrability condition\n\\beq\n(\\nabla^{(\\eta)} R)_{[\\alpha \\beta] \\gamma\\delta}=0.\n\\eeq\n\\end{enumerate}\nIn particular, this encompasses non-quasihomogeneous solutions of\nWDVV, and solutions without a flat identity element.\n\n\\begin{prop}[\\cite{2012arXiv1210.2312R}]\nFor vector fields $X$, $Y$, $Z \\in \\mathfrak{X}(\\mathcal{H}_\\lambda)$, define the\nnon-degenerate symmetric pairing $g$ and quantum product $\\star$ as\n\\bea\n\\label{eq:gmetr}\ng(X,Y) &\\triangleq & \\sum_{P\\in\\mathrm{supp}(\\lambda)}\\Res_P\\frac{X(\\log\\lambda)\n Y(\\log\\lambda)}{\\mathrm{d}_\\pi \\log\\lambda}\\phi^2, \\\\\ng(X,Y \\star Z) &\\triangleq & \\sum_{P\\in\\mathrm{supp}(\\lambda)}\\Res_P\\frac{X(\\log\\lambda)\n Y(\\log\\lambda) Z(\\log\\lambda)}{\\mathrm{d}_\\pi \\log\\lambda}\\phi^2,\n\\label{eq:star}\n\\end{eqnarray}\nwhere $\\mathrm{d}_\\pi$ denotes the relative differential with respect to the\nuniversal family (i.e. the differential in the fiber direction). Then the triple $\\mathcal{F}_{\\lambda,\\phi}=\\l(\\mathcal{H}_{\\lambda}, \\star, g\\r)$ endows\n$\\mathcal{H}_{\\lambda}$ with a weak Frobenius manifold structure.\n\\end{prop}\n\\begin{rmk}\n\\label{rmk:adual}\nEquations \\eqref{eq:gmetr}-\\eqref{eq:star} are the\nDijkgraaf--Verlinde--Verlinde formulae \\cite{Dijkgraaf:1990dj} for a\ntopological Landau--Ginzburg model on a sphere with $\\log\\lambda(q)$ as its\nsuperpotential. The case in which $\\lambda(q)$ itself is used as the\nsuperpotential gives rise to a {\\it different} Frobenius manifold structure,\nwhich is the case originally studied in \\cite[Lecture 5]{Dubrovin:1994hc}; the\nsituation at hand is its Dubrovin-dual\nin the sense of \\cite{MR2070050}, where $g$ plays the role of the\nintersection form and $\\star$ the dual product. \n\\end{rmk}\n\n\\subsubsection[Twisted periods and the QDE]{Twisted periods and the quantum\n differential equation}\n\nThe quantum $D$-module associated to $\\mathcal{F}_{\\lambda, \\phi}$,\n\\beq\n\\nabla^{(g,z)} \\omega =0,\n\\label{eq:QDELG}\n\\eeq\nwhere\n\\beq\n\\label{eq:defconn}\n\\nabla^{(g,z)}_X(Y,z) \\triangleq \\nabla^{(g)}_X Y+z^{-1} X \\star Y\n\\eeq\nenjoys a neat description in terms of the Landau--Ginzburg data $(\\lambda, \\phi)$:\nin particular, flat frames for \\eqref{eq:QDELG} can be computed from the\ntwisted Picard--Lefschetz theory of $\\lambda$ \\cite{MR936695, MR2070050, Brini:2011ff}.\nIn contrast with the classical Picard--Lefschetz theory,\nthis corresponds to\nconsidering cycles $\\gamma \\in H_1(\\mathbb{C} \\setminus H,\n\\mathbf{L})$ in the {\\it complement} of the zero-dimensional hypersurface\n$H=\\lambda^{-1}(0)$ cut\nby $\\lambda$,\nwhere the linear local system $\\mathbf{L}$ is defined by multiplication by\n$\\mathrm{e}^{2\\pi\\mathrm{i}\/z}$ when moving along a simple loop around\nany single point of $H$. Elements $\\gamma$ of the homology group with coefficients twisted by ${\\bf L}$\nare the {\\it twisted cycles} of $\\lambda$. \\\\\n\nOscillating integrals around a basis of twisted cycles of\nthe form\n\\beq\n\\Pi_{\\lambda, \\phi, \\gamma}(z) \\triangleq \\int_\\gamma \\lambda^{1\/z} \\phi\n\\label{eq:periods}\n\\eeq\nare called {\\it\n twisted periods}\\footnote{To be completely consistent with\n \\cite{MR2070050} we should more correctly call these the {\\it\n twisted periods} of $\\mathcal{F}_{\\mathrm{e}^\\lambda,\\phi}$. See Remark \\ref{rmk:adual}.}\nof $\\mathcal{F}_{\\lambda,\\phi}$. Denote by $\\mathrm{Sol}_{\\lambda,\n \\phi}$ the solution space of \\eqref{eq:QDELG},\n\\beq\n\\mathrm{Sol}_{\\lambda,\n \\phi} = \\{s \\in \\mathfrak{X}(\\mathcal{F}_{\\lambda,\\phi}), \\nabla^{(g,z)}s=0 \\}.\n\\eeq\nWe have the following\n\\begin{prop}[Dubrovin, \\cite{MR2070050}]\n\\label{thm:tp}\nThe solution space of the quantum differential equations of\n$\\mathcal{F}_{\\lambda,\\phi}$ is generated by gradients of the twisted periods\n\\eqref{eq:periods}\n\\beq\n\\mathrm{Sol}_{\\lambda, \\phi} = \\mathrm{span}_{\\mathbb{C}((z))}\n\\{\\nabla^{(g)} \\Pi_{\\lambda,\\phi,\\gamma} \\}_{\\gamma \\in H_1(\\mathbb{C} \\setminus H,\n\\mathbf{L})}\n\\eeq\n\\end{prop}\nIn particular, Proposition \\ref{thm:tp} implies that the quantum $D$-modules arising from weak Frobenius structures on genus zero\ndouble Hurwitz spaces are described by systems of period integrals of\ngeneralized hypergeometric type. \\\\\n\n\\begin{rmk}\nSince $\\lambda$ is a genus zero covering map, in an affine chart parametrized by $q\\in\\mathbb{C}$ its logarithm takes the\nform \n\\beq\n\\log\\lambda = \\sum_{i}a_i \\log(q-q_i),\n\\label{eq:logl}\n\\eeq\nwhere $a_i\\in \\mathbb{Z}$. In fact, the existence of the weak Frobenius structure\n\\eqref{eq:gmetr}-\\eqref{eq:star} extends \\cite{phdthesis-romano} to the case where $\\mathrm{d}_\\pi\\log\\lambda$\nis a meromorphic function on $C$; this in particular encompasses the case where $a_i\\in\n\\mathbb{C}$ in \\eqref{eq:logl}. As far as flat coordinates of the deformed connection\n$\\nabla^{(g,z)}$ are concerned, Proposition~\\ref{thm:tp} continues to hold,\n the only proviso being that the locally constant sheaf ${\\bf L}$ be replaced\n with the unique local system specified by the monodromy weights $a_i\/z$ in\n \\eqref{eq:periods}, \\eqref{eq:logl}. \\\\\n\\end{rmk}\n\n\\subsection{A one-dimensional Landau--Ginzburg mirror}\n\nIt is known that the quantum $D$-modules associated to the equivariant Gromov--Witten theory of\nthe $A_n$-singularity $\\mathcal{X}$ and its resolution $Y$ admit a Landau--Ginzburg\ndescription in terms of $n$-dimensional oscillating integrals\n\\cite{MR1408320, MR1328251, MR2700280, MR2529944}. We provide here an alternative description\nvia one-dimensional twisted periods of a genus zero double Hurwitz space\n$\\mathcal{F}_{\\lambda, \\phi}$. \\\\\n\n\nLet $\\mathcal M_A$ be $M_{0,n+3}$. By choosing the last three sections to be the constant sections $0, 1, \\infty$, we realize $\\mathcal M_A$ as an open subset of $\\mathbb{A}^{n}$ and trivialize the universal family. \nIn homogeneous coordinates $[u_0:\\dots:u_n]$ for $\\mathbb{P}^n$,\n\\beq\n\\mathcal M_A= \\mathbb{P}^n\\setminus \\mathrm{Proj} \\frac{\\mathbb{C}[u_0, \\dots, u_n]}{{\\left\\langle\n u_i(u_j-u_k)\\right\\rangle}} \\triangleq \\mathbb{P}^n\\setminus \\mathrm{discr} \\mathcal M_A .\n \\label{eq:discr}\n\\eeq\nLet\n$\\kappa_i=u_i\/u_0$, $i=1, \\dots, n$ be a set of global coordinates on $\\mathcal M_A$\nand $q$ be an affine coordinate on the fibers of the universal\nfamily. We give $\\mathbb{C}\\times \\mathcal M_A$ the structure of a one parameter family of double Hurwitz spaces by specifying the pair $(\\lambda, \\phi)$; we call \n$\\kappa_0$ the coordinate in the first factor, and define \n\\beq\n\\lambda(\\kappa_0, \\ldots \\kappa_n, q) = C_n(\\kappa)\n\\frac{q^{(n+1)\\alpha_1}}{\\left(1-q\\right)^{\\alpha_1+\\alpha_2}} \\prod _{k=1}^{n}\n\\left(1-q\\kappa_k\\right)^{-\\alpha_1-\\alpha_2}, \n\\label{eq:superpot}\n\\eeq\n\\beq\n\\phi(q) = \\frac{1}{\\alpha_1+\\alpha_2}\\frac{\\mathrm{d} q}{q},\n\\label{eq:primeform}\n\\eeq\nand\n\\bea\nC_n(\\kappa) &\\triangleq& \\prod_{j=0}^n \\kappa_j^{\\alpha_1}.\n\\end{eqnarray}\nThen Eqs.~\\eqref{eq:gmetr}-\\eqref{eq:star} and\n\\eqref{eq:superpot}-\\eqref{eq:primeform} define a Frobenius structure $\\mathcal{F}_{\\lambda,\n \\phi}$ on $\\mathbb{C}\\times\\mathcal M_A$; the discriminant ideal in \\eqref{eq:discr}\ncoincides with the locus where the $D$-module \\eqref{eq:defconn} is singular,\nand the irreducible components $V(\\kappa_i-\\kappa_j)$, for $i,j>0$, correspond to the loci where the\n$\\star$-product \\eqref{eq:star} blows-up. We have the following\n\\begin{thm}\n\\label{thm:mirror}\n\\ben\n\\item Let \n\\bea\n\\label{eq:kappa0Y}\n\\kappa_0 &=& \\mathrm{e}^{(t_{n+1}+\\delta_Y)\/\\alpha_1}, \\\\\n\\label{eq:kappaY}\n\\kappa_j &=& \\prod_{i=j}^n \\mathrm{e}^{t_i}, \\quad 1\\leq j\\leq n.\n\\end{eqnarray}\nwhere $\\delta_Y$ is an arbitrary constant. Then, in a neighbourhood $V_Y$ of $\\{ \\mathrm{e}^{t_i}=0\\}$, \n\\beq\n \\mathcal{F}_{\\lambda,\\phi} \\simeq QH_T(Y).\n\\eeq\n\\item Let\n\\bea\n\\label{eq:kappa0X}\n\\kappa_0 &=& \\mathrm{e}^{(x_{n+1}+\\delta_\\mathcal{X})\/\\alpha_1}, \\\\\n\\kappa_j &=& \\exp\\l[-\\frac{2\\mathrm{i}}{n+1}\\l(\\pi j+ \\sum_{k=1}^n\n \\mathrm{e}^{-\\frac{\\mathrm{i} \\pi k (j-1)}{n+1}} \\sin \\left(\\frac{\\pi j\n k}{n+1}\\right)x_k\\r)\\r], \\quad 1\\leq k\\leq n. \n\\label{eq:kappakX}\n\\end{eqnarray}\nwhere $\\delta_\\mathcal{X}$ is an arbitrary constant. Then, in a neighbourhood $V_\\mathcal{X}$ of $\\{x_i=0\\}$,\n\\beq\n \\mathcal{F}_{\\lambda,\\phi} \\simeq QH_T(\\mathcal{X}).\n\\eeq\n\\end{enumerate}\n\\end{thm}\n\\begin{proof} The proof is a straightforward computation from the\n Landau--Ginzburg formulae \\eqref{eq:gmetr}-\\eqref{eq:star}.\n\\ben\n\\item \nConsider the three-point\ncorrelator $R(\\kappa_i {\\partial}_i, \\kappa_j {\\partial}_j, \\kappa_k {\\partial}_k)$, where\n${\\partial}_k \\triangleq \\frac{{\\partial}}{{\\partial} \\kappa_k}$, and define\n\\bea\nR^{(l)}_{i,j,k} &\\triangleq& \\Res_{q=\\kappa_l^{-1}} \\frac{\\kappa_i \\frac{{\\partial} \\ln\\lambda}{{\\partial} \\kappa_i}\n \\kappa_j\\frac{{\\partial} \\ln\\lambda}{{\\partial} \\kappa_j} \\kappa_k \\frac{{\\partial}\n \\ln\\lambda}{{\\partial} \\kappa_k} }{(\\alpha_1+\\alpha_2)^2 q \\frac{{\\partial} \\ln\\lambda}{{\\partial} q}}\\frac{\\mathrm{d} q}{q}.\n\\end{eqnarray}\nInspection shows that $R^{(l)}_{ijk}=0$ unless $l=i=j$, $l=i=k$ or\n$l=j=k$. Assume w.l.o.g. $l=j=i$, and suppose that $i,k>0$. We compute\n\\bea\n\\label{eq:lgqu1}\nR^{(i)}_{i,i,k}\n&=& \n\\frac{\\kappa_i}{\\kappa_k-\\kappa_i}+\\frac{\\alpha_2}{\\alpha_1+\\alpha_2}, \\\\\n\\label{eq:lgqu2}\nR^{(i)}_{i,i,i}\n\\begin{comment}\n&=& \\Res_{q=\\kappa_i^{-1}} \\Bigg\\{\\frac{(\\alpha_1+\\alpha_2\n q\\kappa_i)^3}{(\\alpha_1+\\alpha_2)^3(1-q\\kappa_i)^2}\n \\frac{1}{\\frac{(n+1)\\alpha_1 (1-\\kappa_i\n q)}{\\alpha_1+\\alpha_2}+\\frac{q(1-\\kappa_i q)}{1-q}+\\kappa_i q+\\sum_{l\\neq\n i}^n\\frac{\\kappa_l\n q(1-\\kappa_i q)}{1-\\kappa_l q}}\\frac{\\mathrm{d} q}{q}\\Bigg\\} \\nn \\\\\n&=& \\Res_{q=\\kappa_i^{-1}} \\Bigg\\{\\frac{(\\alpha_1+\\alpha_2\n q\\kappa_i)^3}{(\\alpha_1+\\alpha_2)^3(\\kappa_i^{-1}-q)^2 \\kappa_i^2}\n \\frac{1}{\\kappa_i q^2}\n\\frac{1}{1-(q-\\kappa_i^{-1})\n\\l[ \\frac{(n+1)\\alpha_1}{q(\\alpha_1+\\alpha_2)}\n+\\frac{1}{1-q}\n +\\sum_{l\\neq i}^n\\frac{\\kappa_l}{1-\\kappa_l q}\\r]}\\mathrm{d} q \\Bigg\\} \n\\nn \\\\\n&=& \\Res_{q=\\kappa_i^{-1}} \\Bigg\\{\\frac{(\\alpha_2-2\\alpha_1)(\\alpha_1+\\alpha_2)^2}{(\\alpha_1+\\alpha_2)^3(q-\\kappa_i^{-1})}\n \\mathrm{d} q \n+\\frac{(\\alpha_1+\\alpha_2)^3}{(\\alpha_1+\\alpha_2)^3(q-\\kappa_i^{-1}) \\kappa_i^2}\n \\frac{1}{\\kappa_i q^2} \\nn \\\\\n& & \n\\l[\\frac{(n+1)\\alpha_1}{q(\\alpha_1+\\alpha_2)}\n+\\frac{1}{1-q}\n +\\sum_{l\\neq i}^n\\frac{\\kappa_l}{1-\\kappa_l q}\\r] \\mathrm{d} q \\Bigg\\} \n\\nn \\\\\n\\end{comment}\n&=& \\frac{(n-1) \\alpha_1+\\alpha_2}{\\alpha_1+\\alpha_2}+\n\\sum_{l\\neq i}^{n+1}\\frac{\\kappa_l}{\\kappa_i-\\kappa_l}, \\\\\nR^{(i)}_{0,i,i} \n&=& -\\frac{1}{\\alpha_1+\\alpha_2}.\n\\end{eqnarray}\nMoreover, for all $i$, $j$ and $k$ we have\n\\bea\nR^{(0)}_{i,j,k} &\\triangleq& \\Res_{q=0} \\frac{\\kappa_i \\frac{{\\partial} \\ln\\lambda}{{\\partial} \\kappa_i}\n \\kappa_j\\frac{{\\partial} \\ln\\lambda}{{\\partial} \\kappa_j} \\kappa_k \\frac{{\\partial}\n \\ln\\lambda}{{\\partial} \\kappa_k} }{(\\alpha_1+\\alpha_2)^2 q \\frac{{\\partial} \\ln\\lambda}{{\\partial} q}}\\frac{\\mathrm{d}\n q}{q},\\nn \\\\\n&=& \\frac{\\alpha_1^{2-\\delta_{i,n+1}-\\delta_{j,n+1}-\\delta_{k,n+1}}}{(n+1)(\\alpha_1+\\alpha_2)^2} \\\\\nR^{(\\infty)}_{i,j,k} &\\triangleq& \\Res_{q=\\infty} \\frac{\\kappa_i \\frac{{\\partial} \\ln\\lambda}{{\\partial} \\kappa_i}\n \\kappa_j\\frac{{\\partial} \\ln\\lambda}{{\\partial} \\kappa_j} \\kappa_k \\frac{{\\partial}\n \\ln\\lambda}{{\\partial} \\kappa_k} }{(\\alpha_1+\\alpha_2)^2 q \\frac{{\\partial} \\ln\\lambda}{{\\partial} q}}\\frac{\\mathrm{d}\n q}{q},\\nn \\\\\n&=& -\\frac{(-\\alpha_2)^{2-\\delta_{i,n+1}-\\delta_{j,n+1}-\\delta_{k,n+1}}}{(n+1)(\\alpha_1+\\alpha_2)^2}. \n\\label{eq:resinf}\n\\begin{comment}\n\\\\\nR^{(0)}_{0,0,0} &\\triangleq& \\Res_{q=0} \\frac{1}{(\\alpha_1+\\alpha_2)^2}\\frac{1}{(n+1)\\alpha_1+\\l(\\alpha_1+\\alpha_2\\r) \\sum_{l=1}^{n+1}\\frac{\\kappa_l\n q}{1-\\kappa_l q}}\\frac{\\mathrm{d}\n q}{q},\\nn \\\\\n&=& \\frac{1}{(\\alpha_1+\\alpha_2)^2 (n+1)\\alpha_1} \\\\\nR^{(\\infty)}_{0,0,0} &\\triangleq& \\Res_{q=\\infty} \\frac{1}{(\\alpha_1+\\alpha_2)^2}\\frac{1}{(n+1)\\alpha_1+\\l(\\alpha_1+\\alpha_2\\r) \\sum_{l=1}^{n+1}\\frac{\\kappa_l\n q}{1-\\kappa_l q}}\\frac{\\mathrm{d}\n q}{q},\\nn \\\\\n&=& \\frac{1}{(\\alpha_1+\\alpha_2)^2 (n+1)\\alpha_2}\n\\\\\nR^{(0)}_{0,0,i} &\\triangleq& \\Res_{q=0} \\frac{\\alpha_1+\\alpha_2 q\\kappa_i}{(\\alpha_1+\\alpha_2)^2(1-q\\kappa_i)}\\frac{1}{(n+1)\\alpha_1+\\l(\\alpha_1+\\alpha_2\\r) \\sum_{l=1}^{n+1}\\frac{\\kappa_l\n q}{1-\\kappa_l q}}\\frac{\\mathrm{d}\n q}{q},\\nn \\\\\n&=& \\frac{1}{(\\alpha_1+\\alpha_2)^2 (n+1)} \\\\\nR^{(\\infty)}_{0,0,i} &\\triangleq& \\Res_{q=\\infty} \\frac{\\alpha_1+\\alpha_2 q\\kappa_i}{(\\alpha_1+\\alpha_2)^2(1-q\\kappa_i)}\\frac{1}{(n+1)\\alpha_1+\\l(\\alpha_1+\\alpha_2\\r) \\sum_{l=1}^{n+1}\\frac{\\kappa_l\n q}{1-\\kappa_l q}}\\frac{\\mathrm{d}\n q}{q},\\nn \\\\\n&=& -\\frac{1}{(\\alpha_1+\\alpha_2)^2 (n+1)} \n\\\\\nR^{(0)}_{0,j,i} &\\triangleq& \\Res_{q=0} \\frac{(\\alpha_1+\\alpha_2 q\\kappa_i)(\\alpha_1+\\alpha_2 q\\kappa_j)}{(\\alpha_1+\\alpha_2)^2(1-q\\kappa_i)(1-q\\kappa_j)}\\frac{1}{(n+1)\\alpha_1+\\l(\\alpha_1+\\alpha_2\\r) \\sum_{l=1}^{n+1}\\frac{\\kappa_l\n q}{1-\\kappa_l q}}\\frac{\\mathrm{d}\n q}{q},\\nn \\\\\n&=& \\frac{\\alpha_1}{(\\alpha_1+\\alpha_2)^2 (n+1)} \\\\\nR^{(\\infty)}_{0,j,i} &\\triangleq& \\Res_{q=\\infty} \\frac{(\\alpha_1+\\alpha_2 q\\kappa_i)(\\alpha_1+\\alpha_2 q\\kappa_j)}{(\\alpha_1+\\alpha_2)^2(1-q\\kappa_i)(1-q\\kappa_j)}\\frac{1}{(n+1)\\alpha_1+\\l(\\alpha_1+\\alpha_2\\r) \\sum_{l=1}^{n+1}\\frac{\\kappa_l\n q}{1-\\kappa_l q}}\\frac{\\mathrm{d}\n q}{q},\\nn \\\\\n&=& \\frac{\\alpha_2}{(\\alpha_1+\\alpha_2)^2 (n+1)} \n\\\\\n\\end{comment}\n\\end{eqnarray}\nIt is immediate to see that \\eqref{eq:lgqu1}-\\eqref{eq:lgqu2} under the\nidentification \\eqref{eq:kappaY} imply that the quantum part of the\nthree-point correlator $R({\\partial}_{t_{i_1}}{\\partial}_{t_{i_2}}{\\partial}_{t_{i_3}})$\ncoincides with that of $\\left\\langle\\bra p_{i_1}, p_{i_2}, p_{i_3} \\right\\rangle\\ket^Y_{0}$ in\n\\eqref{eq:yukY}. A tedious, but straightforward computation shows that\n\\eqref{eq:lgqu1}-\\eqref{eq:resinf} yield the expressions\nfor the classical triple intersection numbers of $Y$. \\\\\n\\item This is a consequence of the computation above and Theorem \\ref{thm:crc}.\n\\end{enumerate}\n\\end{proof}\n\\begin{figure}\n\\includegraphics{pochcont.pdf}\n\\caption{The double loop contour $\\gamma_4$ for $n=4$.}\n\\label{fig:pochcont}\n\\end{figure}\n\n\\begin{rmk}\nThe freedom of shift by $\\delta_\\mathcal{X}$ and $\\delta_Y$ respectively along\n$H^0(\\mathcal{X})$ and $H^0(Y)$ in \\eqref{eq:kappa0Y},\n\\eqref{eq:kappa0X} is a consequence of the restriction of the String Axiom to the small phase\nspace. We set $\\delta_\\mathcal{X}=\\delta_Y=0$ throughout this section, but it will turn\nout to be useful to reinstate the shifts in the computations of Section~\\ref{sec:compsymp}.\n\\label{rmk:string}\n\\end{rmk}\n\n\n\\begin{rmk}\nIt should be possible to infer the form of the superpotential\n\\eqref{eq:superpot} from the equivariant GKZ system of $\\mathcal{X}$ and $Y$ by\narguments similar to the non-equivariant case (see e.g. \\cite[Appendix\n A]{MR2510741}). The conceptual path we followed to conjecture the form\n\\eqref{eq:superpot} for a candidate dual Landau--Ginzburg model parallels the study of\nthe equivariant local $\\mathbb{CP}^1$ theory in \\cite{Brini:2011ff}; there, the\nexistence of a relation with a reduction of the 2-dimensional Toda hierarchy allows to derive\na Landau--Ginzburg mirror model through the dispersionless Lax formalism for\n2-Toda. More generally, $(n,m)$-graded reductions \\cite{phdthesis-romano} of $2$-Toda are believed to be relevant for the\nequivariant Gromov--Witten theory of local $\\mathbb{P}(n,m)$ \\cite{agps}; the\ndegenerate limit $m=0$ corresponds to the threefold $A_n$ singularity. In this case, the\ndispersionless 2-Toda Lax function reduces to \\eqref{eq:superpot}. \\\\\n\\end{rmk}\n\n\\subsection{The global quantum $D$-module}\n\\label{sec:globdpic}\n\nAn immediate corollary of Theorem~\\ref{thm:mirror} and Proposition~\\ref{thm:tp} is a concrete description of a global quantum $D$-module $(\\mathcal M_A, F, \\nabla,\nH(,)_g)$ interpolating between $\\mathrm{QDM}(\\mathcal{X})$ and $\\mathrm{QDM}(Y)$. Let\n$F \\triangleq T\\mathcal{F}_{\\lambda, \\phi}$ be\nendowed with the family of connections $\\nabla=\\nabla^{(g,z)}$ as in\n\\eqref{eq:defconn} and for $\\nabla$-flat sections $s_1$, $s_2$ let\n\\beq\nH(s_1, s_2)_g = g(s_1(\\kappa, -z),s_2(\\kappa,z))\n\\eeq\nLet now $V_\\mathcal{X}$ and $V_Y$ be neighbourhoods of $\\{\\kappa_i=\\omega^{-i}\\}$ and\n$\\{\\kappa_i=0\\}$ respectively. Then Theorem~\\ref{thm:mirror} can be rephrased\nas\n\\bea\n(\\mathcal{F}_{\\lambda,\\phi}, T\\mathcal{F}_{\\lambda,\\phi}, \\nabla^{(g,z)},H(,)_{g})|_{V_\\mathcal{X}} &\\simeq &\n\\mathrm{QDM}(\\mathcal{X}), \\\\\n(\\mathcal{F}_{\\lambda,\\phi}, T\\mathcal{F}_{\\lambda,\\phi}, \\nabla^{(g,z)},H(,)_{g})|_{V_Y} &\\simeq &\n\\mathrm{QDM}(Y),\n\\end{eqnarray}\nthat is, the twisted period system of $\\mathcal{F}_{\\lambda,\\phi}$ is a global quantum\n$D$-module connecting the genus zero descendent theory of $\\mathcal{X}$ and $Y$; the\ntwisted periods \\eqref{eq:periods} thus define a global flat frame for the\nquantum differential equations of $\\mathcal{X}$ and $Y$ upon analytic continuation in\nthe $\\kappa$-variables, \n\\beq\n\\mathrm{Sol}_{\\lambda,\\phi}|_{V_\\mathcal{X}} = \\mathcal{S}_\\mathcal{X}, \\quad \n\\mathrm{Sol}_{\\lambda,\\phi}|_{V_Y} = \\mathcal{S}_Y. \n\\eeq\n\n\nA canonical basis of $\\mathrm{Sol}_{\\lambda,\\phi}$ can be\n constructed as follows.\nFor the superpotential \\eqref{eq:superpot}, the twisted homology $H_1(\\mathbb{C}\n \\setminus \\lambda^{-1}(0), \\mathbf{L})$ is generated \\cite{MR1424469} by Pochhammer double\n loop contours $\\{\\xi_i\\}_{i=1}^{n+1}$ encircling the origin\n $q=0$ and $q=\\kappa_i^{-1}$, $i=1, \\dots, n+1$, as in Figure~\\ref{fig:pochcont} (alternatively $\\xi_i=[\\rho_0,\\rho_i]$, where the $\\rho$'s are simple oriented loops around each of the punctures). Then the integrals\n\n\n\\bea\n\\Pi_i^{(n)}(\\kappa,z) & \\triangleq & \\frac{1}{(1-\\mathrm{e}^{2\\pi\\mathrm{i} a})(1-\\mathrm{e}^{-2\\pi\\mathrm{i} b})}\\int_{\\xi_i} \\lambda^{1\/z}(q) \\frac{\\mathrm{d} q}{q} \\nn\n\\\\ &=& \\frac{C_n(\\kappa)^{\\frac{1}{z}}}{(1-\\mathrm{e}^{2\\pi\\mathrm{i} a})(1-\\mathrm{e}^{-2\\pi\\mathrm{i} b})} \\int_{\\xi_i}\nq^{a} (1-q)^{-b} \\prod _{k=1}^n \\left(1-q\n\\kappa_k\\right)^{-b}\\frac{\\mathrm{d} q}{q} \\nn \\\\\n&=& \n\\frac{C_n(\\kappa)^{\\frac{1}{z}} \\kappa_i^{-a}}{(1-\\mathrm{e}^{2\\pi\\mathrm{i} a})(1-\\mathrm{e}^{-2\\pi\\mathrm{i} b})} \n \\int_{\\xi_{n+1}} q^{a} (1-q)^{-b}\\left(1-q\/\\kappa_i\\right)^{-b}\n\\prod_{k\\neq i}^n \\left(1-q \\kappa_k\/\\kappa_i\\right)^{-b} \\frac{\\mathrm{d} q}{q} \\nn \\\\\n\\label{eq:eulerint}\n\\end{eqnarray}\nwhere we defined\n\\bea\na & \\triangleq & \\frac{(n+1) \\alpha_1}{z}, \\\\\nb & \\triangleq & \\frac{\\alpha_1+\\alpha_2}{z},\n\\end{eqnarray}\ngive a basis of twisted periods of $\\mathcal{F}_{\\lambda, \\phi}$; when $\\Re(a)>0$, $\\Re\n(b)<1$ they reduce to line integrals along chains connecting $q=0$ to\n$q=\\kappa_i^{-1}$. \\\\\n\n\nThe integrals \\eqref{eq:eulerint} can be given very explicit expressions in\nterms of known generalized hypergeometric functions \\cite{MR0422713}. Namely, we have\n\\bea\n\\Pi_i^{(n)}(\\kappa,z) &=& \n\\frac{\\Gamma(a)\\Gamma(1-b)}{\\Gamma(1+a-b)} C_n(\\kappa)^{\\frac{1}{z}}\n\\kappa_i^{-a}\n \\nn \\\\\n&\\times & \n\\Phi^{(n)}\\l(a,b,1+a-b;\n\\frac{1}{\\kappa_i}, \\frac{\\kappa_1}{\\kappa_i}, \\dots,\n\\frac{\\kappa_n}{\\kappa_i}\\r), \\quad 1\\leq i\\leq n, \\label{eq:pilaur1} \\\\\n\\Pi_{n+1}^{(n)}(\\kappa,z) &=& \\frac{\\Gamma(a)\\Gamma(1-b)}{\\Gamma(1+a-b)} C_n(\\kappa)^{\\frac{1}{z}}\n\\Phi^{(n)}(a,b,1+a-b;\n\\kappa_1, \\dots, \\kappa_n),\n\\label{eq:pilaur2}\n\\end{eqnarray}\nwhere we defined\n\\beq\n\\label{eq:Phi}\n\\Phi^{(M)}(a, b, c, w_1, \\dots, w_M) \\triangleq F_D^{(M)}(a; b, \\dots, b; c; w_1, \\dots, w_{M}),\n\\eeq\nand $F_D^{(M)}(a; b_1, \\dots, b_M; c; w_1, \\dots, w_M)$ in \\eqref{eq:Phi} is the\ngeneralized hypergeometric Lauricella function of type $D$ \\cite{lauric}:\n\\beq\nF_D^{(M)}(a; b_1, \\dots, b_M; c; w_1, \\dots, w_M) \\triangleq \\sum_{i_1, \\dots, i_M}\n\\frac{(a)_{\\sum_j i_j}}{(c)_{\\sum_j i_j}}\\prod_{j=1}^M \\frac{(b_j)_{i_j} w_j^{i_j}}{i_j!}.\n\\label{eq:FD}\n\\eeq\nIn \\eqref{eq:FD}, we used the Pochhammer symbol $(x)_m$ to denote the ratio $(x)_m=\n\\Gamma(x+m)\/\\Gamma(x)$. \\\\\n\n\n\\begin{rmk}\nThat flat sections of $\\mathrm{QDM}(\\mathcal{X})$ and $\\mathrm{QDM}(Y)$ are solutions of a GKZ-type\nsystem, and therefore take the form of generalized hypergeometric functions in\n$B$-model variables, is a direct consequence of equivariant mirror symmetry for toric\nDeligne--Mumford stacks; see \\cite[Appendix A]{MR2510741} for the case under\nstudy here, and \\cite{ccit2} for the general case. Less expected, however, is the fact that flat sections of $\\mathrm{QDM}(\\mathcal{X})$ and $\\mathrm{QDM}(Y)$ are\nhypergeometric functions in {\\it exponentiated flat variables} for \\eqref{eq:pair},\nthat is, in $A$-model variables. This is a consequence of the particular form\n\\eqref{eq:yukY}, \\eqref{eq:lgqu1}-\\eqref{eq:lgqu2}\nof the quantum product: this depends {\\it rationally} on the variables\nin the K\\\"ahler cone for $Y$ in such a way that the quantum differential equation\n\\eqref{eq:QDE} for $Y$ (and therefore $\\mathcal{X}$, via \\eqref{eq:changevar}) becomes a\ngeneralized hypergeometric system in exponentiated flat coordinates. From the vantage\npoint of mirror symmetry, the rational dependence of the $A$-model three-point\ncorrelators on the quantum parameters can be regarded as an epiphenomenon of the Hard Lefschetz\ncondition, which ensures that the inverse mirror map is a rational\nfunction of the $B$-model variables. \n\\end{rmk}\n\n\\begin{rmk}\nAs a further surprising peculiarity of the\ncase of $A_n$ singularities,\nintegral representations of the flat sections have a {\\it simpler}\ndescription in $A$-model variables: the one-dimensional Euler integrals\n\\eqref{eq:eulerint} replace here the $n$-fold Mellin-Barnes contour integrals\nthat represent solutions of the corresponding GKZ system \\cite{MR2510741,\n MR2700280}. This technical advantage is crucial for our\ncalculations of Section~\\ref{sec:compsymp}.\n The reader may find a comparison of the Hurwitz mirror with the traditional\napproach of toric mirror symmetry in \\cite{bcr2013}. \\\\\n\\end{rmk}\n\n\\subsubsection{Example: $n=2$ and the Appell system}\n\\label{sec:appell}\n\nIn this case the quantum $D$-module has rank three. We factor out the dependence on $C_2(\\kappa)$ in\n\\eqref{eq:pilaur1}-\\eqref{eq:pilaur2} for the flat coordinates of the deformed\nconnection as\n\\beq\nf(\\kappa_1,\\kappa_2,z) \\triangleq (\\kappa_0\\kappa_1\\kappa_2)^{-a\/3} \\tilde t(\\kappa_0,\n\\kappa_1, \\kappa_2, z).\n\\label{eq:fdefn2}\n\\eeq\nThe flatness equations for $\\nabla^{(g,z)}$ for $n=2$ reduce to a\nhypergeometric Appell $F_1$ system \\cite{MR0422713} for $f$:\n\\bea\n\\label{eq:F1eq1}\n(\\kappa_1-\\kappa_2){\\partial}_1 {\\partial}_2 f -b ({\\partial}_1-{\\partial}_2)f &=& 0, \\\\\n\\bigg[\\kappa_1(1-\\kappa_1)\\theta_1^2 +\\kappa_2(1-\\kappa_1){\\partial}_{12} \n+(a+1-2b){\\partial}_1 &+& \\nn \\\\ -(a+1+2b) \\kappa_1 {\\partial}_1 -b \\kappa_2{\\partial}_2 -a b \\bigg]f &=&\n0. \n\\label{eq:F1eq2}\n\\end{eqnarray}\nFor $n=2$, the twisted periods \\eqref{eq:pilaur1}-\\eqref{eq:pilaur2} reduce to\nAppell $F_1$ functions \\cite{MR0422713}\n\\bea\n\\Pi_1^{(2)}(\\kappa_0,\\kappa_1,\\kappa_2,z) &=& \\frac{\\Gamma(a)\\Gamma(1-b)}{\\Gamma(1+a-b)} C_2(\\kappa)^{\\frac{1}{z}}\n\\kappa_1^{-a}\n\\Phi^{(2)}\\l(a,b,b, 1+a-b;\n\\frac{1}{\\kappa_1}, \\frac{\\kappa_2}{\\kappa_1} \\r) \\nn \\\\\n&=& \\frac{\\Gamma(a)\\Gamma(1-b)}{\\Gamma(1+a-b)} (\\kappa_0\\kappa_2)^{a\/3} \\kappa_1^{-a\/3}\n\\, F_1\\l(a,b,b,1+a-b,\\frac{1}{\\kappa_1},\n\\frac{\\kappa_2}{\\kappa_1}\\r) \\label{eq:twistn2a} \\\\\n\\Pi_2^{(2)}(\\kappa_0,\\kappa_1,\\kappa_2,z) &=&\n\\Pi_1^{(2)}(\\kappa_0,\\kappa_2,\\kappa_1,z) \\\\\n\\Pi_2^{(3)}(\\kappa_0,\\kappa_2,\\kappa_1,z)\n&=& \\frac{\\Gamma(a)\\Gamma(1-b)}{\\Gamma(1+a-b)} (\\kappa_0 \\kappa_1 \\kappa_2)^{a\/3} \\,\nF_1\\l(a,b,b, 1+a-b,\\kappa_1, \\kappa_2\\r) \\label{eq:twistn2b}\n\\end{eqnarray}\nwhere\n\\beq\nF_1\\l(a,b_1,b_2, c,x,y\\r) \\triangleq \\sum_{i_1, i_2 \\geq 0}\n\\frac{(a)_{i_1+i_2}}{(c)_{i_1+i_2}}\\frac{(b_1)_{i_1}\n x^{i_1}}{i_1!}\\frac{(b_2)_{i_2} y^{i_2}}{i_2!}.\n\\eeq\nIt is straightforward to check that \\eqref{eq:twistn2a}-\\eqref{eq:twistn2b} yield a complete set of\nsolutions of \\eqref{eq:F1eq1}-\\eqref{eq:F1eq2}. \\\\\n\nIn this case, irreducible components of the discriminant locus are given by the lines $\\kappa_1=\\kappa_2$ and\n$\\kappa_i=0,1,\\infty$, $i=1,2$. Its moduli space is depicted in\nFigure~\\ref{fig:modspace2}. The large radius point of $\\mathcal{X}$\n$(\\kappa_1,\\kappa_2)=(\\mathrm{e}^{4\\pi \\mathrm{i}\/3},\\mathrm{e}^{2\\pi \\mathrm{i}\/3})$, denoted OP in Figure~\\ref{fig:modspace2}, is a regular point of the quantum $D$-module\n\\eqref{eq:F1eq1}-\\eqref{eq:F1eq2}, and the Fuchsian singularities\n$(\\kappa_1,\\kappa_2)=(0,0)$ and $(\\infty, \\infty)$ correspond to two\ncopies of the large radius point (henceforth, LR) of $Y$, referred to as LR1 and LR2 in\nFigure~\\ref{fig:modspace2}. The Frobenius structure induced around the latter two\npoints are canonically isomorphic to $QH_T(Y)$, and they are related to one another\nby the involution $\\kappa_i \\to -\\kappa_i$. In contrast with the $n=1$ case\n\\cite{cavalieri2011open, bcr2013}, where the Appell system reduces to the\nGauss ${}_2 F_1$-system, it is\nimpossible here \\cite{MR0422713} to provide a local solution around LR of the Appell system\n\\eqref{eq:F1eq1}-\\eqref{eq:F1eq2} in terms of Appell $F_1$-functions only;\nsee Appendix~\\ref{sec:anFD} for a discussion of this point. \nRepresenting eigenvectors of the monodromy around LR in\ngeneral in terms of the twisted period basis will be the subject of the first\npart of the proof of Theorem \\ref{thm:sympl} in the next section.\n\n\n\\begin{figure}[t]\n\\includegraphics{modspace2.pdf}\n\\caption{The K\\\"ahler moduli space of the $A_2$ singularity in $A$-model\n coordinates.}\n\\label{fig:modspace2}\n\\end{figure}\n\n\\subsection{Proof of Theorem \\ref{thm:sympl}\n}\n\\label{sec:compsymp}\nLet $\\rho$ be a straight line in $\\mathcal M_A$ connecting the large radius point\n$\\{\\kappa_j=0\\}$ of $Y$ to\nthe one of $\\mathcal{X}$, given by $\\{\\kappa_j=\\omega^{-j}\\}$, with zero winding number\naround all irreducible components of the discriminant locus of $\\mathcal M_A$. We compute the analytic\ncontinuation map $\\mathbb{U}_\\rho^{\\mathcal{X}, Y} : \\mathcal{H}_\\mathcal{X} \\to \\mathcal{H}_Y$ that identifies the\ncorresponding flat frames and Lagrangian cones upon analytic continuation\nalong $\\rho$. \\\\\n\nDefine the period map $\\Omega$:%\n\\beq\n\\bary{ccccc}\n\\Omega &:& H_1\\l(\\mathbb{C} \\setminus (\\lambda), \\mathbf{L}\\r) & \\to &\n\\mathcal{O}_{\\mathcal{F}_{\\lambda, \\phi}}, \\\\\n& & \\xi & \\to & \\int_\\xi \\lambda^{1\/z} \\phi,\n\\eary\n\\label{eq:periodmap}\n\\eeq\nand denote by $\\Pi^{(n)}$ as in \\eqref{eq:eulerint} the image of the\nbasis $\\xi$ of twisted cycles of Section~\\ref{sec:globdpic} under the period map. The horizontality \\eqref{eq:fundsol}-\\eqref{eq:Jfun1} of the $J$-functions of\n$\\mathcal{X}$ and $Y$, the String Equation for $\\mathcal{X}$ and $Y$, and Proposition~\\eqref{thm:tp} together state that $J^\\mathcal{X}$,\n$J^Y$ and $\\Pi^{(n)}$ are three\ndifferent $\\mathbb{C}(\\mathrm{e}^{\\mathrm{i} \\pi a}, \\mathrm{e}^{\\mathrm{i}\\pi b}, z)$-bases of deformed flat coordinates of $\\nabla^{(g,z)}$ under the identifications\n\\eqref{eq:kappa0Y}-\\eqref{eq:kappaY},\n\\eqref{eq:kappa0X}-\\eqref{eq:kappakX}. This entails, for every $\\rho$, the\nexistence of two \n$\\mathbb{C}(\\mathrm{e}^{\\mathrm{i} \\pi a}, \\mathrm{e}^{\\mathrm{i} \\pi b}, z)$-linear maps $A$, $B$\n\\beq\n\\bary{ccccc}\n\\nabla^{(\\eta_Y)} A \\Omega & : & H_1\\l(\\mathbb{C} \\setminus (\\lambda), \\mathbf{L}\\r) & \\to & \\mathcal{S}_Y, \\\\\n\\nabla^{(\\eta_\\mathcal{X})}B^{-1} \\Omega & : & H_1\\l(\\mathbb{C} \\setminus (\\lambda), \\mathbf{L}\\r) & \\to &\n\\mathcal{S}_\\mathcal{X},\n\\label{eq:AB}\n\\eary\n\\eeq\nsuch that\n\\bea\n\\label{eq:pijy}\nA \\Pi^{(n)} &=& J_Y, \\\\\nB J_\\mathcal{X} &=& \\Pi^{(n)}.\n\\label{eq:pijx}\n\\end{eqnarray}\nIn particular,\n\\beq\n\\mathbb{U}_\\rho^{\\mathcal{X}, Y} = A B. \n\\label{eq:UBA}\n\\eeq\n\\\\\n\n$A$ sends the twisted period basis $\\Pi^{(n)}$ to a basis of eigenvectors of the\nmonodromy around the large radius point of $Y$ normalized as in\n\\eqref{eq:Jloc}. We compute $A$\nby investigating the leading asymptotics of the twisted periods\n\\eqref{eq:pilaur1}-\\eqref{eq:pilaur2} around the large radius point of $Y$; as in the example of\nSection~\\ref{sec:appell}, we denote the latter by LR. \\\\\n\n\nIn $\\mathbb{C}^m$ with coordinates $(w_1, \\dots, w_m)$, let $\\chi_i$, for\nevery $i=1, \\dots, m$, be a path connecting the point at\ninfinity $W^\\infty_i$,\n\\beq\nW_i^\\infty\\triangleq(\\overbrace{0,\\dots, 0}^{\\text{$i$ times}}, \\overbrace{\\infty,\\dots,\n \\infty}^{\\text{$m-i$ times}}),\n\\eeq\nwith zero winding number along $w_i=w_j$ ($i \\neq j$) and $w_i=0,1$. We want\nto compute the analytic continuation along $\\chi_i$ of the\nLauricella function \n$F_D^{(m)}(a,b_1, \\dots, b_n, c, w_1, \\dots, w_i, w_{i+1}^{-1}, \\dots, w_m^{-1})$\nfrom an open ball centered on $W^\\infty_i$ to the origin \n$W^\\infty_0=(0, \\dots, 0)$ in the sector where $w_i \\ll 1$,\n$w_i\/w_j \\ll 1$ for $ii$ appearing in \\eqref{eq:FD} through an iterated\nuse of Goursat's identity \\eqref{eq:2F1conn}. The final result is\n\\eqref{eq:fdinf}; we refer the reader to Appendix \\ref{sec:anFD} for the\ndetails of the derivation. \\\\\n\nIn our case, Eq.~\\eqref{eq:fdinf} (see also Remark~\\ref{rmk:relabel}) implies, around $w_i=\\infty$, that\n\\bea\n\\Phi^{(m)}(a, b, c; w_1, \\dots, w_m) & \\sim & \n\\sum_{j=0}^{m-1}\\Gamma\\l[\\bary{ccc}c, & a-j b, & (j+1)\n b-a \\\\ a, & b, & c-a \\eary\\r] \\nn \\\\ & & \\prod_{i=1}^j (-w_{m-i+1})^{-b}\n(-w_{m-j})^{-a+j b}\\nn \\\\ &+& \\prod_{j=1}^m (-w_j)^{-b}\n\\Gamma\\l[\\bary{cc}c, & a-m b \\\\ a, & c-m b \\eary\\r].\n\\end{eqnarray}\nwhen $w_i \\sim 0$, $w_i\/w_j \\sim 0$ for $j>i$. In particular, at the level of twisted periods this entails\n\\bea\n\\Pi_{n-k}^{(n)} & \\sim & C_n(\\kappa)^{\\frac{1}{z}}\n\\kappa_{n-k}^{-a}\n\\frac{\\Gamma(a)\\Gamma(1-b)}{\\Gamma(1+a-b)} \n\\Phi^{(k+1)}\\l(a,b,1+a-b, \\frac{\\kappa_{n-k+1}}{\\kappa_{n-k}},\n \\dots, \\frac{\\kappa_{n}}{\\kappa_{n-k}}, \\frac{1}{\\kappa_{n-k}} \\r)\n\\nn \\\\\n& \\sim & \nC_n(\\kappa)^{\\frac{1}{z}}\n\\kappa_{n-k}^{-a}\n\\Bigg\\{\n\\frac{\\Gamma(a)\\Gamma(b-a)}{\\Gamma(b)} \\l(-\\kappa_{n-k}\\r)^{a} \\nn \\\\ &+& \n\\sum_{j=1}^{k}\\frac{\\Gamma(a-j b)\\Gamma((j+1) b-a)}{\\Gamma(b)}\n\\l(-\\frac{\\kappa_{n+1-j}}{\\kappa_{n-k}}\\r)^{-a+j b} (-\\kappa_{n-k})^{b} \\prod_{i=1}^{j-1} \\l(-\\frac{\\kappa_{n+1-i}}{\\kappa_{n-k}}\\r)^{-b}\n\\nn \\\\ &+& \\kappa_{n-k}^{(k+1) b} \n\\frac{\\Gamma(1-b)\\Gamma(a-(k+1) b)}{\\Gamma(1+a-(k+2) b)} \\prod_{j=n-k+1}^{n} (-\\kappa_j)^{-b}\\Bigg\\}\n\\nn \\\\\n& \\sim & \nC_n(\\kappa)^{\\frac{1}{z}}\n\\Bigg\\{\n\\sum_{j=0}^{k}\\frac{\\Gamma(a-j b)\\Gamma((j+1) b-a)}{\\Gamma(b)} (-1)^a \\l(\\kappa_{n+1-j}\\r)^{-a+j b} \\prod_{i=1}^{j-1} \\l(\\kappa_{n+1-i}\\r)^{-b}\n\\nn \\\\ &+& (-1)^{(k+1) b}\n\\frac{\\Gamma(1-b)\\Gamma(a-(k+1) b)}{\\Gamma(1+a-(k+2) b)}\\kappa_{n-k}^{(k+1)\n b-a} \\prod_{j=n-k+1}^{n} (\\kappa_j)^{-b} \\Bigg\\}.\n\\label{eq:dectp}\n\\end{eqnarray}\nin a neighbourhood of $\\kappa=0$ given by $|\\kappa_i| \\ll 1$,\n$\\kappa_i\/\\kappa_j \\ll 1$ for $j>i$; notice that in cohomology coordinates\n\\eqref{eq:kappaY} for $Y$, this becomes an actual open ball $|q|\\ll 1$ around the point\nof classical limit $q_i = \\mathrm{e}^{t_i}=0$. Now, from the discussion of\nSection~\\ref{sec:GIT} and Eqns.~\\eqref{eq:Jred}, \\eqref{eq:kappaY}, around\nthe limit point of classical cohomology the $J$-function of $Y$ behaves as\n\\beq\nJ^Y_{p_i} = z C_n(\\kappa)^{\\frac{1}{z}} \\kappa_i^{(n-i+1)b-a}\\prod_{j=i+1}^n(\\kappa_j)^{-b}\\l(1+\\mathcal{O}(\\mathrm{e}^{t})\\r).\n\\label{eq:Jred2}\n\\eeq\nThen we can read off from \\eqref{eq:dectp}-\\eqref{eq:Jred2} the decomposition of each twisted period\n$\\Pi_i^{(n)}$ in terms of eigenvectors of the monodromy around LR, and in\nparticular, in terms of the localized components of the $J$-function. Explicitly,\n\\beq\n\\Pi^{(n)} = A^{-1} J^Y,\n\\eeq\nwhere\n\\beq\nA^{-1}_{ji} = \\left\\{\\bary{cl} (-1)^{(n-i+1) b}\n\\frac{\\Gamma(1-b)\\Gamma(a-(n-i+1) b)}{z \\Gamma(1+a-(n-i+2) b)} & \\mathrm{for}\n\\quad i=j, \\\\\n(-1)^a\\frac{\\Gamma(a-(n-i+1) b)\\Gamma((n-i+2) b-a)}{z \\Gamma(b)} & \\mathrm{for}\n\\quad ji.\n \\eary\\right. \n\\label{eq:matrAINV}\n\\eeq\nIts inverse reads\n\\beq\nA_{ij} = \\left\\{\\bary{cl} \\mathrm{e}^{\\pi\\mathrm{i} (n-i+1) b}\n\\frac{z \\Gamma(1+a-(n-i+2) b)\\Gamma(1-a+(n-i+1) b)\\sin(a+(n-i+1) b)}{\\Gamma(1-b)\\pi} & i=j, \\\\\n\\mathrm{e}^{-i \\pi (a-b (2 n-2j+3))} \\frac{z \\sin (\\pi b) \\Gamma (1-a+b (n+1-i)) \\Gamma (1+a-b (n-i+2))}{\\pi \\Gamma (1-b)}\n & j>i, \\\\\n0 & j1$; in doing so, we fix a path of\nanalytic continuation by choosing the principal branch for both the power functions\n$(-z)^{-a}$ and $(-z)^{-b}$ in \\eqref{eq:2F1conn} and continue $\\,\n_2F_1(a,b;c;z)$ to $|z>1|$ along a path that has winding number zero around\nthe Fuchsian singularity at $z=1$. As a power series in $w_N$ the analytic continuation\nof \n\\eqref{eq:FD2F1} around $w_N=\\infty$ then reads\n\\bea\n& & F_D^{(N)}(a; b_1, \\dots, b_N; c; w_1, \\dots, w_N) = (-w_N)^{-a}\n\\Gamma\\l[\\bary{cc}c, & b_N-a \\\\ b_N, & c-a \\eary\\r] \\nn \\\\ & & F_D^{(N)}\\l(a; b_1, \\dots,\nb_{N-1}, 1-c+a; 1-b_N+a,\\frac{w_1}{w_N}, \\dots, \\frac{1}{w_N}\\r) \n+\n(-w_N)^{-b_N}\n\\Gamma\\l[\\bary{cc}c, & a-b_N \\\\ a, & c-b_N \\eary\\r] \\nn \\\\ & & C_N^{(N-1)}\\l(b_1, \\dots,\nb_{N}, 1-c+b_N; a-b_N,-w_1,-w_2, \\dots, \\frac{1}{w_N}\\r),\n\\label{eq:FDcont1}\n\\end{eqnarray}\nwhere we defined \\cite[Chapter~3]{MR0422713}\n\\bea\nC_N^{(k)}\\l(b_1, \\dots,\nb_{N}, a; a',x_1, \\dots, x_N\\r) & \\triangleq & \\sum_{i_1, \\dots, i_{N}}\n(a)_{\\alpha_N^{(k)}(\\mathbf{i})}(a')_{-\\alpha_N^{(k)}(\\mathbf{i})} \\prod_{j=1}^{N}\n\\frac{(b_j)_{i_j} w_j^{i_j}}{i_j!} \n\\end{eqnarray}\nand \n\\bea\n\\alpha_N^{(k)}(\\mathbf{i}) &\\triangleq& \\sum_{j=k+1}^{N} i_j-\\sum_{j=1}^{k} i_j,\\\\\n\\Gamma\\l[\\bary{ccc}a_1, & \\dots, & a_m \\\\ b_1, & \\dots, & b_n \\eary\\r] &\\triangleq&\n\\frac{\\prod_{i=1}^m \\Gamma(a_i)}{\\prod_{i=1}^l\\Gamma(b_i)}.\n\\end{eqnarray}\nNow, notice that the $F_D^{(N-1)}$ function in the r.h.s. of \\eqref{eq:FDcont1} is\nanalytic in $\\Omega_N$; there is nothing more that should be done there. The\nanalytic continuation of the $C_N^{(N-1)}$ function is instead much more involved (see\n\\cite{MR0422713} for a complete treatment of the $N=3$ case); but as all we\nare interested in is the leading term of the expansion around $P$ in\n$\\Omega_N$ we isolate the $\\mathcal{O}(1)$ term in its $1\/w_N$ expansion to find\n\\bea\n& & C_N^{(N-1)}\\l(b_1, \\dots,\nb_{N}, 1-c+b_N; a-b_N,-w_1,-w_2, \\dots, \\frac{1}{w_N}\\r) = \\nn \\\\\n&=& F_D^{(N-1)}\\l(a-b_N,b_1, \\dots,\nb_{N-1}, c-b_N; w_1, \\dots, w_{N-1}\\r)+ \\mathcal{O}\\l(\\frac{1}{w_{N}}\\r)\n\\label{eq:CNFD}\n\\end{eqnarray}\nWe are done: by \\eqref{eq:CNFD}, the form of the leading terms in the expansion of\n$F_D^{(N)}$ inside $\\Omega_N$ can be found recursively by iterating $N$ times the\nprocedure we have followed in \\eqref{eq:FD2F1}-\\eqref{eq:CNFD}; as at each step \\eqref{eq:2F1conn}-\\eqref{eq:CNFD} generate\none additional term, we end up with a sum of $N+1$ monomials each having\npower-like monodromy around $P$. Explicitly:\n\\bea\nF_D^{(N)}(a; b_1, \\dots, b_N; c; w_1, \\dots, w_N) & \\sim & \n\\sum_{j=0}^{N-1}\\Gamma\\l[\\bary{ccc}c, & a-\\sum_{i=N-j+1}^N b_i, & \\sum_{i=n-j}^N\n b_i-a \\\\ a, & b_{N-j}, & c-a \\eary\\r] \\nn \\\\ & & \\prod_{i=1}^j (-w_{N-i+1})^{-b_{N-i+1}}\n(-w_{N-j})^{-a+\\sum_{i=N-j+1}^Nb_i}\\nn \\\\ &+& \\prod_{i=1}^N (-w_i)^{-b_i}\n\\Gamma\\l[\\bary{cc}c, & a-\\sum_{i=1}^N b_j \\\\ a, & c-\\sum_{i=1}^N b_j \\eary\\r].\n\\label{eq:fdinf}\n\\end{eqnarray}\n\n\\begin{rmk}\nThe analytic continuation to some other sectors of the ball $B(P,\\epsilon)$ is\nstraightforward. In particular we can replace the condition $w_i\/w_j \\sim 0$\nfor $j>i$ by its reciprocal $w_j\/w_i \\sim 0$; this amounts to relabeling $b_i\n\\to b_{N-i+1}$ in \\eqref{eq:fdinf}.\n\\label{rmk:relabel}\n\\end{rmk}\n\n\\begin{rmk} When $a=-d$ for $d\\in \\mathbb{Z}^+$, the function $F_D^{(N)}$ reduces to\n a polynomial in $w_1, \\dots, w_N$. In this case the arguments above reduce\n to a formula of Toscano \\cite{MR0340663} for Lauricella polynomials:\n\\bea\n& & F_D^{(N)}(-d; b_1, \\dots, b_N; c; w_1, \\dots, w_N) \\nn \\\\ &=& (-w_N)^{d}\\frac{(b)_d}{(c)_d}\n F_D^{(N)}\\l(-d; b_1, b_2 \\dots,\nb_{N-1};1-d-c, 1-d-b_N,\\frac{w_1}{w_N}, \\dots, \\frac{1}{w_N}\\r).\n\\label{eq:FDtosc}\n\\end{eqnarray}\n\\label{rmk:toscano}\n\\end{rmk}\n\\begin{comment}\n\\section{GKZ mirror symmetry}\n\\label{sec:tms}\n\\subsection{$I$-functions and the Picard--Fuchs system}\n\nIt is instructive to compare the hypergeometric form of the $\\nabla$-flat\nsections \\eqref{eq:pilaur1}-\\eqref{eq:pilaur2} in $A$-model variables with the\ngeneralized hypergeometric functions that arise from solutions of the\nmirror GKZ system. \\\\\n\nLet us start from $n=1$. If we dualize \\eqref{eq:divclass},\n\\beq\n0\\longrightarrow\\mathbb{Z}^3\\stackrel{\\l(\\bary{ccc} 1 & 0 & 0 \\\\ 1 & 1 & 0 \\\\ 1 & 2 & 0 \\\\\n 0 & 0 & 1\\eary\\r)}{\\longrightarrow}\\mathbb{Z}^{4}\\stackrel{\\l( \\bary{cccc} 1 & -2 &\n 1 & 0 \\eary\\r)}{\\longrightarrow}\\mathbb{Z}\\longrightarrow\n0,\n\\label{eq:divclass2}\n\\eeq\ngives us the chamber decomposition depicted in Figure~\\ref{fig:bmod1} (the\n{\\it secondary fan} of $\\mathcal{X}$ and $Y$).\n\\begin{figure}[h]\n\\centering\n\\includegraphics{bmod1.pdf}\n\\caption{The secondary fan for the case $n=1$.}\n\\label{fig:bmod1}\n\\end{figure}\nThe {\\it B-model moduli space} $\\mathcal M_B$ is the toric orbifold corresponding to the\none-dimensional fan in Figure~\\ref{fig:bmod1}. From \\eqref{eq:divclass2}, we have\n$\\mathcal M_B \\simeq \\mathbb{P}(1,2)$: the northern (smooth) hemisphere of the orbifold projective line\ncorresponds to the right hand chamber in Figure~\\ref{fig:bmod1}, which gives rise\nto the resolution $Y = (K\\oplus\\mathcal{O})_{\\mathbb{P}^1}$; conversely for the southern\nhemisphere and the singularity $\\mathcal{X}=[\\mathbb{C}^3\/\\mathbb{Z}_2]$. Let $y_s$ be a coordinate\npatch for the smooth patch, and $y_o$ for the orbifold patch, so that\n$y_s=1\/y_o^2$; the large volume points for $Y$ and $\\mathcal{X}$ correspond to $y_s=0$\nand $y_o=0$ respectively. The Picard--Fuchs operator $\\mathcal{D}$ for $\\mathcal{X}$ and $Y$ reads \\cite{MR2510741}\n\\bea\n\\label{eq:GKZX1}\n\\mathcal{D} &=& z \\theta_o (z\\theta_o -z)- y_o^2 \\l(\\alpha_1-\\frac{z}{2}\n\\theta_o\\r)\\l(\\alpha_2-\\frac{z}{2} \\theta_o\\r), \\qquad \\theta_o =\n\\frac{{\\partial}}{{\\partial} y_o}, \\\\\n&=& 2 z \\theta_s (2 z\\theta_s +z)- y_s^{-1}\\l(\\alpha_1+z \\theta_s\\r)\\l(\\alpha_2+z \\theta_s\\r), \\qquad \\theta_s =\n\\frac{{\\partial}}{{\\partial} y_s}.\n\\label{eq:GKZY1}\n\\end{eqnarray}\nConsider the patch of $\\mathcal M_B$ first and define the $I$-function\nof $Y$ as the cohomology valued series\n\\bea\nI^{K_{\\mathbb{P}^1}\\oplus\\mathcal{O}_{\\mathbb{P}^1}}(y_s,z) & \\triangleq & z y_s^{p\/z}\\left[1+\\sum_{d>0}\n2 p y_s^{d} \\frac{\\Gamma(\\frac{2 p}{z}+2d)}{\\Gamma(\\frac{2\n p}{z}+1)}\n\\frac{\\Gamma(\\frac{p+z+\\alpha_1}{z})}{\\Gamma(\\frac{p+z+\\alpha_1}{z}+d)} \n\\frac{\\Gamma(\\frac{p+z+\\alpha_2}{z})}{\\Gamma(\\frac{p+z+\\alpha_2}{z}+d)}\\right],\n\\nn \\\\\n&=& y_s^{\\alpha_1\/z}\\left[z+2\\alpha_1\\sum_{d>0}\n \\frac{y_s^{d}}{d!} \\frac{\\Gamma(2d+\\frac{2\\alpha_1}{z})}{\\Gamma(1+\\frac{2\n \\alpha_1}{z})}\n\\frac{\\Gamma(\\frac{z-\\alpha_2+\\alpha_1}{z})}{\\Gamma(\\frac{z-\\alpha_2+\\alpha_1}{z}+d)}\\right]\nP_2 + (1 \\leftrightarrow 2), \\nn \\\\\n&=& z y_s^{\\alpha_1\/z} \\, _2F_1\\l(\\frac{\\alpha_1}{z}, \\frac{1}{2}+\\frac{\\alpha_1}{z}, \\frac{\\alpha_1-\\alpha_2}{z}, 4 y_s\\r) P_2\n+ (1 \\leftrightarrow 2),\n\\label{eq:IY2F1}\n\\end{eqnarray}\nwhere $p=c_1(\\mathcal{O}_{\\mathbb{P}^1}(1))=\\alpha_2 P_1+\\alpha_1 P_2$ is the hyperplane class and\n$\\{P_1, P_2\\}$ is the localized basis for $H_T(Y)$ of Section~\\ref{sec:GIT}\n(that is, the equivariant classes corresponding to the North and the South\npole of the base $\\mathbb{P}^1$). Then the components of $I^Y$ are a basis of\nsolutions of \\eqref{eq:GKZY1}, and under the mirror map\n\\beq\ny_s = \\frac{\\mathrm{e}^{t}}{(1+\\mathrm{e}^{t})^2}\n\\label{eq:mirmap1}\n\\eeq\nit gives \\cite{MR2276766} a family of elements of the cone $\\mathcal{L}_Y$ of $Y$ such that\n\\beq\nI^Y(y_s(t),-z) = -z + t + \\mathcal{O}\\l(\\frac{1}{z}\\r).\n\\eeq\nThen\n\\beq\nJ^Y(t,z)|_{t_0=0} = I^Y(y_s(t),z),\n\\eeq\nas can be verified upon plugging \\eqref{eq:mirmap1} into \\eqref{eq:IY2F1},\nusing\n\\beq\n _2F_1(a,b;a-b+1;z)=(1-z)^{-a} \\,\n _2F_1\\left(\\frac{a}{2},\\frac{a+1}{2}-b;a-b+1;-\\frac{4\n z}{(1-z)^2}\\right)\n\\eeq\nand comparing with \\eqref{eq:t1LR1}-\\eqref{eq:t1LR2}. \\\\\n\nSimilarly, in the orbifold chamber we define\n\\bea\nI^{[\\mathbb{C}^3\/\\mathbb{Z}_2]}(y_o,z) & \\triangleq & z \\Bigg[\\sum_{k\\geq 0}\n \\frac{y_o^{2k}}{(2k)!}\\prod_{r=0}^{k-1}\\l(\\frac{\\alpha_1}{z}-r\\r)\\l(\\frac{\\alpha_2}{z}-r\\r) \\mathbf{1}_0\n \\nn \\\\\n&+& \\sum_{k\\geq 0}\n \\frac{y_o^{2k+1}}{(2k+1)!}\\prod_{r=0}^{k-1}\\l(\\frac{\\alpha_1}{z}-r-\\frac{1}{2}\\r)\\l(\\frac{\\alpha_2}{z}-r-\\frac{1}{2}\\r) \\mathbf{1}_{1\/2}\n\\Bigg], \\nn \\\\\n &=& z \\, _2F_1\\l(\\alpha_1, \\alpha_2, \\frac{1}{2}, \\frac{y_o^2}{4}\\r) \\mathbf{1}_0 + y_o\n\\, _2F_1\\l(\\alpha_1+\\frac{1}{2}, \\alpha_2+\\frac{1}{2}, \\frac{3}{2}, \\frac{y_o^2}{4}\\r) \\mathbf{1}_{1\/2}.\n\\end{eqnarray}\nAs before,\n\\beq\nJ^\\mathcal{X}(x,z)|_{x_0=0} = I^\\mathcal{X}(y_o(x),z).\n\\eeq\n\\begin{rmk}\nNotice that the orbifold point $y_o=0$ in the $B$-model moduli space is a\nFuchsian singularity for \\eqref{eq:GKZX1} with critical exponents\n$(0,1\/2)$, and therefore $\\mathbb{Z}_2$ monodromy. This is unlike the orbifold point in the $A$-model, which is a\nsmooth point for the Dubrovin connection, since the projection map\n$y_s:\\mathcal M_A\\to\\mathcal M_B$ realizes $\\mathcal M_A$ as a double cover of $\\mathcal M_B$ branched at the conifold and the orbifold point.\n\\end{rmk}\nFor general $n$, the $B$-model moduli space is the projective toric\nDeligne--Mumford stack associated to the simplicial stacky fan with rays given\nby the columns of $M$ in \\eqref{eq:MN}. The large volume chamber,\ncorresponding to the stability condition \\eqref{resgit}, is the chamber with\nmaximal cones \n\\beq\n\\sigma_i=\\l\\{ \\bary{ccc} M_{\\bullet,i} & \\mathrm{for} & 1\\leq i 0$, the\nresulting series converges for $|y_{s,n}|R_{d_1, \\dots, d_{n-1}}$. Iterating this procedure gives analytic\ncontinuation formulae for $I^Y(y_s,z)$ to other chambers of $\\mathcal M_B$. \\\\\n\nFor example, start from $n=1$ and rewrite \n\\beq\nI^{K_{\\mathbb{P}^1}\\oplus\\mathcal{O}_{\\mathbb{P}^1}}(y_s,z) =\n\\Theta(\\alpha_1,\\alpha_2,z) y_s^{-\\alpha_1\/z} \n \\sum_{d\\geq 0}\n \\frac{y_s^{d}}{d!}\n \\frac{\\Gamma(2d-\\frac{2\\alpha_1}{z})}{\\Gamma(\\frac{z+\\alpha_2-\\alpha_1}{z}+d)}\n P_2 + (1 \\leftrightarrow 2)\n\\eeq\nwhere $\\Theta(\\alpha_1,\\alpha_2,z)=-2 \\alpha_1 \n \\frac{\\Gamma(\\frac{z+\\alpha_2-\\alpha_1}{z})}{\\Gamma(1-\\frac{2\n \\alpha_1}{z})}$. This converges when $|y_s|<1\/4$. Let $C$ be the contour in\n the complex $s$-plane depicted in Fig. \\ref{fig:contc3z2}, and consider the integral\n\\beq\n\\mathcal{I}(y_s,z) = \n\\int_{C} \\mathrm{d} s\\frac{y_s^{s}}{\\Gamma(s+1)} \\frac{\\Gamma(s)\\Gamma(1-s)\n \\Gamma(2s-\\frac{2\\alpha_1}{z})}{\\Gamma(\\frac{z+\\alpha_2-\\alpha_1}{z}+s)} P_2 + (1 \\leftrightarrow 2)\n\\eeq\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.8]{analytic.pdf}\n\\caption{The contour $C$.}\n\\label{fig:contc3z2}\n\\end{figure}\nWhen $|y_s|<1\/4$, we close the contour to the right and pick up residues at\n$s=n\\geq 0$. This gives\n$I^N_{K_{\\mathbb{P}^1}\\oplus\\mathcal{O}_{\\mathbb{P}^1}}(y_s,z) y_s^{\\alpha_1\/z} \/\n\\Theta(\\alpha_1,\\alpha_2,z)$. When $|y_s|>1\/4$, we close the contour to the right\nand pick up residues at $s=-n\/2+\\alpha_1\/z$. We obtain\n\\bea\n\\tilde I_{K_{\\mathbb{P}^1}\\oplus\\mathcal{O}_{\\mathbb{P}^1}}(y_s,z) &=& -\\frac{\\Theta(\\alpha_1,\\alpha_2,z)}{2}\n \\sum_{n\\geq 0}\n \\frac{(-y_o)^{n}}{n!} \\frac{\\Gamma(n\/2-\\alpha_1\/z)}{\\Gamma(1-n\/2+\\alpha_2\/z)} P_2 + (1 \\leftrightarrow 2)\n\\end{eqnarray}\nThe symplectomorphism $\\mathbb{U}^{\\mathcal{X}, Y}: \\mathcal{L}_{[\\mathbb{C}^3\/\\mathbb{Z}_2]}\\to\n\\mathcal{L}_{(K+\\mathcal{O})_{\\mathbb{P}^1}}$ relating the cones of $\\mathcal{X}$ and $Y$ is designed so that\n$U(I^{[\\mathbb{C}^3\/\\mathbb{Z}_2]}(y_o, -z)) = \\tilde\nI^{K_{\\mathbb{P}^1}\\oplus\\mathcal{O}_{\\mathbb{P}^1}}(y_s,-z)$. As $U$ is\nindependent of $y_o$, we equate the first few powers of $y_o$ on both sides. We\nhave\n\\beq\nI^{[\\mathbb{C}^3\/\\mathbb{Z}_2]}(y_o, -z) = -z \\mathbf{1}_0 - z y_o \\l(\\frac{\\alpha_1}{z}+\\frac{1}{2}\\r)\\l(\\frac{\\alpha_2}{z}+\\frac{1}{2}\\r)\\mathbf{1}_{\\frac{1}{2}}+\\mathcal{O}\\l(y_o^2\\r)\n\\eeq\nand thus\n\\bea\n\\mathbb{U}^{\\mathcal{X},Y}\\l(\\mathbf{1}_0\\r) &=& \n\\frac{\\alpha_1 \\Gamma \\left(\\frac{\\alpha_1}{z}\\right) \\Gamma \\left(\\frac{z+\\alpha_1-\\alpha_2}{z}\\right)}{\\Gamma \\left(\\frac{2 \\alpha_1}{z}+1\\right) \\Gamma \\left(1-\\frac{\\alpha_2}{z}\\right)}P_2+\\frac{\\alpha_2\n \\Gamma \\left(\\frac{\\alpha_2}{z}\\right) \\Gamma \\left(\\frac{z-\\alpha_1+\\alpha_2}{z}\\right)}{\\Gamma\n \\left(1-\\frac{\\alpha_1}{z}\\right) \\Gamma \\left(\\frac{2\n \\alpha_2}{z}+1\\right)} P_1, \\\\\n\\mathbb{U}^{\\mathcal{X},Y}\\l(\\mathbf{1}_{\\frac{1}{2}}\\r) &=& \n\\sqrt{\\pi } z^2 \\left(\\frac{4^{-\\frac{\\alpha_1}{z}} \\Gamma \\left(\\frac{z+\\alpha_1-\\alpha_2}{z}\\right)}{\\Gamma\n \\left(\\frac{\\alpha_1}{z}\\right) \\Gamma \\left(\\frac{1}{2}-\\frac{\\alpha_2}{z}\\right)}P_2+\\frac{ 4^{-\\frac{\\alpha_2}{z}}\n \\Gamma \\left(\\frac{z-\\alpha_1+\\alpha_2}{z}\\right)}{\\Gamma \\left(\\frac{1}{2}-\\frac{\\alpha_1}{z}\\right) \\Gamma\n \\left(\\frac{\\alpha_2}{z}\\right)}P_1\\right)\n\\end{eqnarray}\n\n\\begin{rmk}\nWorking parametrically in $n$, the analytic continuation of the $n$-fold Mellin--Barnes\nintegral representation of \\eqref{eq:IYn} becomes practically unwieldy, as it\nwould require to iterate this procedure for every chamber that is crossed in\nthe process (which are $n$ in our case). \n\\end{rmk}\n\\end{comment}\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{secIntro}\nThe class of adaptive importance sampling (AIS) methods is a key Monte Carlo methodology for estimating integrals that cannot be obtained in closed form \\citep{robert2004monte}. This problem arises in many settings, such as Bayesian signal processing and machine learning \\citep{bugallo2015adaptive, bugallo2017adaptive} or optimal control, \\citep{kappen2016adaptive} where the quantities of interest are usually defined as intractable expectations. Adaptive importance samplers are versions of classical importance samplers (IS) which iteratively improve the proposals to generate samples better suited to the estimation problem at hand. Its variants include, for example, \\textit{population Monte Carlo} methods \\citep{cappe2004population} and adaptive mixture importance sampling \\citep{cappe2008adaptive}. Since there has been a surge of papers on the topic of AIS recently, a comprehensive review is beyond the scope of this article; see e.g. \\cite{bugallo2017adaptive} for a recent review.\n\nDue to the popularity of the adaptive importance samplers, their theoretical performance has also received attention in the past few years. The same as conventional IS methods, AIS schemes enjoy the classical $\\mathcal{O}(1\/\\sqrt{N})$ convergence rate of the $L_2$ error, where $N$ is the number of Monte Carlo samples used in the approximations, see e.g. \\cite{robert2004monte} and \\cite{agapiou2017importance}. However, since an adaptation is performed over the iterations and the goal of this adaptation is to improve the proposal quality, an insightful convergence result would provide a bound which explicitly depends on the number of iterations, $t$, (which sometimes we refer to as \\textit{time}) and the number of samples, $N$. Although there are convergence results of adaptive methods (see \\cite{douc2007convergence} for a convergence theory for population Monte Carlo based on minimizing Kullback-Leibler divergence), none of the available results yields an explicit bound of the error in terms of the number of iterations and the number of particles at the same time.\n\nOne difficulty of proving such a result for adaptive mixture samplers is that the adaptive mixtures form an interacting particle system and it is unclear what kind of adaptation they perform or whether the adapted proposals actually get closer to the target for some metric. An alternative to adaptation using mixtures is the idea of minimizing a cost function in order to adapt the proposal. This idea has been popular in the literature, in particular, minimizing the variance of the weight function has received significant attention, see, e.g., \\citet{arouna2004adaptative,arouna2004robbins, kawai2008adaptive, lapeyre2011framework, ryu2014adaptive, kawai2017acceleration, kawai2018optimizing}. Relevant to us, in particular, is the work of \\citet{ryu2014adaptive}, who have have proposed an algorithm called Convex Adaptive Monte Carlo (Convex AdaMC). This scheme is based on minimizing the variance of the IS estimator, which is a quantity related to the $\\chi^2$ divergence between the target and the proposal. \\citet{ryu2014adaptive} have shown that the variance of the IS estimator is a convex function of the parameters of the proposal when the latter is chosen within the exponential family. Based on this observation, \\citet{ryu2014adaptive} have formulated Convex AdaMC, which draws one sample at each iteration and construct the IS estimator, which requires access to the normalised target. They proved a central limit theorem (CLT) for the resulting sampler. The idea has been further extended for self-normalised importance samplers by \\citet{ryu2016convex}, who considered minimising the $\\alpha$-divergence between the target and an exponential family. Similarly, \\citet{ryu2016convex} proved a CLT for the resulting sampler. Similar ideas were also considered by \\citet{kawai2017acceleration, kawai2018optimizing}, who also aimed at minimizing the variance expression. Similarly, \\citet{kawai2018optimizing} showed that the variance of the weight function is convex when the proposal family is suitably chosen and provided general conditions for such proposals. \\citet{kawai2018optimizing} has also developed an adaptation technique based on the stochastic approximation, which is similar to the scheme we analyse in this paper. {There have been other results also considering $\\chi^2$ divergence and relating it to the necessary sample size of the IS methods, see, e.g., \\citet{sanz2018importance}. Following the approach of \\citet{chatterjee2018sample}, \\citet{sanz2018importance} considers and ties the necessary sample size to $\\chi^2$-divergence, in particular, shows that the necessary sample size grows with $\\chi^2$-divergence, hence implying that minimizing it can lead to more efficient importance sampling procedures.}\n\nIn this work, we develop and analyse a family of adaptive importance samplers, coined \\textit{optimised adaptive importance samplers} (OAIS), which relies on a particular adaptation strategy based on convex optimisation. We adapt the proposal with respect to a quantity (essentially the $\\chi^2$-divergence between the target and the proposal) that also happens to be the constant in the error bounds of the IS (see, e.g., \\citep{agapiou2017importance}). Assuming that proposal distributions belong to the exponential family, we recast the adaptation of the proposal as a convex optimisation problem and then develop a procedure which essentially optimises the $L_2$ error bound of the algorithm. By using results from convex optimisation, we obtain error rates depending on the number of iterations, denoted as $t$, and the number of Monte Carlo samples, denoted as $N$, together. In this way, we explicitly display the trade-off between these two essential quantities. To the best of our knowledge, none of the papers on the topic provides convergence rates depending explicitly on the number of iterations and the number of particles together, as we do herein.\n\nThe paper is organised as follows. In Sec.~\\ref{sec:AISintro}, we introduce the problem definition, the IS and the AIS algorithms. In Sec.~\\ref{sec:theAlg}, we introduce the OAIS algorithms. In Sec.~\\ref{sec:analysis}, we provide the theoretical results regarding optimised AIS and show its convergence using results from convex optimisation. Finally, we make some concluding remarks in Sec.~\\ref{sec:conc}.\n\n\\subsection*{Notation}\n\nFor $L\\in{\\mathbb N}$, we use the shorthand $[L] = \\{1,\\ldots,L\\}$. We denote the state space as ${\\mathsf X}$ and assume ${\\mathsf X} \\subseteq {\\mathbb R}^{d_x}$, $d_x \\ge 1$. The space of bounded real-valued functions and the set of probability measures on space ${\\mathsf X}$ are denoted as $B({\\mathsf X})$ and ${\\mathcal P}({\\mathsf X})$, respectively. Given $\\varphi\\in B({\\mathsf X})$ and $\\pi\\in{\\mathcal P}({\\mathsf X})$, the expectation of $\\varphi$ with respect to (w.r.t.) $\\pi$ is written as $(\\varphi,\\pi) = \\int \\varphi(x) \\pi(\\mbox{d}x)$ or ${\\mathbb E}_\\pi[\\varphi(X)]$. The variance of $\\varphi$ w.r.t. $\\pi$ is defined as $\\var_\\pi(\\varphi) = (\\varphi^2,\\pi) - (\\varphi,\\pi)^2$. If $\\varphi\\in B({\\mathsf X})$, then $\\|\\varphi\\|_\\infty = \\sup_{x\\in{\\mathsf X}} |\\varphi(x)| < \\infty$. The unnormalised density associated to $\\pi$ is denoted with $\\Pi(x)$. We denote the proposal as $q_\\theta \\in {\\mathcal P}({\\mathsf X})$, with an explicit dependence on the parameter $\\theta\\in\\Theta$. The parameter space is assumed to be a subset of $d_\\theta$-dimensional Euclidean space, i.e., $\\Theta \\subseteq {\\mathbb R}^{d_\\theta}$. \n\nWhenever necessary we denote both the probability measures, $\\pi$ and $q_\\theta$, and their densities with the same notation. To be specific, we assume that both $\\pi(\\mbox{d}x)$ and $q_\\theta(\\mbox{d}x)$ are absolutely continuous with respect to the Lebesgue measure and we denote their associated densities as $\\pi(x)$ and $q_\\theta(x)$. The use of either the measure or the density will be clear from both the argument (sets or points, respectively) and the context.\n\n\\section{Background}\\label{sec:AISintro}\n\nIn this section, we review importance and adaptive importance samplers.\n\n\\subsection{Importance sampling}\n\nConsider a target density $\\pi \\in {\\mathcal P}({\\mathsf X})$ and a bounded function $\\varphi \\in B({\\mathsf X})$. Often, the main interest is to compute an integral of the form\n\\begin{align}\\label{eq:ProbDefn}\n(\\varphi,\\pi) = \\int_{\\mathsf X} \\varphi(x) \\pi(x) \\mbox{d}x.\n\\end{align}\nWhile perfect Monte Carlo can be used to estimate this expectation when it is possible to sample exactly from $\\pi(x)$, this is in general not tractable. Hereafter, we consider the cases when the target can be evaluated exactly and up to a normalising constant, respectively.\n\nImportance sampling (IS) uses a proposal distribution which is easy to sample and evaluate. The method consists in weighting these samples, in order to correct the discrepancy between the target and the proposal, and finally constructing an estimator of the integral. To be precise, let $q_\\theta\\in{\\mathcal P}({\\mathsf X})$ be the proposal which is parameterized by the vector $\\theta\\in\\Theta$. The unnormalised target density is denoted as $\\Pi:{\\mathsf X} \\to {\\mathbb R}_+$. Therefore, we have\n\\begin{align*}\n\\pi(x) = \\frac{\\Pi(x)}{Z_\\pi},\n\\end{align*}\nwhere $Z_\\pi :=\\int_{\\mathsf X} \\Pi(x) {\\mathrm{d}} x < \\infty$. Next, we define functions $w_\\theta, W_\\theta:{\\mathsf X} \\times \\Theta \\to {\\mathbb R}_+$ as\n\\begin{align*}\nw_\\theta(x) = \\frac{\\pi(x)}{q_\\theta(x)} \\quad \\textnormal{and} \\quad W_\\theta(x) = \\frac{\\Pi(x)}{q_\\theta(x)},\n\\end{align*}\nrespectively. For a chosen proposal $q_\\theta$, the IS proceeds as follows. First, a set of independent and identically distributed (iid) samples $\\{x^{(i)}\\}_{i=1}^N$ is generated from $q_\\theta$. When $\\pi(x)$ can be evaluated, one constructs the empirical approximation of the probability measure $\\pi$, denoted $\\pi_\\theta^N$, as\n\\begin{align*}\n\\pi_\\theta^N(\\mbox{d}x) = \\frac{1}{N} \\sum_{i=1}^N w_\\theta(x^{(i)})\\delta_{x^{(i)}}(\\mbox{d}x),\n\\end{align*}\nwhere $\\delta_{x'}(\\mbox{d}x)$ denotes the Dirac delta measure that places unit probability mass at $x=x'$. For this case, the IS estimate of the integral in \\eqref{eq:ProbDefn} can be given as\n\\begin{align}\\label{eq:ISestimate}\n(\\varphi,\\pi^N_\\theta) = \\frac{1}{N} \\sum_{i=1}^N w_\\theta(x^{(i)}) \\varphi(x^{(i)}).\n\\end{align}\nHowever, in most practical cases, the target density $\\pi(x)$ can only be evaluated up to an unknown normalizing proportionality constant (i.e., we can evaluate $\\Pi(x)$ but not $Z_\\pi$). In this case, we construct the empirical measure $\\pi^N_\\theta$ as\n\\begin{align*}\n\\pi_\\theta^N(\\mbox{d}x) = \\sum_{i=1}^N \\mathsf{w}_\\theta^{(i)} \\delta_{x^{(i)}}(\\mbox{d}x),\n\\end{align*}\nwhere\n\\begin{align*}\n\\mathsf{w}_\\theta^{(i)} = \\frac{W_\\theta(x^{(i)})}{\\sum_{j=1}^N W_\\theta(x^{(j)})}.\n\\end{align*}\nFinally this construction leads to the so called self-normalizing importance sampling (SNIS) estimator\n\\begin{align}\\label{eq:SNISestimate}\n(\\varphi,\\pi^N_\\theta) = \\sum_{i=1}^N \\mathsf{w}_\\theta^{(i)} \\varphi(x^{(i)}).\n\\end{align}\nAlthough the IS estimator \\eqref{eq:ISestimate} is unbiased, the SNIS estimator \\eqref{eq:SNISestimate} is in general biased. However, the bias and the MSE vanish with a rate $\\mathcal{O}(1\/N)$, therefore providing guarantees of convergence as $N\\to\\infty$. Crucially for us, the MSE of both estimators. {For clarity, below we present an MSE bound for the (more general) SNIS estimator \\eqref{eq:SNISestimate} which is adapted from \\citet{agapiou2017importance}.}\n\\begin{thm}\\label{thm:ISfund}\nAssume that $(W_\\theta^2,q_\\theta) < \\infty$. Then for any $\\varphi\\in B({\\mathsf X})$, we have\n\\begin{align}\n{\\mathbb E}\\left[\\left((\\varphi,\\pi) - (\\varphi,\\pi_{\\theta}^N)\\right)^2\\right] \\leq \\frac{c_\\varphi \\rho(\\theta)}{{N}},\n\\label{eqThm1-1}\n\\end{align}\nwhere $c_\\varphi = 4\\|\\varphi\\|_\\infty^2$ and the function $\\rho:\\Theta \\to [\\rho^\\star,\\infty)$ is defined as\n\\begin{align}\n\\rho(\\theta) = {\\mathbb E}_{q_\\theta}\\left[\\frac{\\pi^2(X)}{q^2_\\theta(X)}\\right],\n\\label{eqThm1-2}\n\\end{align}\nwhere $\\rho^\\star := \\inf_{\\theta\\in\\Theta} \\rho(\\theta) \\geq 1$.\n\\end{thm}\n\\begin{proof}\nSee Appendix \\ref{app:proofIS} for a self-contained proof.\n\\end{proof}\n\\begin{rem} For the IS estimator \\eqref{eq:ISestimate}, this bound can be improved so that $c_\\varphi~=~\\|\\varphi\\|_\\infty^2$. However, this improvement does not effect our results in this paper, hence we present a single bound of the form in \\eqref{eqThm1-1} for the estimators \\eqref{eq:ISestimate} and \\eqref{eq:SNISestimate} for conciseness. $\\square$\n\\end{rem}\n\\begin{rem}\\label{rem:relationToChi} As pointed out by \\cite{agapiou2017importance}, the function $\\rho$ is essentially the $\\chi^2$ divergence between $\\pi$ and $q_\\theta$, i.e.,\n\\begin{align*}\n\\rho(\\theta) := \\chi^2(\\pi || q_\\theta) + 1.\n\\end{align*}\nNote that $\\rho(\\theta)$ can also be expressed in terms of the variance of the weight function $w_\\theta$, which coincides with the $\\chi^2$-divergence, i.e.,\n\\begin{align*}\n\\rho(\\theta) = \\var_{q_\\theta}(w_\\theta(X)) + 1.\n\\end{align*}\nTherefore, minimizing $\\rho(\\theta)$ is equivalent to minimizing $\\chi^2$-divergence and the variance of the weight function $w_\\theta$, i.e., $\\var_{q_\\theta}(w_\\theta(X))$. $\\square$\n\\end{rem}\n\\begin{rem} Remark~\\ref{rem:relationToChi} implies that, when both $\\pi$ and $q_\\theta$ belong {to the same parametric family (i.e., there exists $\\theta \\in \\Theta$ such that $\\pi=q_\\theta$),} one readily obtains\n\\begin{align*}\n\\rho^\\star := \\inf_{\\theta\\in\\Theta} \\rho(\\theta) = 1. \\quad \\square\n\\end{align*}\n\\end{rem}\n\\begin{rem} For the IS estimator \\eqref{eq:ISestimate}, the bound in Theorem~\\ref{thm:ISfund} can be modified so that it holds for unbounded test functions $\\varphi$ as well; see, e.g. \\citet{ryu2014adaptive}. Therefore, a similar quantity to $\\rho(\\theta)$, which includes $\\varphi$ whilst still retaining convexity, can be optimised for this case. Unfortunately, obtaining such a bound is not straightforward for the SNIS estimator \\eqref{eq:SNISestimate} as shown by \\citet{agapiou2017importance}. In order to significantly simplify the presentation, we restrict ourselves to the class of bounded test functions, i.e., we assume $\\|\\varphi\\|_\\infty < \\infty$. $\\square$\n\\end{rem}\n{Finally, we present a bias result from \\citet{agapiou2017importance}.\n\\begin{thm}\\label{thm:SNISbias}\nAssume that $(W_\\theta^2,q_\\theta) < \\infty$. Then for any $\\varphi\\in B({\\mathsf X})$, we have\n\\begin{align*}\n\\left| {\\mathbb E}\\left[(\\varphi,\\pi_{\\theta}^N)\\right] - (\\varphi,\\pi) \\right| \\leq \\frac{\\bar{c}_\\varphi \\rho(\\theta)}{{N}},\n\\end{align*}\nwhere $\\bar{c}_\\varphi = 12\\|\\varphi\\|_\\infty^2$ and the function $\\rho:\\Theta \\to [\\rho^\\star,\\infty)$ is the same as in Theorem~\\ref{thm:ISfund}.\n\\end{thm}\n\\begin{proof}\nSee Theorem 2.1 in \\citet{agapiou2017importance}.\n\\end{proof}}\n\\subsection{Parametric adaptive importance samplers}\nStandard importance sampling may be inefficient in practice when the proposal is poorly calibrated with respect to the target. In particular, as implied by the error bound provided in Theorem~\\ref{thm:ISfund}, the error made by the IS estimator can be high if the $\\chi^2$-divergence between the target and the proposal is large. Therefore, it is more common to employ an iterative version of importance sampling, also called as \\textit{adaptive importance sampling} (AIS). The AIS algorithms are importance sampling methods which aim at iteratively improving the proposal distributions. More specifically, the AIS methods specify a sequence of proposals $(q_t)_{t\\geq 1}$ and perform importance sampling at each iteration. The aim is to improve the proposal so that the samples are better matched with the target, which results in less variance and more accuracy in the estimators. There are several variants, the most popular one being population Monte Carlo methods \\citep{cappe2004population} which uses previous samples in the proposal.\n\n\\begin{algorithm}[t]\n\\begin{algorithmic}[1]\n\\caption{Parametric AIS}\\label{alg:ParametricAIS}\n\\State Choose a parametric proposal $q_{\\theta}$ with initial parameter $\\theta=\\theta_0$.\n\\For{$t\\geq 1$}\n\\State Adapt the proposal,\n\\begin{align*}\n\\theta_t = \\mathcal{T}_t(\\theta_{t-1}),\n\\end{align*}\n\\State Sample,\n\\begin{align*}\nx_t^{(i)} \\sim q_{\\theta_t}, \\quad \\textnormal{for } i = 1,\\ldots,N,\n\\end{align*}\n\\State Compute weights,\n\\begin{align*}\n\\mathsf{w}_{\\theta_t}^{(i)} = \\frac{W_{\\theta_t}(x_t^{(i)})}{\\sum_{i=1}^N W_{\\theta_t}(x_t^{(i)})}, \\quad \\textnormal{where} \\quad W_{\\theta_t}^{(i)} = \\frac{\\Pi(x_t^{(i)})}{q_{\\theta_t}(x^{(i)})}.\n\\end{align*}\n\\State Report the point-mass probability measure\n\\begin{align*}\n{\\pi}_{\\theta_t}^N({\\mathrm{d}} x) = \\sum_{i=1}^N \\mathsf{w}_{\\theta_t}^{(i)} \\delta_{x_t^{(i)}}({\\mathrm{d}} x),\n\\end{align*}\nand the estimator\n\\begin{align*}\n(\\varphi,{\\pi}_{\\theta_t}^N) = \\sum_{i=1}^N \\mathsf{w}_{\\theta_t}^{(i)} \\varphi(x_t^{(i)}).\n\\end{align*}\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\nIn this section, we review one particular AIS, which we refer to as \\textit{parametric AIS}. In this variant, the proposal distribution is a parametric distribution, denoted $q_\\theta$. Over time, this parameter $\\theta$ is updated (or \\textit{optimised}) with respect to a predefined criterion resulting in a sequence $(\\theta_t)_{t\\geq 1}$. This yields a sequence of proposal distributions denoted as $(q_{\\theta_t})_{t\\geq 1}$.\n\nOne iteration of the algorithm goes as follows. Assume at time $t-1$ we are given a proposal distribution $q_{\\theta_{t-1}}$. At time $t$, we first update the parameter of this proposal,\n\\begin{align*}\n\\theta_t = \\mathcal{T}_t(\\theta_{t-1}),\n\\end{align*}\nwhere $\\{\\mathcal{T}_t:\\Theta \\to \\Theta, t\\geq 1\\}$, is a sequence of (deterministic or stochastic) maps, e.g., gradient mappings, constructed so that they minimise a certain cost function. Then, in the same way we have done in conventional IS, we sample\n\\begin{align*}\nx_t^{(i)} \\sim q_{\\theta_t}({\\mathrm{d}} x), \\quad \\textnormal{for } i = 1,\\ldots,N,\n\\end{align*}\ncompute weights\n\\begin{align*}\n\\mathsf{w}_{\\theta_t}^{(i)} = \\frac{W_{\\theta_t}(x_t^{(i)})}{\\sum_{i=1}^N W_{\\theta_t}(x_t^{(i)})},\n\\end{align*}\nand finally construct the empirical measure\n\\begin{align*}\n\\pi_{\\theta_t}^N({\\mathrm{d}} x) = \\sum_{i=1}^N \\mathsf{w}_{\\theta_t}^{(i)} \\delta_{x_t^{(i)}}({\\mathrm{d}} x).\n\\end{align*}\nThe estimator of the integral \\eqref{eq:ProbDefn} is then computed as in Eq. \\eqref{eq:SNISestimate}. \n\nThe full procedure of the parametric AIS method is summarized in Algorithm~\\ref{alg:ParametricAIS}. Since this is a valid IS scheme, this algorithm enjoys the same guarantee provided in Theorem~\\ref{thm:ISfund}. In particular, we have the following theorem.\n\\begin{thm}\\label{thm:ISfundAIS}\nAssume that, given a sequence of proposals $(q_{\\theta_t})_{t\\geq 1} \\in {\\mathcal P}({\\mathsf X})$, we have $(W_{\\theta_t}^2,q_{\\theta_t}) < \\infty$ for every $t$. Then for any $\\varphi\\in B({\\mathsf X})$, we have\n\\begin{align*}\n{\\mathbb E}\\left[\\left|(\\varphi,\\pi) - (\\varphi,\\pi_{\\theta_t}^N)\\right|^2\\right] \\leq \\frac{c_\\varphi \\rho(\\theta_t)}{{N}},\n\\end{align*}\nwhere $c_\\varphi = 4 \\|\\varphi\\|_\\infty^2$ and the function $\\rho(\\theta_t):\\Theta \\to [\\rho^\\star,\\infty)$ is defined as in Eq. \\eqref{eqThm1-2}.\n\\end{thm}\n\\begin{proof}\nThe proof is identical to the proof of Theorem~\\ref{thm:ISfund}. We have just re-stated the result to introduce the iteration index $t$.\n\\end{proof}\nHowever, this theorem does not give an insight of what happens as the number of iterations increases, i.e., when $t\\to\\infty$, with the bound. Ideally, the adaptation of the AIS should improve this bound with time. In other words, in the ideal case, the error should decrease as $t$ grows. Fortunately, Theorem~\\ref{thm:ISfundAIS} suggests that the maps $\\mathcal{T}_t:\\Theta\\to\\Theta$ can be chosen so that the function $\\rho$ is minimised over time. More specifically, the sequence $(\\theta_t)_{t\\geq 1}$ can be chosen so that it leads to a decreasing sequence (at least in expectation) $(\\rho(\\theta_t))_{t\\geq 1}$. In the following sections, we will summarize the deterministic and stochastic strategies to achieve this aim.\n\\begin{rem}\\label{remR} We define the unnormalised version of $\\rho(\\theta)$ and denote it as $R(\\theta)$. It is characterised as follows\n\\begin{align*}\n\\rho(\\theta) = \\frac{R(\\theta)}{Z_\\pi^2} \\quad \\textnormal{where} \\quad Z_\\pi = \\int_{\\mathsf X} \\Pi(x) {\\mathrm{d}} x < \\infty.\n\\end{align*}\nHence, $R(\\theta)$ can also be expressed as\n\\begin{align}\\label{eq:Rtheta}\nR(\\theta) = {\\mathbb E}_{q_{\\theta}} \\left[\\frac{\\Pi^2(X)}{q_{\\theta}^2(X)}\\right].\n\\end{align} $\\square$\n\\end{rem}\n\n\n\\subsection{AIS with exponential family proposals}\\label{sec:expFamily}\n\nFollowing \\cite{ryu2014adaptive}, we note that when $q_\\theta$ is chosen as an exponential family density, the function $\\rho(\\theta)$ is convex. In particular, we define\n\\begin{align}\\label{eq:PropDefineExp}\nq_\\theta(x) = \\exp(\\theta^\\top T(x) - A(\\theta)) h(x),\n\\end{align}\nwhere $A: {\\mathbb R}^{d_\\theta}\\to{\\mathbb R} \\cup \\{\\infty\\}$ is the log of the normalization constant, i.e.,\n\\begin{align*}\nA(\\theta) = \\log \\int \\exp(\\theta^\\top T(x)) h(x) \\mbox{d}x,\n\\end{align*}\nwhile $T:{\\mathbb R}^{d_x}\\to{\\mathbb R}^{d_\\theta}$ and $h:{\\mathbb R}^{d_x}\\to{\\mathbb R}_+$. Then we have the following lemma adapted from \\cite{ryu2014adaptive}.\n\\begin{lem}\\label{prop:rhoconvex} Let $q_\\theta$ be chosen as in \\eqref{eq:PropDefineExp}. Then $\\rho:\\Theta \\to [\\rho^\\star,\\infty)$ is convex, i.e., for any $\\theta_1,\\theta_2\\in\\Theta$ and $\\lambda \\in [0,1]$, the following inequality holds\n\\begin{align*}\n\\rho(\\lambda\\theta_1 + (1-\\lambda) \\theta_2) \\leq \\lambda \\rho(\\theta_1) + (1-\\lambda) \\rho(\\theta_2).\n\\end{align*}\n\\end{lem}\n\\begin{proof}\nSee Appendix \\ref{app:proofLemma1} for a self-contained proof.\n\\end{proof}\nLemma~\\ref{prop:rhoconvex} shows that $\\rho$ is a convex function, therefore, optimising it could give us provably convergent algorithms (as $t$ increases). Next lemma, borrowed from \\citet{ryu2014adaptive}, shows that $\\rho$ is differentiable and its gradient can indeed be computed as an expectation.\n{\\begin{lem}\\label{lem:GradientRho} The gradient $\\nabla\\rho(\\theta)$ can be written as\n\\begin{align}\\label{eq:gradRho}\n\\nabla \\rho(\\theta) = {\\mathbb E}_{q_\\theta} \\left[(\\nabla A(\\theta) - T(X)) \\frac{\\pi^2(X)}{q_\\theta^2(X)}\\right].\n\\end{align}\n\\end{lem}}\n\\begin{proof}\nThe proof is straightforward since $q_\\theta$ is from an exponential family and $A(\\theta)$ is differentiable.\n\\end{proof}\n\\begin{rem} Note that Eqs.~\\eqref{eq:Rtheta} and \\eqref{eq:gradRho} together imply that\n\\begin{align}\\label{eq:gradR}\n\\nabla R(\\theta) = {\\mathbb E}_{q_\\theta} \\left[(\\nabla A(\\theta) - T(X)) \\frac{\\Pi^2(X)}{q_\\theta^2(X)}\\right].\n\\end{align}\nWe also note (see Remark~\\ref{remR}) that\n\\begin{align}\\label{eq:RelGrads}\n\\nabla R(\\theta) = Z_\\pi^2 \\nabla \\rho(\\theta).\n\\end{align}\n$\\square$\n\\end{rem}\nIn the following sections, we assume that $\\rho(\\theta)$ is a convex function. Thus Lemma~\\ref{prop:rhoconvex} constitutes an important motivation for our approach. We leave general proposals which lead to nonconvex $\\rho(\\theta)$ for future work.\n\n\n\\section{Algorithms}\\label{sec:theAlg}\n\nIn this section, we describe adaptation strategies based on minimizing $\\rho(\\theta)$. In particular, we design maps $\\mathcal{T}_t:\\Theta\\to\\Theta$, for $t\\geq 1$, for scenarios where\n\\begin{itemize}\n\\setlength{\\itemindent}{2em}\n\\item[(i)] the gradient of $\\rho(\\theta)$ can be exactly computed,\n\\item[(ii)] an unbiased estimate of the gradient of $\\rho(\\theta)$ can be obtained, and\n\\item[(iii)] an unbiased estimate of the gradient of $R(\\theta)$ can be obtained.\n\\end{itemize}\nScenario (i) is unrealistic in practice but gives us a guideline in order to further develop the idea. {In particular, the error bounds for the more complicated cases follow the same structure as this case. Therefore, the results obtained in case (i) provide a good qualitative understanding of the results introduced later.} Scenario (ii) can be realized in cases where it is possible to evaluate $\\pi(x)$, in which case the IS leads to unbiased estimators. Scenario (iii) is what a practitioner would most often encounter: the target can only be evaluated up to the normalizing constant, i.e., $\\Pi(x)$ can be evaluated but $\\pi(x)$ cannot.\n\n{We finally remark that, for the cases where we assume a stochastic gradient can be obtained for $\\rho$ and $R$ (namely, the case (ii) and the case (iii) respectively), we consider two possible algorithms to perform adaptation. The first method is a \\textit{vanilla} SGD algorithm \\citep{bottou2016optimization} and the second method is a SGD scheme with iterate averaging \\citep{schmidt2017minimizing}. While vanilla SGD is easier to implement and algorithmically related to population-based Monte Carlo methods, iterate averaged SGD results in a better theoretical bound and it has some desirable variance reduction properties.}\n\n\n\n\\subsection{Exact gradient OAIS}\n\nWe first introduce the OAIS scheme where we assume that the exact gradients of $\\rho(\\theta)$ are available. Since $\\rho$ is defined as an expectation (an integral), this assumption is unrealistic. However, the results we can prove for this procedure shed light onto the results that will be proved for practical scenarios in the following sections.\n\nIn particular, in this scheme, given $\\theta_{t-1}$, we specify $\\mathcal{T}_t$ as\n\\begin{align}\\label{eq:exactOAIS}\n\\theta_t = \\mathcal{T}_t(\\theta_{t-1}) = \\mathsf{Proj}_\\Theta(\\theta_{t-1} - \\gamma \\nabla \\rho(\\theta_{t-1})),\n\\end{align}\nwhere $\\gamma > 0$ is the step-size parameter of the map and $\\mathsf{Proj}_\\Theta$ denotes projection onto the compact parameter space $\\Theta$. This is a classical gradient descent scheme on $\\rho(\\theta)$. In Section \\ref{ssErrorsExactGrad}, we provide non-asymptotic results for this scheme. However, as we have noted, this idea does not lead to a practical scheme and cannot be used in most cases in practice as the gradients of $\\rho$ in exact form are rarely available.\n\\begin{rem} {We use a projection operator in Eq.~\\eqref{eq:exactOAIS} because we assume throughout the analysis in Section~\\ref{sec:analysis} that the parameter space $\\Theta$ is compact.}\n$\\square$\n\\end{rem}\n\n{\\subsection{Stochastic gradient OAIS}}\n\n{Although it has a nice and simple form, exact-gradient OAIS is often intractable as, in most practical cases, the gradient can only be estimated. In this section, we first look at the case where $\\pi(x)$ can be evaluated, which means that an unbiased estimate of $\\nabla \\rho(\\theta)$ can be obtained. Then we consider the general case, where one can only evaluate $\\Pi(x)$ and can obtain an unbiased estimate of $\\nabla R(\\theta)$.}\n\n{In the following subsections, we consider an algorithm where the gradient is estimated using samples which can also be used to construct importance sampling estimators. The procedure is outlined in Algorithm \\ref{alg:SGDAIS} for the case in which only $\\Pi(x)$ can be evaluated and $\\nabla R(\\theta)$ is estimated.}\n\n{\\subsubsection{Normalised case}}\n\n{If we assume that the density $\\pi(x)$ can be evaluated exactly, then the algorithm can be described as follows. Given $(\\theta_k)_{1\\leq k\\leq t-1}$, at iteration $t$ we compute the next parameter iterate as\n\\begin{align*}\n\\theta_t = \\mathsf{Proj}_{\\Theta}(\\theta_{t-1} - \\gamma_t g_t),\n\\end{align*}\nwhere $g_t$ is an unbiased estimator of $\\nabla \\rho(\\theta_{t-1})$. We note that, due to the analytical form of $\\nabla \\rho$ (see Eq. \\eqref{eq:gradRho}), the samples and weights generated at iteration $t-1$, i.e., $\\left\\{ x_{t-1}^{(i)}, w_{\\theta_{t-1}}(x_{t-1}^{(i)}) \\right\\}_{i=1}^N$ can be reused to estimate the gradient. This makes an algorithmic connection to the population Monte Carlo methods where previous samples and weights are used to adapt the proposal \\citep{cappe2004population}.}\n\n{Given the updated parameter $\\theta_t$, the algorithm first samples from the updated proposal $x_t^{(i)} \\sim q_{\\theta_t}$, $i=1, \\ldots, N$, and then proceeds to construct the IS estimator as in \\eqref{eq:ISestimate}. Namely, \n\\begin{align}\n(\\varphi,\\pi^N_{{\\theta}_t}) = \\frac{1}{N} \\sum_{i=1}^N w_{{\\theta}_t}({x}_t^{(i)}) \\varphi({x}_t^{(i)}).\n\\label{eqEstimator_1}\n\\end{align}}\n\n{\\begin{algorithm}[tb!]\n\\begin{algorithmic}[1]\n\\caption{Stochastic gradient OAIS}\\label{alg:vanillaSGDAIS}\n\\State Choose a parametric proposal $q_{\\theta}$ with initial parameter $\\theta=\\theta_0$.\n\\For{$t\\geq 1$}\n\\State Update the proposal parameter,\n\\begin{align*}\n\\theta_t = \\mathsf{Proj}_\\Theta(\\theta_{t-1} - \\gamma_t \\tilde{g}_t)\n\\end{align*}\nwhere $\\tilde{g}_t$ is computed by approximating the expectation in Eq. \\eqref{eq:gradR} using the samples $x_{t-1}^{(i)}$ and weights $\\mathsf{w}_{\\theta_{t-1}}^{(i)} = \\Pi( x_{t-1}^{(i)} ) q_{\\theta_{t-1}}(x_{t-1}^{(i)})^{-1}$, $i=1, ..., N$.\n\\State Sample,\n\\begin{align*}\n{x}_t^{(i)} \\sim q_{{\\theta}_t}, \\quad \\textnormal{for } i = 1,\\ldots,N,\n\\end{align*}\n\\State Compute weights,\n\\begin{align*}\n\\mathsf{w}_{{\\theta}_t}^{(i)} = \\frac{W_{{\\theta}_t}({x}_t^{(i)})}{\\sum_{i=1}^N W_{{\\theta}_t}({x}_t^{(i)})}.\n\\end{align*}\n\\State Report,\n\\begin{align*}\n{\\pi}_{{\\theta}_t}^N({\\mathrm{d}} x) = \\sum_{i=1}^N \\mathsf{w}_{{\\theta}_t}^{(i)} \\delta_{{x}_t^{(i)}}({\\mathrm{d}} x),\n\\end{align*}\nand\n\\begin{align*}\n(\\varphi,{\\pi}_{{\\theta}_t}^N) = \\sum_{i=1}^N \\mathsf{w}_{{\\theta}_t}^{(i)} \\varphi({x}_t^{(i)}).\n\\end{align*}\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}}\n\n{\\subsubsection{Self-normalised case}}\n\n{For the general case, where we can only evaluate $\\Pi(x)$, the algorithm proceeds similarly. Given $(\\theta_k)_{1\\leq k\\leq t-1}$, the method proceeds by first updating the parameter\n\\begin{align*}\n\\theta_t = \\mathsf{Proj}_{\\Theta}(\\theta_{t-1} - \\gamma_t \\tilde{g}_t),\n\\end{align*}\nwhere $\\tilde{g}_t$ is an unbiased estimator of $\\nabla R(\\theta_{t-1})$. Given the updated parameter, we first sample $x_t^{(i)} \\sim q_{\\theta_t}$, $i=1, ..., N$, and then construct the SNIS estimate as in \\eqref{eq:SNISestimate}, i.e., \n\\begin{align*}\n(\\varphi,\\pi^N_{{\\theta}_t}) = \\sum_{i=1}^N \\mathsf{w}^{(i)}_{{\\theta}_t} \\varphi({x}_t^{(i)}).\n\\end{align*}\nwhere\n\\begin{align*}\n\\mathsf{w}_{{\\theta}_t}^{(i)} = \\frac{W_{{\\theta}_t}({x}^{(i)})}{\\sum_{j=1}^N W_{{\\theta}_t}({x}^{(j)})},\n\\end{align*}}\n\n\\subsection{Stochastic gradient OAIS with averaged iterates}\n\n{Next, we describe a variant of the stochastic gradient OAIS that uses averages of the iterates generated by the SGD scheme \\citep{schmidt2017minimizing} in order to compute the proposal densities, generate samples and compute weights. In Section \\ref{sec:analysis} we show that the convergence rate for this method is better than the rate that can be guaranteed for Algorithm \\ref{alg:vanillaSGDAIS}.}\n\n\\subsubsection{Normalised case}\n\nWe assume first that the density $\\pi(x)$ can be evaluated. At the beginning of the $t$-th iteration, the algorithm has generated the sequence $(\\theta_k)_{1\\leq k \\leq t-1}$. First, in order to perform the adaptive importance sampling steps, we set\n\\begin{align}\\label{eq:AveragingSGD}\n\\bar{\\theta}_t = \\frac{1}{t}\\sum_{k=0}^{t-1} \\theta_k\n\\end{align}\nand sample $\\bar{x}_{t}^{(i)} \\sim q_{\\bar{\\theta}_t}$ for $i = 1,\\ldots,N$. Following the standard parametric AIS procedure (Algorithm~\\ref{alg:ParametricAIS}), we obtain the estimate of $(\\varphi,\\pi)$ as,\n\\begin{align*}\n(\\varphi,\\pi^N_{\\bar{\\theta}_t}) = \\frac{1}{N} \\sum_{i=1}^N w_{\\bar{\\theta}_t}(\\bar{x}_t^{(i)}) \\varphi(\\bar{x}_t^{(i)}).\n\\end{align*}\nNext, we update the parameter vector using the projected stochastic gradient step\n\\begin{align}\\label{eq:recSgdAdaptNorm}\n\\theta_t = \\mathcal{T}_t(\\theta_{t-1}) = \\mathsf{Proj}_\\Theta(\\theta_{t-1} - \\gamma_t g_t),\n\\end{align}\nwhere $g_t$ is an unbiased estimate of $\\nabla\\rho(\\theta_{t-1})$, i.e., ${\\mathbb E}[g_t] = \\nabla \\rho(\\theta_{t-1})$ and $\\mathsf{Proj}_\\Theta$ denotes projection onto the set $\\Theta$. Note that in order to estimate this gradient using \\eqref{eq:gradRho}, we sample $x_t^{(i)} \\sim q_{\\theta_{t-1}}$ for $i = 1, \\ldots, N$, and estimate the expectation in \\eqref{eq:gradRho}. It is worth noting that the samples $\\{ x_t^{(i)} \\}_{i=1}^M$ are different from the samples $\\{ \\bar x_t^{(i)} \\}_{i=1}^N$ used to estimate $(\\varphi,\\pi)$.\n\n\\subsubsection{Self-normalised case}\n\n\\begin{algorithm}[tb!]\n\\begin{algorithmic}[1]\n\\caption{Stochastic gradient OAIS with averaged iterates}\\label{alg:SGDAIS}\n\\State Choose a parametric proposal $q_{\\theta}$ with initial parameter $\\theta = \\theta_0$.\n\\For{$t\\geq 1$}\n\\State Compute the average parameter vector\n\\begin{align*}\n\\bar{\\theta}_t = \\frac{1}{t} \\sum_{k=0}^{t-1} \\theta_k\n\\end{align*}\n\\State Sample,\n\\begin{align*}\n\\bar{x}_t^{(i)} \\sim q_{\\bar{\\theta}_t}, \\quad \\textnormal{for } i = 1,\\ldots,N,\n\\end{align*}\n\\State Compute weights,\n\\begin{align*}\n\\mathsf{w}_{\\bar{\\theta}_t}^{(i)} = \\frac{W_{\\bar{\\theta}_t}(\\bar{x}_t^{(i)})}{\\sum_{i=1}^N W_{\\bar{\\theta}_t}(\\bar{x}_t^{(i)})}.\n\\end{align*}\n\\State Report the point-mass probability measure\n\\begin{align*}\n{\\pi}_{\\bar{\\theta}_t}^N({\\mathrm{d}} x) = \\sum_{i=1}^N \\mathsf{w}_{\\bar{\\theta}_t}^{(i)} \\delta_{\\bar{x}_t^{(i)}}({\\mathrm{d}} x),\n\\end{align*}\nand the estimator\n\\begin{align*}\n(\\varphi,{\\pi}_{\\bar{\\theta}_t}^N) = \\sum_{i=1}^N \\mathsf{w}_{\\bar{\\theta}_t}^{(i)} \\varphi(\\bar{x}_t^{(i)}).\n\\end{align*}\n\\State Update the parameter vector,\n\\begin{align*}\n\\theta_t = \\mathsf{Proj}_\\Theta(\\theta_{t-1} - \\gamma_t \\tilde{g}_t)\n\\end{align*}\nwhere $\\tilde g_t$ is an estimate of $\\nabla R(\\theta_{t-1})$ computed by approximating the expectation in Eq. \\eqref{eq:gradR} using a set of iid samples ${x}_t^{(i)} \\sim q_{\\theta_{t-1}}$, $i=1, ..., N$.\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\nIn general, $\\pi(x)$ cannot be evaluated exactly, hence a stochastic unbiased estimate of $\\nabla\\rho(\\theta)$ cannot be obtained. When the target can only be evaluated up to a normalisation constant, i.e., only $\\Pi(x)$ can be computed, we can use the SNIS procedure as explained in Section~\\ref{sec:AISintro}. Therefore, we introduce here the most general version of the stochastic method, coined \\textit{stochastic gradient OAIS}, which uses the averaged iterates in \\eqref{eq:AveragingSGD} to construct the proposal functions. The scheme is outlined in Algorithm \\ref{alg:SGDAIS}.\n\nTo run this algorithm, given the parameter vector $\\bar{\\theta}_t$ in \\eqref{eq:AveragingSGD}, we first generate a set of samples $\\{\\bar{x}_t^{(i)}\\}_{i=1}^N$ from the proposal $q_{\\bar{\\theta}_t}$. Then the integral estimate given by the SNIS can be written as,\n\\begin{align*}\n(\\varphi,\\pi^N_{\\bar{\\theta}_t}) = \\sum_{i=1}^N \\mathsf{w}_{\\bar{\\theta}_t}^{(i)} \\varphi(\\bar{x}_t^{(i)}),\n\\end{align*}\nwhere\n\\begin{align*}\n\\mathsf{w}_{\\bar{\\theta}_t}^{(i)} = \\frac{W_{\\bar{\\theta}_t}(\\bar{x}^{(i)})}{\\sum_{j=1}^N W_{\\bar{\\theta}_t}(\\bar{x}^{(j)})}.\n\\end{align*}\nFinally, for the adaptation step, we obtain the unbiased estimate of the gradient $\\nabla R(\\theta)$ and adapt the parameter as\n\\begin{align}\\label{eq:SgdUnnormalizedAdapt}\n\\theta_t = \\mathsf{Proj}_\\Theta(\\theta_{t-1} - \\gamma_t \\tilde{g}_t)\n\\end{align}\nwhere $\\tilde{g}_t$ is an unbiased estimate of $\\nabla R(\\theta_{t-1})$, i.e., ${\\mathbb E}[\\tilde{g}_t] = \\nabla R(\\theta_{t-1})$. Note that, as in the normalised case, this gradient is estimated by approximating the expectation in \\eqref{eq:gradR} using iid samples $x_t^{(i)} \\sim q_{\\theta_{t-1}}$, $i = 1,\\ldots,N$. These samples are different, again, from the set $\\{ \\bar x_t^{(i)} \\}_{i=1}^N$ employed to estimate $(\\varphi,\\pi)$.\n{\n\\begin{rem}\nIn Algorithm \\ref{alg:SGDAIS} the samples $\\{ \\bar x_t^{(i)} \\}_{i=1}^N$ drawn from the proposal distribution $q_{\\bar \\theta_{t-1}}({\\mathrm{d}} x)$ are \\textit{not} used to compute the gradient estimator $\\tilde g_t$ which, in turn, is needed to generate the next iterate $\\theta_t$. Therefore, if we can afford to generate $T$ iterates, $\\theta_0, \\ldots, \\theta_{T-1}$, with $T$ known before hand, and we are only interested in the estimator $(\\varphi,\\pi_{\\bar \\theta_T}^N)$ obtained at the last iteration (once the proposal function has been optimized) then it is be possible to skip steps 3--6 in Algorithm \\ref{alg:SGDAIS} up to time $T-1$. Only at time $t=T$, we would compute the average parameter vector $\\bar \\theta_T$, sample $\\bar x_T^{(i)}$ from the proposal $q_{\\bar \\theta_T}({\\mathrm{d}} x)$ and generate the point-mass probability measure $\\pi_{\\bar \\theta_T}^N$ and the estimator $(\\varphi,\\pi_{\\bar \\theta_T}^N)$ .\n\\end{rem}\n}\n\n\\section{Analysis}\\label{sec:analysis}\n\nTheorem~\\ref{thm:ISfund} yields an intuitive result about the performance of IS methods in terms of the divergence between the target $\\pi$ and the proposal $q_\\theta$. We now apply ideas from convex optimisation theory in order to minimize $\\rho(\\theta)$ and obtain finite-time, finite-sample convergence rates for the AIS procedures outlined in Section \\ref{sec:theAlg}.\n\n\\subsection{Convergence rate with exact gradients} \\label{ssErrorsExactGrad}\n\nLet us first assume that we can compute the gradient of $\\rho(\\theta)$ exactly. In particular, we consider the update rule in Eq. \\eqref{eq:exactOAIS}. For the sake of the analysis, we impose some regularity assumptions on the $\\rho(\\theta)$.\n\n{\n\\begin{assumption}\\label{ass:LipschitzCont}\nLet $\\rho(\\theta)$ be a convex function with Lipschitz derivatives in the compact space $\\Theta$. To be specific, $\\rho$ is convex and differentiable, and there exists a constant $L<\\infty$ such that\n\\begin{eqnarray}\n\\|\\nabla \\rho(\\theta) - \\nabla \\rho(\\theta')\\|_2 &\\leq& L \\|\\theta - \\theta'\\|_2 \\nonumber\n\\end{eqnarray}\nfor any $\\theta,\\theta' \\in \\Theta$.\n\\end{assumption}\n}\n\n{\n\\begin{rem} Assumption \\ref{ass:LipschitzCont} holds when the density $q_\\theta(x)$ belongs to an exponential family (see Section~\\ref{sec:expFamily}) and $\\Theta$ is compact \\citep{ryu2014adaptive}, even if it may not hold in general for $\\theta \\in {\\mathbb R}^{d_\\theta}$. $\\square$\n\\end{rem}\n}\n\n\\begin{lem}\\label{lem:GDconv} If Assumption~\\ref{ass:LipschitzCont} holds and we set a step-size $\\gamma \\leq 1\/L$, then the inequality\n\\begin{align}\n\\rho(\\theta_t) - \\rho^\\star \\leq \\frac{\\|\\theta_0 - \\theta^\\star\\|^2}{2\\gamma t},\n\\label{eqLem3-1}\n\\end{align}\nis satisfied for the sequence $\\{\\theta_t\\}_{t\\ge 0}$ generated by the recursion \\eqref{eq:exactOAIS} where $\\theta^\\star$ is a minimum of $\\rho$.\n\\end{lem}\n\\begin{proof}\nSee, e.g., \\cite{nesterov2013introductory}.\n\\end{proof}\n\nThis rate in \\eqref{eqLem3-1} is one of the most fundamental results in convex optimisation. Lemma \\ref{lem:GDconv} enables us to prove the following result for the MSE of the AIS estimator adapted using exact gradient descent in Eq. \\eqref{eqEstimator_1}.\n\n\\begin{thm}\\label{thm:GD} Let Assumption~\\ref{ass:LipschitzCont} hold and construct the sequence $(\\theta_t)_{t\\geq 1}$ using recursion \\eqref{eq:exactOAIS}, where $(q_{\\theta_t})_{t\\geq 1}$ is the sequence of proposal distributions. Then, the inequality\n\\begin{align}\\label{ineq:gOAISbound}\n{\\mathbb E}\\left[\\left((\\varphi,\\pi) - (\\varphi,\\pi_{\\theta_t}^N)\\right)^2\\right] &\\leq \\frac{c_\\varphi \\|\\theta_0 - \\theta^\\star\\|_2^2}{2 \\gamma t {N}} + \\frac{c_\\varphi \\rho^\\star}{{N}}\n\\end{align}\nis satisfied, where $c_\\varphi = 4 \\|\\varphi\\|_\\infty^2$, $0 < \\gamma \\leq 1\/L$ and $L$ is the Lipschitz constant of the gradient $\\nabla\\rho(\\theta)$ in Assumption \\ref{ass:LipschitzCont}.\n\\end{thm}\n\n\\begin{proof}\nSee Appendix \\ref{app:thm:GD}.\n\\end{proof}\n\n\\begin{rem}\\label{remGDasymptote} Theorem~\\ref{thm:GD} sheds light onto several facts. We first note that $\\rho^\\star$ in the error bound \\eqref{ineq:gOAISbound} can be interpreted as an indicator of the quality of the parametric proposal. We recall that $\\rho^\\star = 1$ when both $\\pi$ and $q_\\theta$ belong to the same exponential family. For this special case, Theorem~\\ref{thm:GD} implies that\n\\begin{align*}\n\\lim_{t\\to\\infty} \\left\\|(\\varphi,\\pi) - (\\varphi,\\pi_{\\theta_t}^N)\\right\\|_2 \\leq \\mathcal{O}\\left(\\frac{1}{\\sqrt{N}}\\right).\n\\end{align*}\nIn other words, when the target and the proposal are both from the exponential family, this adaptation strategy is leading to an \\textit{asymptotically optimal} Monte Carlo estimator (optimal meaning that we attain the same rate as a Monte Carlo estimator with $N$ iid samples from $\\pi$). On the other hand, when $\\pi$ and $q_\\theta$ do not belong to the same family, we obtain\n\\begin{align*}\n\\lim_{t\\to\\infty} \\left\\|(\\varphi,\\pi) - (\\varphi,\\pi_{\\theta_t}^N)\\right\\|_2 \\leq \\mathcal{O}\\left(\\sqrt{\\frac{\\rho^\\star}{N}}\\right),\n\\end{align*}\ni.e., the $L_2$ rate is again asymptotically optimal, but the constant in the error bound is worse (bigger) by a factor $\\sqrt{\\rho^\\star}>1$. $\\square$\n\\end{rem}\n\nThis bound shows that as $t\\to\\infty$, what we are left with is essentially the minimum attainable IS error for a given parametric family $\\{q_\\theta\\}_{\\theta\\in\\Theta}$. Intuitively, when the proposal $q_\\theta$ is from a different parametric family than $\\pi$, the gradient OAIS optimises the error bound in order to obtain the best possible proposal. In particular, the MSE has two components: First an $\\mathcal{O}(1\/tN)$ component which can be made to vanish over time by improving the proposal and a second $\\mathcal{O}(1\/N)$ component which is related to $\\rho^\\star$. The quantity $\\rho^\\star$ is related to the minimum $\\chi^2$-divergence between the target and proposal. This means that the discrepancy between the target and \\textit{optimal} proposal (according to the $\\chi^2$-divergence) can only be tackled by increasing $N$. This intuition is the same for the schemes we analyse in the next sections, although the rate with respect to the number of iterations necessarily worsens because of the uncertainty in the gradient estimators.\n\n\\begin{rem} When $\\gamma = 1\/L$, Theorem~\\ref{thm:GD} implies that if $t = \\mathcal{O}({L}\/{\\rho^\\star})$ and $N = \\mathcal{O}(\\rho^\\star \/ \\varepsilon)$, for some $\\varepsilon>0$, then we have\n\\begin{align*}\n{\\mathbb E}\\left[\\left((\\varphi,\\pi) - (\\varphi,\\pi_{\\theta_t}^N)\\right)^2\\right] &\\leq \\mathcal{O}(\\varepsilon).\n\\end{align*}\nWe remark that once we choose the number of samples $N = \\mathcal{O}(\\rho^\\star\/\\varepsilon)$, the number of iterations $t$ for adaptation is independent of $N$ and $\\varepsilon$. $\\square$\n\\end{rem}\n\n\\begin{rem} One can use different maps $\\mathcal{T}_t$ for optimisation. For example, one can use Nesterov's accelerated gradient descent (which has more parameters than just a step size), in which case, one could prove (by a similar argument) the inequality \\citep{nesterov2013introductory}\n\\begin{align*}\n{\\mathbb E}\\left[\\left((\\varphi,\\pi) - (\\varphi,\\pi_{\\theta_t}^N)\\right)^2\\right] &\\leq \\mathcal{O}\\left(\\frac{1}{t^2 {N}} + \\frac{\\rho^\\star}{{N}}\\right).\n\\end{align*}\nThis is an improved convergence rate, going from $\\mathcal{O}(1\/t)$ to $\\mathcal{O}(1\/t^2)$ in the first term of the bound. $\\square$\n\\end{rem}\n\n\\subsection{Convergence rate with averaged SGD iterates} \\label{ssConvergence-Averaged-Iterates}\n\nWhile, for the purpose of analysis, it is convenient to assume that the minimization of $\\rho(\\theta)$ can be done deterministically, this is rarely the case in practice. The `best' realistic case is that we can obtain an unbiased estimator of the gradient. {Hence, we address this scenario, under the assumption that the actual gradient functions $\\nabla \\rho$ and $\\nabla R$ are bounded in $\\Theta$ (i.e., $\\rho(\\theta)$ is Lipschitz in $\\Theta$).}\n\n{\\begin{assumption}\\label{ass:BoundedGradient} The gradient functions $\\nabla \\rho(\\theta)$ and $\\nabla R(\\theta)$ are bounded in $\\Theta$. To be specific, there exist finite constants $G_\\rho$ and $G_R$ such that\n\\begin{eqnarray}\n\\sup_{\\theta\\in\\Theta} \\|\\nabla \\rho(\\theta)\\|_2 &<& G_\\rho <\\infty \\quad \\text{and} \\nonumber\\\\\n\\sup_{\\theta\\in\\Theta} \\|\\nabla R(\\theta)\\|_2 &<& G_R < \\infty. \\nonumber\n\\end{eqnarray}\n\\end{assumption}}\n\n{We note that this is a mild assumption in the case of interest in this paper, where $\\Theta \\subset {\\mathbb R}^{d_\\theta}$ is assumed to be compact.}\n\n\n\\subsubsection{Normalised target}\\label{sec:NormalisedIS}\n\nFirst, we assume that we can evaluate $\\pi(x)$, which means that at iteration $t$, we can obtain an unbiased estimate of $\\nabla \\rho(\\theta_{t-1})$, denoted $g_t$. We use the optimisation algorithms called \\textit{stochastic gradient} methods, which use stochastic and unbiased estimates of the gradients to optimise a given cost function \\citep{RobbinsMonro}. Particularly, we focus on optimised samplers using stochastic gradient descent (SGD) as an adaptation strategy.\n\n{We start proving that the stochastic gradient estimates $(g_t)_{t\\geq 0}$ have a finite mean-squared error (MSE) w.r.t. the true gradients. To prove this result, we need an additional regularity condition.}\n{\\begin{assumption}\\label{ass:SupSupBound} \nThe normalised target and proposal densities satisfy the inequality \n\\begin{align*}\n\\sup_{\\theta\\in\\Theta} {\\mathbb E}_{q_\\theta}\\left[ \\left|\\frac{\\pi^2(X)}{q_\\theta^2(X)} \\frac{\\partial \\log q_\\theta}{\\partial \\theta_j}(X) \\right|^2 \\right] =: D_\\pi^j < \\infty.\n\\end{align*}\nfor $j=1, \\ldots, d_\\theta$. We denote $D_\\pi := \\max_{j \\in \\{1,\\ldots,d_\\theta\\}} D_\\pi^j < \\infty$.\n\\end{assumption}}\n\n{\n\\begin{rem} \\label{remSupSup}\nLet us rewrite $D_\\pi^j$ in Assumption \\ref{ass:SupSupBound} in terms of the weight function, namely\n\\begin{align*}\nD_\\pi^j = \\sup_{\\theta\\in\\Theta} {\\mathbb E}_{q_\\theta} \\left[ \\left| w_\\theta^2(X) \\frac{\\partial \\log q_\\theta}{\\partial \\theta_j}(X) \\right|^2 \\right].\n\\end{align*}\nWhen $q_\\theta(x)$ belongs to the exponential family, we readily obtain\n\\begin{align*}\nD_\\pi^j = \\sup_{\\theta\\in\\Theta} {\\mathbb E}_{q_\\theta} \\left[ w_\\theta^4(X) \\left( \\frac{\\partial A(\\theta)}{\\partial \\theta_i} -T_i(X) \\right)^2 \\right],\n\\end{align*}\nwhere $T_i(X)$ is the $i$-th sufficient statistic for $q_\\theta(x)$. Let us construct a bounding function for the weights of the form\n$$\nK(\\theta) := \\sup_{x \\in {\\sf X}} w_\\theta(x).\n$$\nIf we choose the compact space $\\Theta$ in such a way that $K(\\theta)$ is bounded, then we readily have\n\\begin{align*}\nD_\\pi^j &\\le \\sup_{\\theta\\in\\Theta} K^4(\\theta) {\\mathbb E}_{q_\\theta} \\left[ \\left( \\frac{\\partial A(\\theta)}{\\partial \\theta_i} -T_i(X) \\right)^2 \\right] \\\\\n&\\le \\| K \\|_\\infty^4 \\text{Var}( T_i(X) ),\n\\end{align*}\nwhere we have used the fact that $\\frac{\\partial^m A(\\theta)}{\\partial \\theta_i} = {\\mathbb E}_{q_{\\theta}}\\left[ T_i^m(X) \\right]$. Therefore, if the weights remain bounded in $\\Theta$, a sufficient condition for Assumption \\ref{ass:SupSupBound} to hold is that the sufficient statistics of the proposal distribution all have finite variances, i.e., $\\max_{i \\in \\{1, \\ldots, d_\\theta\\} } T_i(X) < \\infty$.\n\\\\ \\\\\n{There are alternative conditions that, when satisfied, lead to Assumption \\ref{ass:SupSupBound} holding true. As an example, in Appendix \\ref{apRho2} we provide an alternative sufficient condition in terms of the function $\\rho_2(\\theta):={\\mathbb E}[ w_\\theta^4(X) ]$.}\n\\end{rem}\n}\n\nNow we show that when $g_t$ is an iid Monte Carlo estimator of $\\nabla \\rho$, we have the following finite-sample bound for the MSE.\n{\\begin{lem}\\label{lem:GradientMonteCarlo}\nIf Assumption~\\ref{ass:SupSupBound} holds, the following inequality holds,\n\\begin{align*}\n{\\mathbb E}[\\| g_t - \\nabla \\rho(\\theta_{t-1})\\|_2^2] \\leq \\frac{d_\\theta c_{\\rho} D_\\pi}{N},\n\\end{align*}\nwhere $d_\\theta$ is the parameter dimension and $c_\\rho, D_\\pi < \\infty$ are constant w.r.t. $N$.\n\\end{lem}\n\\begin{proof}\nSee Appendix~\\ref{app:lem:GradientMonteCarlo}.\n\\end{proof}}\n\nIn order to obtain convergence rates for the estimator $(\\varphi,\\pi_{\\bar \\theta_t}^N)$ we first recall a classical result for the SGD (see, e.g., \\cite{bubeck2015convex}).\n\\begin{lem}\\label{lem:SGDconv} \n{Let Assumptions \\ref{ass:BoundedGradient} and \\ref{ass:SupSupBound} hold, apply recursion \\eqref{eq:recSgdAdaptNorm} and let $(g_t)_{t\\geq 0}$ be the stochastic gradient estimates in Lemma~\\ref{lem:GradientMonteCarlo}}. If we choose the step-size sequence $\\gamma_k = \\alpha \/ \\sqrt{k}$, $1\\leq k \\leq t$, for any $\\alpha > 0$, then\n{\\begin{align}\\label{eq:SGDrate}\n{\\mathbb E}[\\rho(\\bar{\\theta}_t) - \\rho^\\star] \\leq \\frac{{\\mathbb E}\\|\\theta_0 - \\theta^\\star\\|_2^2}{2\\alpha\\sqrt{t}} + \\frac{\\alpha d_\\theta c_\\rho D_\\pi}{\\sqrt{t} N} + \\frac{\\alpha G^2_\\rho}{\\sqrt{t}},\n\\end{align}}\nwhere $\\bar{\\theta}_t = \\frac{1}{t}\\sum_{k=0}^{t-1} \\theta_k$.\n\\end{lem}\n\n\\begin{proof}\nSee Appendix \\ref{app:lem:SGDconv} for a self-contained proof.\n\\end{proof}\n\nWe can now state the first core result of the paper, which is the convergence rate for the AIS algorithm using a SGD adaptation of the parameter vectors $\\theta_t$.\n\\begin{thm}\\label{thm:SGDAIS} \nLet Assumptions \\ref{ass:BoundedGradient} and \\ref{ass:SupSupBound} hold, let the sequence $(\\theta_t)_{t\\geq 1}$ be computed using \\eqref{eq:recSgdAdaptNorm} and construct the averaged iterates $\\bar{\\theta}_t = \\frac{1}{t} \\sum_{k=0}^{t-1} \\theta_k$. Then, the sequence of proposal distributions $(q_{\\bar{\\theta}_t})_{t\\geq 1}$ satisfies the inequality\n{\\begin{align}\\label{eq:rateSGDAIS}\n{\\mathbb E}\\left[\\left((\\varphi,\\pi) - (\\varphi,\\pi_{\\bar{\\theta}_t}^N)\\right)^2\\right] &\\leq \\frac{c_1}{\\sqrt{t}N} + \\frac{c_2}{\\sqrt{t} N^2} + \\frac{c_3}{\\sqrt{t} N} + \\frac{c_4}{N}\n\\end{align}\nfor $t \\ge 1$ and any $\\varphi \\in B({\\sf X})$, where\n\\begin{align*}\nc_1 &= \\frac{c_\\varphi {\\mathbb E}\\|\\theta_0 - \\theta^\\star\\|_2^2}{2 \\alpha}, \\\\ \nc_2 &= {c_\\varphi c_\\rho \\alpha d_\\theta D_\\pi}, \\\\\nc_3 &={c_\\varphi \\alpha G_\\rho^2}, \\\\\nc_4 &={c_\\varphi \\rho^\\star},\n\\end{align*}\nand $c_\\varphi = 4\\|\\varphi\\|_\\infty^2$ are finite constants independent of $t$ and $N$.}\n\\end{thm}\n\n\\begin{proof}\nSee Appendix \\ref{app:thm:SGDAIS}.\n\\end{proof}\n\n\\begin{rem} \nNote that the expectation on the left hand side of \\eqref{eq:rateSGDAIS} is taken w.r.t. the distribution of the measure-valued random variable $\\pi_{\\bar \\theta_t}^N$. $\\square$\n\\end{rem}\n\nTheorem~\\ref{thm:SGDAIS} can be interpreted similarly to Theorem~\\ref{thm:GD}. One can see that the overall rate of the MSE bound is $\\mathcal{O}\\left({1}\/{\\sqrt{t}N} + {1}\/{N}\\right)$. This means that, as $t\\to\\infty$, we are only left with a rate that is optimal for the AIS for a given parametric proposal family. In particular, again, $\\rho^\\star$ is related to the minimal $\\chi^2$-divergence between the target $\\pi$ and the parametric proposal $q_\\theta$. When the proposal and the target are from the same family, we are back to the case $\\rho^\\star = 1$, thus the adaptation leads to the optimal Monte Carlo rate $\\mathcal{O}(1\/\\sqrt{N})$ for the $L_2$ error within this setting as well.\n\n\\subsubsection{Self-normalised estimators}\n\nWe have noted that it is possible to obtain an unbiased estimate of $\\nabla\\rho(\\theta)$ when the normalised target $\\pi(x)$ can be evaluated. However, if we can only evaluate the unnormalised density $\\Pi(x)$ instead of $\\pi(x)$ and use the self-normalized IS estimator, the estimate of $\\nabla\\rho(\\theta)$ is no longer unbiased. We refer to Sec.~5 of \\cite{tadic2017asymptotic} for stochastic optimisation with biased gradients for adaptive Monte Carlo, where the discussion revolves around minimizing the Kullback-Leibler divergence rather than the $\\chi^2$-divergence. The results presented in \\cite{tadic2017asymptotic}, however, are asymptotic, while herein we are interested in finite-time bounds. Due to the structure of the AIS scheme, it is possible to avoid working with biased gradient estimators. In particular, we can implement the proposed AIS schemes using unbiased estimators of $\\nabla R(\\theta)$ instead of biased estimators of $\\nabla \\rho(\\theta)$. Since optimizing the unnormalised function $R(\\theta)$ leads to the same minima as optimizing the normalised function $\\rho(\\theta)$, we can simply use $\\nabla R(\\theta)$ for the adaptation in the self-normalised case.\n\nSimilar to the argument in Section \\ref{sec:NormalisedIS}, we first start the assumption below, which is the obvious counterpart of Assumption \\ref{ass:SupSupBound}.\n{\\begin{assumption}\\label{ass:SupSupBoundPi} \nThe unnormalized target $\\Pi(x)$ and the proposal densities $q_\\theta(x)$ satisfy the inequalities\n\\begin{align*}\n\\sup_{\\theta\\in\\Theta} {\\mathbb E}_{q_\\theta} \\left[ \\left| \\frac{\\Pi^2(X)}{q_\\theta^2(X)} \\frac{\\partial \\log q_\\theta}{\\partial \\theta_j}(X) \\right|^2 \\right] =: D_\\Pi^j < \\infty\n\\end{align*}\nfor $j=1, \\ldots, d_\\theta$. We denote $D_\\Pi : = \\frac{1}{d_\\theta} \\sum_{j=1}^{d_\\theta} D_\\Pi^j < \\infty$.\n\\end{assumption}}\n\nRemark \\ref{remSupSup} holds directly for Assumption \\ref{ass:SupSupBoundPi} as long as $Z_\\pi<\\infty$. {Next, we prove an MSE bound for the stochastic gradients $(\\tilde{g}_t)_{t\\geq 0}$ employed in recursion \\eqref{eq:SgdUnnormalizedAdapt}, i.e., the unbiased stochastic estimates of $\\nabla R(\\theta)$.}\n\n{\n\\begin{lem}\\label{lem:GradientMonteCarloR}\nIf Assumptions \\ref{ass:BoundedGradient} and \\ref{ass:SupSupBoundPi} hold, the inequality\n\\begin{align*}\n{\\mathbb E}[\\| \\tilde{g}_t - \\nabla R(\\theta_{t-1})\\|_2^2] \\leq \\frac{d_\\theta c_R D_\\Pi}{N},\n\\end{align*}\nis satisfied, where $c_R, D_\\Pi < \\infty$ are constants w.r.t. of $N$.\n\\end{lem}\n\\begin{proof}\nThe proof is identical to the proof of Lemma~\\ref{lem:GradientMonteCarlo}.\n\\end{proof}\n}\n\nWe can now obtain explicit rates for the convergence of the unnormalized function $R(\\bar \\theta_t)$, evaluated at the averaged iterates $\\bar \\theta_t$. \n\n\\begin{lem}\\label{lem:SGDBiasedconv} \nIf Assumptions \\ref{ass:BoundedGradient} and \\ref{ass:SupSupBoundPi} hold and the sequence $(\\theta_t)_{t\\ge 1}$ is computed via recursion \\eqref{eq:SgdUnnormalizedAdapt}, with step-sizes $\\gamma_k = \\beta \/ \\sqrt{k}$ for $1\\leq k \\leq t$ and $\\beta > 0$, we obtain the inequality\n{\\begin{align}\\label{eq:UnnormRate}\n{\\mathbb E}[R(\\bar{\\theta}_t) - R ^\\star] \\leq \\frac{{\\mathbb E} \\|\\theta_0 - \\theta^\\star\\|_2^2}{2 \\beta \\sqrt{t}} + \\frac{\\beta d_\\theta c_R D_\\Pi}{\\sqrt{t} N} + \\frac{\\beta G^2_R}{\\sqrt{t}}\n\\end{align}}\nwhere $c_R,D_\\Pi<\\infty$ are constants w.r.t. $t$ and $N$. Relationship \\ref{eq:UnnormRate} implies that\n{\\begin{align}\\label{eq:NormRateWithNormConsts}\n{\\mathbb E}[\\rho(\\bar{\\theta}_t) - \\rho^\\star] \\leq \n\\frac{{\\mathbb E} \\|\\theta_0 - \\theta^\\star\\|_2^2}{2 \\beta Z_\\pi^2 \\sqrt{t}} + \\frac{\\beta d_\\theta c_R D_\\Pi}{Z_\\pi^2 \\sqrt{t} N} + \\frac{\\beta G^2_R}{Z_\\pi^2 \\sqrt{t}}.\n\\end{align}}\n\\end{lem}\n\\begin{proof} The proof of the rate in \\eqref{eq:UnnormRate} is identical to the proof of Lemma~\\ref{lem:SGDconv}. The rate in \\eqref{eq:NormRateWithNormConsts} follows by observing that $\\rho(\\theta) = R(\\theta) \/ Z_\\pi^2$ for every $\\theta\\in\\Theta$.\n\\end{proof}\n\nFinally, using Lemma~\\ref{lem:SGDBiasedconv}, we can state our main result: an explicit error rate for the MSE of Algorithm~\\ref{alg:SGDAIS} as a function of the number of iterations $t$ and the number of samples $N$.\n\n\\begin{thm}\\label{thm:SGDAISUN} \nLet Assumptions \\ref{ass:BoundedGradient} and \\ref{ass:SupSupBoundPi} hold and let the sequence $(\\theta_t)_{t\\ge 1}$ be computed via recursion \\eqref{eq:SgdUnnormalizedAdapt}, with step-sizes $\\gamma_k = \\beta \/ \\sqrt{k}$ for $1\\leq k \\leq t$ and $\\beta > 0$. We have the following inequality for the sequence of proposal distributions $(q_{\\bar{\\theta}_t})_{t\\geq 1}$,\n{\\begin{align}\\label{eq:rateUnnormAIS}\n{\\mathbb E}\\left[\\left((\\varphi,\\pi) - (\\varphi,\\pi_{\\bar{\\theta}_t}^N)\\right)^2\\right] &\\leq \\frac{C_1}{\\sqrt{t} N} + \\frac{C_2}{\\sqrt{t}N^2} + \\frac{C_3}{\\sqrt{t} N} + \\frac{C_4}{N},\n\\end{align}\nwhere\n\\begin{align*}\nC_1 &= \\frac{c_\\varphi {\\mathbb E}\\|\\theta_0 - \\theta^\\star\\|_2^2}{2 \\beta Z_\\pi^2}, \\\\\nC_2 &= \\frac{c_\\varphi \\beta c_R d_\\theta D_\\Pi}{Z_\\pi^2}, \\\\\nC_3 &= \\frac{c_\\varphi \\beta G^2_R}{Z_\\pi^2}, \\\\\nC_4 &= {c_\\varphi \\rho^\\star},\n\\end{align*}\nand $c_\\varphi = 4\\|\\varphi\\|_\\infty^2$ are finite constants independent of $t$ and $N$.}\n\\end{thm}\n\\begin{proof}\nThe proof follows from Lemma~\\ref{lem:SGDBiasedconv} and mimicking the exact same steps as in the proof of Theorem~\\ref{thm:SGDAIS}.\n\\end{proof}\n\n\\begin{rem} \nTheorem~\\ref{thm:SGDAISUN}, as in Remark~\\ref{remGDasymptote}, provides relevant insights regarding the performance of the stochastic gradient OAIS algorithm. In particular, for a general target $\\pi$, we obtain\n\\begin{align*}\n\\lim_{t\\to\\infty} \\left\\|(\\varphi,\\pi) - (\\varphi,\\pi_{\\bar{\\theta}_t}^N)\\right\\|_2 = \\mathcal{O}\\left(\\sqrt{\\frac{\\rho^\\star}{N}}\\right).\n\\end{align*}\nThis result shows that the $L_2$ error is asymptotically optimal. As in previous cases, if the target $\\pi$ is in the exponential family, then the asymptotic convergence rate is $\\mathcal{O}(1\/\\sqrt{N})$ as $t \\to \\infty$. $\\square$\n\\end{rem}\n\n\\begin{rem} \nTheorem~\\ref{thm:SGDAISUN} also yields a practical heuristic to tune the step-size and the number of particles together. Assume that $0 < \\beta < 1$ and let $N = 1\/\\beta$ (which we assume to be an integer without loss of generality). In this case, the rate \\eqref{eq:rateUnnormAIS} simplifies into\n\\begin{align*}\n{\\mathbb E}\\left[\\left((\\varphi,\\pi) - (\\varphi,\\pi_{\\bar{\\theta}_t}^N)\\right)^2\\right] &\\leq \\frac{c_\\varphi {\\mathbb E}\\|\\theta_0 - \\theta^\\star\\|_2^2}{2 Z_\\pi^2 \\sqrt{t}} + \\frac{c_\\varphi \\beta^3 c_R d_\\theta D_\\Pi}{Z_\\pi^2 \\sqrt{t}} + \\frac{c_\\varphi \\beta^2 G^2_R}{Z_\\pi^2\\sqrt{t}} + c_\\varphi \\rho^\\star \\beta\n\\end{align*}\nNow, if we let $t = \\mathcal{O}(1\/\\beta^2)$ we readily obtain\n\\begin{align*}\n{\\mathbb E}\\left[\\left((\\varphi,\\pi) - (\\varphi,\\pi_{\\bar{\\theta}_t}^N)\\right)^2\\right] &\\leq \\mathcal{O}(\\beta).\n\\end{align*}\nTherefore, one can control the error using the step-size of the optimisation scheme provided that other parameters of the algorithm are chosen accordingly. The same argument also holds for Theorem~\\ref{thm:SGDAIS}. $\\square$\n\\end{rem}\n\n\\begin{rem} \n{It is not straightforward to compare the rates in inequality \\eqref{eq:rateUnnormAIS} (for the unnormalized target $\\Pi(x)$) and inequality \\eqref{eq:rateSGDAIS} (for the normalized target $\\pi(x)$). Even if \\eqref{eq:rateUnnormAIS} may ``look better'' by a constant factor compared to the rate in \\eqref{eq:rateSGDAIS}, this is usually not the case. Indeed, the variance of the errors in the unnormalised gradient estimators is typically higher and this reflects on the variance of the moment estimators. Another way to look at this issue is to realise that, very often, $Z_\\pi << 1$, which makes the bound in \\eqref{eq:rateUnnormAIS} much greater than the bound in \\eqref{eq:rateSGDAIS}.}\n\\end{rem}\n\n{Finally, we can adapt Theorem~\\ref{thm:SNISbias} to our case, providing a convergence rate of the bias of the importance sampler given by Algorithm~\\ref{alg:SGDAIS}.}\n\n{\\begin{thm}\\label{thm:SGDAISbias}\nUnder the setting of Theorem~\\ref{thm:SGDAISUN}, we have\n\\begin{align}\\label{eq:rateUnnormAISbias}\n\\left| {\\mathbb E}\\left[(\\varphi,\\pi_{\\bar{\\theta}_t}^N)\\right] - (\\varphi, \\pi)\\right| &\\leq \\frac{3C_1}{\\sqrt{t} N} + \\frac{3C_2}{\\sqrt{t}N^2} + \\frac{3C_3}{\\sqrt{t} N} + \\frac{3C_4}{N},\n\\end{align}\nwhere $C_1,C_2,C_3,C_4$ are finite constants given in Theorem~\\ref{thm:SGDAISUN} and independent of $t$ and $N$.\n\\end{thm}}\n\\begin{proof}\nThe proof follows from Theorem~\\ref{thm:SNISbias} and mimicking the same proof technique used to prove Theorem~\\ref{thm:SGDAISUN}.\n\\end{proof}\n\n\\subsection{Convergence rate with vanilla SGD}\n\n{The arguments of Section \\ref{ssConvergence-Averaged-Iterates} can be carried over to the analysis of Algorithm \\ref{alg:vanillaSGDAIS}, where the proposal functions $q_{\\theta_t}(x)$ are constructed using the iterates $\\theta_t$ rather than the averages $\\bar \\theta_t$. Unfortunately, achieving the optimal $\\mathcal{O}(1\/\\sqrt{t})$ rate for the vanilla SGD is difficult in general. The best available rate without significant restrictions on the step-size is given by \\citet{shamir2013stochastic}. In particular, we can adapt \\citet[Theorem~2]{shamir2013stochastic} to our setting in order to state the following lemma.\n\\begin{lem}\\label{lem:vanillaSGDconv} \nApply recursion \\eqref{eq:SgdUnnormalizedAdapt} for the computation of the iterates $(\\theta_t)_{t\\ge 1}$, choose the step-sizes $\\gamma_k = \\beta \/ \\sqrt{k}$ for $1\\leq k \\leq t$, where $\\beta > 0$, and let Assumptions \\ref{ass:BoundedGradient} and \\ref{ass:SupSupBoundPi} hold. Then, we have the inequality\n\\begin{align}\n{\\mathbb E}[R({\\theta}_t) - R ^\\star] \\leq \\left(\\frac{D^2}{\\beta \\sqrt{t}} + \\frac{\\beta d_\\theta c_R D_\\Pi}{\\sqrt{t} N} + \\frac{\\beta G^2_R}{\\sqrt{t}}\\right) (2 + \\log t),\n\\end{align}\nwhere $D := \\sup_{\\theta,\\theta' \\in \\Theta} \\|\\theta - \\theta'\\| < \\infty$. This in turn implies that\n{\\begin{align}\n{\\mathbb E}[\\rho({\\theta}_t) - \\rho^\\star] \\leq \n\\left(\\frac{D^2}{\\beta \\sqrt{t}} + \\frac{\\beta d_\\theta c_R D_\\Pi}{\\sqrt{t} N} + \\frac{\\beta G^2_R}{\\sqrt{t}}\\right)\\frac{(2 + \\log t)}{Z_\\pi^2}.\n\\end{align}}\n\\end{lem}}\n\\begin{proof}\nIt is straightforward to prove this result using \\citet[Theorem~2]{shamir2013stochastic} and the proof of Lemma~\\ref{lem:SGDconv}.\n\\end{proof}\n{The obtained rate is, in general, $\\mathcal{O}\\left( \\frac{\\log t}{\\sqrt{t}}\\right)$. This is known to be suboptimal and it can be improved to the {information-theoretical optimal} $\\mathcal{O}(1\/\\sqrt{t})$ rate by choosing a specific step-size scheduling, see, e.g., \\citet{jain2019making}. {However, in this case, the scheduling of $(\\gamma_t)_{t\\geq 1}$ depends directly on the total number of iterates to be generated, in such a way that the error $\\mathcal{O}(1\/\\sqrt{t})$ is guaranteed only for the {\\em last} iterate, at the final time $t$.}}\n\nWe can extend Lemma \\ref{lem:vanillaSGDconv} to obtain the following result.\n\n{\\begin{thm}\\label{thm:vanillaSGDAIS} \nApply recursion \\eqref{eq:SgdUnnormalizedAdapt} for the computation of the iterates $(\\theta_t)_{t\\ge 1}$, choose the step-sizes $\\gamma_k = \\beta \/ \\sqrt{k}$ for $1\\leq k \\leq t$, where $\\beta > 0$, and let Assumptions \\ref{ass:BoundedGradient} and \\ref{ass:SupSupBoundPi} hold. If we construct the sequence of proposal distributions $(q_{{\\theta}_t})_{t\\geq 1}$ be the sequence of proposal distributions we obtain the following MSE bounds\n\\begin{align}\n{\\mathbb E}\\left[\\left((\\varphi,\\pi) - (\\varphi,\\pi_{{\\theta}_t}^N)\\right)^2\\right] &\\le \\left(\n\t\\frac{C_1}{\\sqrt{t} N} + \\frac{C_2}{\\sqrt{t}N^2} + \n\t\\frac{C_3}{\\sqrt{t} N}\n\\right)(2 + \\log t) + \\frac{C_4}{N},\n\\label{eq:rateUnnormAIS-2}\n\\end{align}\nwhere\n\\begin{align*}\nC_1 &= \\frac{c_\\varphi D^2}{2 \\beta Z_\\pi^2}, \\\\\nC_2 &= \\frac{c_\\varphi \\beta c_R d_\\theta D_\\Pi}{Z_\\pi^2}, \\\\\nC_3 &= \\frac{c_\\varphi \\beta G^2_R}{Z_\\pi^2}, \\\\\nC_4 &= {c_\\varphi \\rho^\\star},\n\\end{align*}\nand $c_\\varphi = 4\\|\\varphi\\|_\\infty^2$ are finite constants independent of $t$ and $N$.\n\\end{thm}}\n\\begin{proof}\nThe proof follows from Lemma~\\ref{lem:vanillaSGDconv} with the exact same steps as in the proof of Theorem~\\ref{thm:SGDAIS}.\n\\end{proof}\n{Finally, it is also straightforward to adapt the bias result in Theorem~\\ref{thm:SGDAISbias} to this case, which results in the similar bound. We skip it for space reasons and also because it has the same form as in Theorem~\\ref{thm:SGDAISbias} with an extra $\\log t$ factor.}\n\n\\section{Conclusions}\\label{sec:conc}\nWe have presented and analysed \\textit{optimised} parametric adaptive importance samplers and provided non-asymptotic convergence bounds for the MSE of these samplers. Our results display the precise interplay between the number of iterations $t$ and the number of samples $N$. In particular, we have shown that the optimised samplers converge to an optimal proposal as $t\\to\\infty$, leading to an asymptotic rate of $\\mathcal{O}(\\sqrt{\\rho^\\star\/N})$. This intuitively shows that the number of samples $N$ should be set in proportion to the minimum $\\chi^2$-divergence between the target and the exponential family proposal, as we have shown that the adaptation (in the sense of minimising $\\chi^2$-divergence or, equivalently, the variance of the weight function) cannot improve the error rate beyond $\\mathcal{O}(\\sqrt{\\rho^\\star\/N})$. The error rates in this regime may be dominated by how close the target is to the exponential family.\n\nNote that the algorithms we have analysed require constant computational load at each iteration and the computational load does not increase with $t$ as we do not re-use the samples in past iterations. Such schemes, however, can also be considered and analysed in the same manner. More specifically, in the present setup the computational cost of each iteration depends on the cost of evaluating $\\Pi(x)$.\n\nOur work opens up several other paths for research. One direction is to analyse the methods with more advanced optimisation algorithms. Another challenging direction is to consider more general proposals than the natural exponential family, which may lead to non-convex optimisation problems of adaptation. Analysing and providing guarantees for this general case would provide foundational insights for general adaptive importance sampling procedures. Also, as shown by \\citet{ryu2016convex}, similar theorems can also be proved for $\\alpha$-divergences.\n\nAnother related piece of work arises from variational inference \\citep{wainwright2008graphical}. In particular, \\citet{dieng2017variational} have recently considered performing variational inference by minimising the $\\chi^2$-divergence, which is close to the setting in this paper. In particular, the variational approximation of the target distribution in the variational setting coincides with the proposal distribution we consider within the importance sampling context in this paper. This also implies that our results may be used to obtain finite-time guarantees for the expectations estimated using the variational approximations of target distributions.\n\nFinally, the adaptation procedure can be modified to handle the non-convex case as well. In particular, the SGD step can be converted into a stochastic gradient Langevin dynamics (SGLD) step. The SGLD method can be used as a global optimiser when $\\rho$ and $R$ are non-convex and a global convergence rate can be obtained using the standard SGLD results, see, e.g., \\citet{raginsky2017non,zhang2019nonasymptotic}. Global convergence results for other adaptation schemes such as stochastic gradient Hamiltonian Monte Carlo (SGHMC) can also be achieved using results from nonconvex optimisation literature, see, e.g., \\citet{akyildiz2020nonasymptotic}.\n\\section*{Acknowledgements}\n\\\"O.~D.~A. is funded by the Lloyds Register Foundation programme on Data Centric Engineering through the London Air Quality project. This work was supported by The Alan Turing Institute for Data Science and AI under EPSRC grant EP\/N510129\/1. J.~M. acknowledges the support of the Spanish \\textit{Agencia Estatal de Investigaci\\'on} (awards TEC2015-69868-C2-1-R ADVENTURE and RTI2018-099655-B-I00 CLARA) and the Office of Naval Research (award no. N00014-19-1-2226).\n\n\n\n\n\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Statement of Results}\nIt has long been known that, given an ergodic invertible probability measure preserving system, a Rohlin tower may be constructed with base independent of a given partition of the underlying space(\\cite{Roh:52}, \\cite{Roh:65}). In \\cite{Alp:79}, meanwhile, S. Alpern proved a `multiple' Rohlin tower theorem (see \\cite{EP:97} for an easy proof) whose full statement we will not give, but which has the following corollary of interest: \n\n\\begin{thm}\\label{thm:alp}\nLet $N \\in \\ensuremath{\\mathbb N} $ and \\ensuremath{\\epsilon > 0\\;} be given. For any ergodic invertible measure-preserving transformation $T$ of a Lebesgue probability space $(X, \\mathcal B, \\mu)$, there exists a Rohlin tower of height $N$ with base $B$ and error set $E$ with $\\mu(E) < \\ensuremath{\\epsilon} $, so that $T(E) \\subset B$. \n\\end{thm}\n\nA {\\em Rohlin tower of height} $N$ {\\em with base} $B$ {\\em and error set} $E$ is characterized by the collection of sets \n$\\{B, TB, \\dots, T^{N-1}B, E\\}$ forming a partition of $X$. If in addition $T(E) \\subset B$, we shall say {\\bf Alpern Tower}.\nIt is our goal to show that for ergodic transformations on $(X, \\ensuremath{\\mathcal B} , \\mu)$, given a finite measurable partition $\\mathbb P$ of $X$, an Alpern tower may be constructed with base $B$ independent of $\\mathbb P$. Precisely:\n\n\\begin{bigthm}\n\\label{mainthm}\nLet $(X, \\mathcal B, \\mu)$ be a Lebesgue probability space, and suppose $\\mathbb P$ is a finite measurable partition of $X$. For any ergodic invertible measure-preserving transformation $T$ of $X$, $N \\in \\ensuremath{\\mathbb N} $, \nthere exists a Rohlin tower of height $N$ with base $B$ and error set $E$ such that $T(E) \\subset B$ and $B$ is independent of $\\mathbb P$. \n\\end{bigthm}\n\nWe do not specify the size of the error set; but the process of constructing our tower makes it clear that the error set may be made arbitrarily small. \n\n\n\\section{Proof of main result}\n\nFor the remainder of the paper, $(X, \\mathcal B, \\mu)$ will be a fixed Lebesgue probability space and $T:X \\to X$ will be an invertible ergodic measure-preserving transformation on $X$. All mentioned sets will be measurable and we will adopt a cavalier attitude toward null sets. In particular, ``partition'' will typically mean ``measurable partition modulo null sets''. \n\\begin{defn}\nBy a {\\em tower over B} we will mean a set $B \\subset X$, called the {\\em base}, and a countable partition $B = B_1 \\cup B_2 \\cup \\cdots$, together with their images $T^iB_j$, $0 \\le i < j$, such that the family $\\{T^iB_j : 0 \\le i < j\\}$ consists in pairwise disjoint sets. If this family partitions $X$, we will say that the tower is {\\em exhaustive}. \n\\end{defn}\n\nIf a tower over $B$ is exhaustive and $B = B_N \\cup B_{N+1}$, we shall speak of an {\\em exhaustive Alpern tower of height} $\\{N, N+1\\}$, as in such a case, \n$\\{B, TB, \\ldots, T^{N-1}B, E=T^N B_{N+1}\\}$ partitions $X$ with $T(E) \\subset B$. So \nwe may re-phrase Theorem 1 as: \\medskip\n\n\\noindent {\\em {\\bf Theorem \\ref{mainthm}:} Let $(X, \\mathcal B, \\mu)$ be a Lebesgue probability space and suppose $\\mathbb P$ is a finite measurable partition of $X$. For any ergodic invertible measure-preserving transformation $T$ of $X$, $N \\in \\ensuremath{\\mathbb N} $, one may find an exhaustive Alpern tower of height $\\{N, N+1\\}$ having base independent of $\\mathbb P$.} \\medskip\n\n\\noindent We require a lemma (and a corollary). \n\n\n\n\\begin{lem}\\label{lem:m}\nLet $M \\in \\ensuremath{\\mathbb N} $ and let $\\ensuremath{\\mathbb P} = \\{P_1, \\dots, P_t\\}$ be a partition of $X$ with $\\mu(P_i) > 0$ for each $i$. There exists a set $S$ of positive measure so that if $x \\in S$ with first return $n(x) = n$, say, then $|\\{x, Tx, \\dots, T^{n-1}x\\} \\cap P_i | \\ge M, 1 \\le i \\le t$. \n\\end{lem}\n\\proof For almost every $x$ we may find $K(x)$ so that for each $i$ between $1$ and $t$ we have $|\\{x, Tx, \\dots, T^{K(x) - 1}x\\} \\cap P_i| \\ge M$. Since almost all of $X$ is the countable union (over $k \\in \\ensuremath{\\mathbb N} $) of $\\{x: K(x) = k\\}$, there exists some fixed $K$ so that the set $A = \\{x: K(x) \\le K\\}$ has positive measure. If $C \\subset A$ has very small measure ($\\mu(C) < 1\/K$) then the average first-return time of $x \\in C$ to $C$ is $\\frac{1}{\\mu(C)} > K$, so we can find $S \\subset C$ with $\\mu(S) > 0$ so that $S, TS, \\dots, T^{K - 1}S$ are pairwise disjoint. \\hfill\\ensuremath{_\\blacksquare} \n\n\n\\begin{cor}\\label{cor:B}\n Let $M \\in \\ensuremath{\\mathbb N} $ and $\\ensuremath{\\mathbb P} = \\{P_1, \\dots, P_t\\}$ be a partition of $X$ with $\\mu(P_i) > 0$ for each $i$. There is a tower having base \n$S = S_{tM} \\cup S_{tM+1} \\cup \\cdots$ where for each $x \\in S_r$, $|\\{x, Tx, \\dots, T^{r - 1}x\\} \\cap P_i| \\ge M$ for all $1 \\le i \\le t$. \n\\end{cor}\n\\proof Let $S$, $K$ be as in Lemma \\ref{lem:m} and choose any $k \\ge K$. \\hfill\\ensuremath{_\\blacksquare} \\medskip\n\nWe turn now to the proof of Theorem \\ref{mainthm}. Fix a partition $\\ensuremath{\\mathbb P} = \\{P_1, \\dots, P_t\\}$, an arbitrary natural number $N$, and $\\epsilon >0$. Set $m_i = \\mu(P_i)$, and assume (without loss of generality) that $0 < m_1 \\le m_2 \\le \\dots \\le m_t$. Select and fix $M > \\frac{3N^3t}{m_1}$. Let $S$ be as in Corollary \\ref{cor:B} for this $M$; hence $S = S_{tM} \\cup S_{tM+1} \\cup \\cdots$. (Some $S_i$ may be empty, of course.) For each non-empty $S_R$, partition $S_R$ by $\\ensuremath{\\mathbb P} $-name of length $R$. (Recall that $x, y$ in $S_R$ have the same $\\ensuremath{\\mathbb P} $-name of length $R$ if $T^ix$ and $T^iy$ lie in the same cell of \\ensuremath{\\mathbb P} for $0 \\le i < R$.) Let $C$ be the base of one of the resulting columns; hence, every $x \\in C$ has the same $\\ensuremath{\\mathbb P} $-name of length $R$ (for some $R\\ge tM$), and the length $R$ orbit of each $x \\in C$ meets each $P_i$ at least $M$ times. \n\nPartition $C$ into pieces $C^{(1)}, C^{(2)}\\dots$, $C^{(t)}$ whose measures will be determined later. Then partition each $C^{(i)}$ into $N$ equal measure pieces, $C^{(i)} = C^{(i)}_1 \\cup C^{(i)}_2 \\cup \\dots \\cup C^{(i)}_N$. \n\nNow we fix $(R,C)$ and focus our attention on the height $R$ {\\em column} over a single $C^{(i)}$ and its height $R$ {\\em subcolumns} over $C^{(i)}_j$, $1\\leq j\\leq N$. We refer to the sets $T^rC^{(i)}$, $0 \\le r < R$, as {\\em levels} and to the sets $T^rC^{(i)}_j$ as {\\em rungs}. We are going to build a portion of $B$ by carefully selecting some rungs from the subcolumns under consideration. As we move through the various subcolumns, we need to have gaps of length $N$ or $N+1$ between selections. Now to specifics. We want \nto have our $\\ensuremath{C^{(i)}}$-selections form a ``staircase'' of height $N$ starting at level $N^2 - N$. That is, at height $(N-1)N$, the rung over $C^{(i)}_1$ is the only one selected; at height $N(N-1) + 1$, the rung over $C^{(i)}_2$ is the only one selected; etc., so that at height $N^2-1$, the rung over $C^{(i)}_N$ is the only one selected. \n\nThis is easy to accomplish. First, we select each base rung $C^{(i)}_j$, $j = 1, 2, \\dots, N$ (i.e., the rungs in the zeroth level). Over $C^{(i)}_1$, we then select $N-1$ additional rungs with gaps of length $N$; that is, we select the rungs at heights $N$, $2N, \\dots, (N-1)N$. Over $C^{(i)}_2$ we select $N-2$ rungs with gap $N$, then a rung with gap $N+1$. We continue in this fashion, choosing one less gap of length $N$ and one more of length $N+1$ in each subsequent subcolumn. In the last subcolumn (that over $C^{(i)}_N$) we are thus choosing rungs with gaps of length $N+1$ a total of $N-1$ times. See the left side of \nFigure \\ref{pic:bottom} for the case $N = 4$.\n\nNow we perform a similar procedure moving down from the top, so as to obtain a staircase starting at height $R-(N^2-1)$. \nNote that there are either $N$ or $N-1$ unselected rungs at the top of each subcolunm. See the right side of Figure \\ref{pic:bottom}. \n\n\n\n\n\\begin{figure}[hbtp]\n\n\\caption[Bottom of Tower for $N = 4$]\n {Bottom, Top of Tower for $N = 4$}\n\\setlength{\\unitlength}{.2in}\n\n\\begin{picture}(20,20)(0,0) \n\\label{pic:bottom}\n\\put(0, 0){$C^{(i)}_1 \\hspace{4mm} C^{(i)}_2 \\hspace{4mm} C^{(i)}_3 \\hspace{3.3mm} C^{(i)}_4$}\n\\multiput(0,2)(0,1){16}{\\line(1,0){1}}\n\\multiput(2,2)(0,1){16}{\\line(1,0){1}}\n\\multiput(4,2)(0,1){16}{\\line(1,0){1}}\n\\multiput(6,2)(0,1){16}{\\line(1,0){1}}\n\\put(4,18){\\vdots}\n\\linethickness{1mm}\n\\multiput(0,2)(0,4){4}{\\line(1,0){1}}\n\\multiput(2, 2)(0,4){3}{\\line(1,0){1}}\n\\put(2, 15){\\line(1,0){1}}\n\\multiput(4, 2)(0,4){2}{\\line(1,0){1}}\n\\multiput(4, 11)(0, 5){2}{\\line(1, 0){1}}\n\\put(4, 16){\\line(1,0){1}}\n\\multiput(6, 2)(0,5){4}{\\line(1,0){1}}\n\n\\linethickness{.2mm}\n\\put(13, 1.5){\\vdots}\n\\multiput(10,3)(0,1){15}{\\line(1,0){1}}\n\\multiput(12,3)(0,1){15}{\\line(1,0){1}}\n\\multiput(14,3)(0,1){15}{\\line(1,0){1}}\n\\multiput(16,3)(0,1){15}{\\line(1,0){1}}\n\\linethickness{1mm}\n\\multiput(10, 3)(0,5){3}{\\line(1,0){1}}\n\\multiput(12, 4)(0,5){3}{\\line(1,0){1}}\n\\multiput(14, 5)(0,5){2}{\\line(1,0){1}}\n\\put(14, 14){\\line(1,0){1}}\n\\multiput(16,6)(0,4){2}{\\line(1,0){1}}\n\\put(16,14){\\line(1,0){1}}\n\n\\end{picture}\n\n\\end{figure} \\medskip\n\nNext, we want to select rungs through the middle of the tower so as to iterate the staircase pattern all the way up, except that we will skip certain levels (i.e. not select any of their rungs), continuing the staircase pattern where we left off with the following rung. As we want to match stride with the staircase already selected at the top, the total number of levels skipped in the middle section will be constrained to a certain residue class modulo $N$, and as we want the selected rungs to form a portion of an Alpern tower of height $\\{N, N+1\\}$, we cannot skip any two levels with fewer than $N$ levels between them.\n\nSome terminology: an {\\em appearance} of $P_j$ in \\ensuremath{C^{(i)}}\\ is just a level of \\ensuremath{C^{(i)}}\\ that is contained in $P_j$. \nA {\\em selection} of $P_j$ is just a selected rung in a subcolumn of \\ensuremath{C^{(i)}}\\ that is contained in $P_j$. The {\\em net skips} of $P_j$ in the tower over \n\\ensuremath{C^{(i)}}\\ is defined as \\[ S_j(\\ensuremath{C^{(i)}}) = (\\# \\textnormal{ of appearances of $P_j$) } - (\\# \\textnormal{ of selections of $P_j$)}.\\] \nFor example, looking at Figure \\ref{pic:bottom}, one sees that $4$ zeroth level rungs are selected. So if the zeroth level belongs to $P_j$, the zeroth level \ncontribution to $S_j(\\ensuremath{C^{(i)}})$ is $-3$ (one appearance and 4 selections).\n\n \n \n\nLet $\\delta = 2(N-1)(N-2)$ and choose $\\gamma$ with \n\\begin{displaymath}{\\delta\\over m_1}+N > \\gamma \\ge {\\delta\\over m_1} \\quad \\mbox{and} \\quad (t-1)\\delta + \\gamma \\equiv R ~(\\hspace{-2.2ex}\\mod N).\n\\end{displaymath} \nOver \\ensuremath{C^{(i)}}, we skip a quantity of ``middle'' levels belonging to each $P_j$ (for $j \\neq i$) sufficient to ensure that $S_j(\\ensuremath{C^{(i)}})=\\delta$ for $j\\neq i$ and $S_i(\\ensuremath{C^{(i)}})=\\gamma$. (Note that $P_j$ cannot have been skipped more than $\\delta$ times in the outer rungs.) This is not delicate; one can just enact the selection greedily. That is to say, travel up the tower, beginning at level $N^2$, skipping rungs that belong to cells requiring additional skips whenever there's been no too-recent skip. Since each $P_j$ appears at least $M>{3N^3t\\over m_1}$ times, and we need only $\\gamma +(t-1)\\delta\\le {2N^2t\\over m_1}$ net skips, we'll find all the skips we need. \n\n\nWe have not specified the relative masses of the bases of the columns $\\ensuremath{C^{(i)}}$. Set\n\\begin{equation}\n\\label{bi} b_j = \\frac{\\mu(P_j)(\\gamma + (t-1)\\delta) - \\delta}{\\gamma - \\delta} \\; \n\\end{equation}\nand put $\\mu(\\ensuremath{C^{(i)}})= b_i \\mu(C)$, $1\\leq i\\leq t$. Our choice of $\\gamma$ ensures that $b_i\\ge 0$ for each $i$, and one easily checks that $\\sum b_i=1$, so this is coherent.\n\nLet $B_{C}$ be the union of the rungs selected from the columns over $C$ \n(this includes each of the rungs selected from each of the $N$ subcolumns over $\\ensuremath{C^{(i)}}$, $1\\le i\\le t$) \nand put $B=\\bigcup_{C} B_{C}$. (Here $C$ runs over the bases of the columns corresponding to every $\\ensuremath{\\mathbb P} $-name of length $R$ for every $R\\geq tM$.) It\nis clear that $B$ forms the base of an Alpern tower of height $\\{N, N+1\\}$. It remains to show that $B$ is independent of $\\ensuremath{\\mathbb P} $, which we will do by constructing a set $A$, disjoint from $B$, such that both $A$ and $A\\cup B$ can be shown to be independent of $\\ensuremath{\\mathbb P} $. \n \nHere is how $A$ is constructed. Consider again the tower over $\\ensuremath{C^{(i)}}$. This tower had $R$ levels and $RN$ rungs, some of which were selected for the base $B$. We now choose $\\gamma +(t-1)\\delta$ additional rungs for the set $A$. For each $j\\neq i$, $\\delta$ of these rungs should be contained in $P_j$, with the remaining $\\gamma$ contained in $P_i$. (We don't worry about gaps and whatnot; just choose any such collection of rungs disjoint from the family of $B$ selections.) \nDenote the union of the these additional rungs (in all of the columns over $\\ensuremath{C^{(i)}}$, $1\\leq i\\leq t$) by $A_{C}$. Finally, put $A=\\bigcup _{C} A_{C}$. \n\nThat $A\\cup B$ is independent of $\\ensuremath{\\mathbb P} $ is a consequence of the fact that for each $\\ensuremath{C^{(i)}}$, the number appearances of $P_j$ in the column over $\\ensuremath{C^{(i)}}$ is \nprecisely the number of $B$-selections from $P_j$ plus the number of $A$-selections from $P_j$. Accordingly, the relative masses of the cells of $\\ensuremath{\\mathbb P} $ restricted to $A\\cup B$ are equal to the relative frequencies of the appearances of the cells of $\\ensuremath{\\mathbb P} $ in the column over $\\ensuremath{C^{(i)}}$. Therefore, since the proportion of the column that is selected for $A\\cup B$ is independent of $\\ensuremath{C^{(i)}}$ (in fact is always equal to ${1\\over N}$), and since the columns over the various $\\ensuremath{C^{(i)}}$ exhaust $X$, $A\\cup B$ is independent of $\\ensuremath{\\mathbb P} $ (in fact $\\mu\\big(P_j \\cap (A\\cup B)\\big) = {1\\over N} \\mu(P_j)$, $1\\leq j\\leq t$).\n\nThat $A$ is independent of $\\ensuremath{\\mathbb P} $, meanwhile, is a consequence of equation (\\ref{bi}). Fixing $C$ and recalling that $b_i = {\\mu(\\ensuremath{C^{(i)}})\\over \\mu(C)}$, \nthat there were $\\delta$ \n$P_j$-rungs in the column over $\\ensuremath{C^{(i)}}$ selected for $A$, $i\\neq j$, and that there were $\\gamma$ $P_i$-rungs in the column over $\\ensuremath{C^{(i)}}$ selected for $A$, \nthe relative mass of $P_i$ among the $A$-selections in the tower over $C$ is \n\\[ r_i = \\frac{b_i \\gamma + (1-b_i)\\delta}{\\gamma + (t-1)\\delta} .\\]\nBut, solving for $\\mu(P_i)$ in equation (\\ref{bi}), one gets that\n\\[ \\mu(P_i) = \\frac{b_i \\gamma + (1-b_i)\\delta}{\\gamma + (t-1)\\delta} \\]\nas well. So the intersection of $A$ with the column over $C$ is independent of $\\ensuremath{\\mathbb P} $. That this is true for every $C$ gives independence of $A$ from $\\ensuremath{\\mathbb P} $ simpliciter. \\hfill\\ensuremath{_\\blacksquare} \n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\bf Introduction}\n\\bigskip\n\nDenote by $S^{1} = \\mathbb{R}\/\\mathbb{Z}$ the circle and $p :\n\\mathbb{R}\\longrightarrow S^{1}$ the canonical projection. Let $f$\n be an orientation preserving homeomorphism of $S^{1}$. The homeomorphism $f$ admits a lift $\\widehat{f} :\n\\mathbb{R}\\longrightarrow \\mathbb{R}$ that is an increasing\nhomeomorphism of $\\mathbb{R}$ such that $p\\circ\\widehat{f} =\nf\\circ p$. Conversely, the projection of such a homeomorphism of\n$\\mathbb{R}$ is an orientation preserving homeomorphism of\n$S^{1}$. Let $x\\in S^{1}$. We call \\emph{orbit} of $x$ by $f$\nthe subset $O_{f}(x) = \\{f^{n}(x): n\\in\\mathbb{Z} \\}$. Historically, the study of the dynamics of \ncircle homeomorphisms was\ninitiated by Poincar\\'e (\\cite{hP}, 1886), who introduced the\nrotation number of a homeomorphism $f$ of $S^{1}$ as $\\rho (f) =\n\\underset{n\\to +\\infty}{\\lim}\\frac{\\widehat{f}^{n}(\\widehat{x}) -\n\\widehat{x}}{n}~(\\textrm{mod } 1)$, where $\\widehat{x}\\in\n\\mathbb{R}$ such that $p(\\widehat{x})= x$. Poincar\\'e shows that\nthis limit exists and does not depend on neither $x$ nor the lift\n$\\widehat{f}$ of $f$. We say that $f$ is semi-conjugate to the\nrotation $R_{\\rho(f)}$ if there exists an orientation preserving\nsurjective continuous map $h: S^{1}\\longrightarrow S^{1}$ of degree\none such that $h\\circ f = R_{\\rho(f)}\\circ h$.\n\n{\\it Poincar\\'e's theorem.} Let $f$ be an homeomorphism of $S^{1}$\nwith irrational rotation number $\\rho(f)$. Then $f$ is\nsemi-conjugate to the rotation $R_{\\rho(f)}$.\n\nA natural question is whether the semi-conjugation $h$ could be\nimproved to be a conjugation, that is $h$ to be an homeomorphism. In\nthis case, we say that $f$ is topologically conjugate to the\nrotation $R_{\\rho(f)}$. In this direction, Denjoy (\\cite{aD32})\nproves the following:\n\\medskip\n\n{\\it Denjoy's theorem \\cite{aD32}}. Every $C^{2}$-diffeomorphism $f$\nof $S^{1}$ with irrational rotation number $\\rho(f)$ is\ntopologically conjugate to the rotation $R_{\\rho(f)}$.\n\\medskip\n\nDenjoy asked whether or not $C^{2}$-diffeomorphisms $f$ of $S^{1}$\nare ergodic with respect to the Lebesgue measure $m$ ($f$ is said to\nbe ergodic with respect to $m$ if any $f$-invariant measurable set\n$A$ has measure $m(A)$ equal to $0$ or $1$). Simultaneously, Herman\nand Katok gave a positive answer to this question:\n\\medskip\n\n{\\it Herman-Katok's theorem} (\\cite{yKdO89}, \\cite{bHaK95}). Every\n$C^{2}$-diffeomorphism $f$ with irrational rotation number is\nergodic with respect to the Lebesgue measure $m$.\n\\medskip\n\nIt is well known that an homeomorphism $f$ of $S^{1}$ with\nirrational rotation number preserves a unique normalized measure on\n$S^{1}$, denoted by $\\mu_{f}$. If $\\widehat{h}$ is the lift of\n$h$, by taking $\\widehat{h}(0) = 0$, the conjugating\nhomeomorphism $h$ is unique and related to $\\mu_{f}$ by\n$h(x)=p(\\mu_{f}([0,x])) \\in S^{1}$ for $x \\in S^{1}$. Uniqueness of\n$\\mu_{f}$ implies that $\\mu_{f}$ is either singular, or absolutely\ncontinuous with respect to $m$, in this second case, $h$ is an\nabsolutely continuous function. Recall that $\\mu_{f}$ is said to be\n\\textit{singular} with respect to the Lebesgue measure $m$ on\n$S^{1}$ if there exists a measurable subset $E$ of $S^{1}$ such that\n$\\mu_{f}(E) = 1$ and $m(E) = 0$.\n In fact, if $\\mu_{f}$ is absolutely continuous with respect to\n the Lebesgue measure $m$ and $f$ is a $C^{2}$-diffeomorphism,\n $\\mu_{f}$ is necessarily equivalent to $m$ as a consequence of Herman-Katok's theorem\n above (i.e. $m$ is absolutely continuous with respect to $\\mu_{f}$\n and conversely).\n\\medskip\nIn the sequel we deote by $\\mathbb{R}^{\\ast}=\\mathbb{R}\\backslash\n\\{0\\}$ and $\\mathbb{N}^{\\ast}=\\mathbb{N}\\backslash \\{0\\}$.\n\n\\textit{Definition}. A real number $\\alpha\\in ]0, 1[$ is called\n\\textit{Diophantine with exponent $\\delta\\geq 0$} if there is a\nconstant $c(\\alpha)>0$ such that\n$$ \\ (1) \\qquad \\qquad \\arrowvert\\alpha -\\dfrac{p}{q}\\arrowvert\\geq \\dfrac{c(\\alpha)}{q^{2+\\delta}}\\qquad \\textrm{ for any }\\ \\dfrac{p}{q}\\in \\mathbb{Q}.$$\n\\medskip\n\\\n\\\\\nA number that is neither rational nor Diophantine is called a\n\\textit{Liouville number}. \\\n\\\\\n Every real number $\\alpha\\in ]0, 1[$ has a continued fraction expansion\nrepresented by\n$$\\alpha = \\frac{1}{a_{1}+ \\frac{1}{a_{2}+ \\dots}}: =\n[a_{1},a_{2},\\dots,a_{n},\\dots]$$ \\\n\\\\\nwhere $a_{m} \\in \\mathbb{N}^{\\ast}$, $m\\in \\mathbb{N}^{*}$ are\ncalled \\textit{partial quotients} of $\\alpha$. When ($a_{m})_{m\\in\n\\mathbb{N}}$ is bounded, $\\alpha$ is said to be of \\textit{bounded\ntype}. This is equivalent to the fact that $(1)$ holds with $\\delta\n= 0$.\n\\medskip\n\nThe problem of smoothness of the conjugacy $h$ of smooth\ndiffeomorphisms to rotations is now very well understood (see for\ninstance \\cite{mH79}, \\cite{yKdO89}, \\cite{kmKyS89}, \\cite{KT09},\n\\cite{jcY84}).\n\n\n\n\\medskip\n\n\n\nWe refer the reader to the books \\cite{aN10} and \\cite{wDvS91} for a\nthorough account on circle homeomorphisms.\n\\smallskip\n\nThe situation is more complicated for circle homeomorphisms with\nbreak points or shortly, class $P$-homeomorphisms (see the\ndefinition below). This class are known to satisfy the conclusion of\nDenjoy's theorem (Corollary 2.6) (see also \\cite{yKdO89};\n\\cite{mH79}, chapter VI) and, with additional regularity, of\nHerman-Katok's theorem (see \\cite{kaA06}). However,\nKatznelson-Ornstein's theorem \\cite{yKdO89} cannot be extended in\ngeneral to class $P$. The study of the regularity of the invariant\nmeasures of class $P$-homeomorphisms arises then naturally.\n\\bigskip\n\n\\textbf{Class $P$-homeomorphisms} \\\\\nThe following definition is du to M.R. Herman.\n\\begin{defn}(see \\cite{mH79}, p.74) An orientation preserving homeomorphism $f$ of $S^{1}$ is called a \\emph{class $P$}-homeomorphism if it is differentiable\nexcept at finitely many points, the so called \\textit{break points}\nof $f$, at which left and right derivatives (denoted, respectively,\nby $\\textrm{Df}_{-}$ and $\\textrm{Df}_{+}$) exist and such that the\nderivative $\\textrm{Df}: S^{1} \\longrightarrow \\mathbb{R}_{+}^{*}$\nhas the following properties:\n\\begin{itemize}\n\\item [\\textbullet] there exist two constants $01$. Assume that:\n\\begin{itemize}\n \\item [(i)] The rotation numbers $\\rho(f_{i})$ of $f_{i},~i=1,2$\nare irrational of bounded type and coincide\n$\\rho(f_{1})=\\rho(f_{2})=\\rho, ~\\rho \\in \\mathbb{R}\\backslash\n\\mathbb{Q}$,\n \\item [(ii)] $\\sigma_{f_{1}}(a_{1}) \\notin\n\\{\\sigma_{f_{2}}(a_{2}),~\\sigma_{f_{2}}(b_{2})\\}$,\n \\item [(iii)]\n$\\sigma_{f_{1}}(a_{1})~\\sigma_{f_{1}}(b_{1})=\\sigma_{f_{1}}(a_{2})~\\sigma_{f_{1}}(b_{2})$,\n \\item [(iv)] The break points of $f_{i},~i=1,2$ do not lie on the\nsame orbit,\n\\end{itemize}\nThen the map $h$ conjugating $f_{1}$ to $f_{2}$ is singular.\n\\end{cor}\n\\medskip\n\n\\begin{cor}\\label{c:15}\nLet $f$ and $g$ satisfy the assumptions of the main theorem. Then:\n\\begin{itemize}\n \\item [(i)] If $g$ does not have the $(D)-$ property and $f$ has\nthe $(D)-$ property then the map conjugating $f$ to $g$ is\nsingular.\n \\item [(ii)] If $f$ and $g$ have the $(D)-$property and $D^{2}f,~D^{2}g \\in\nL^{p}(S^{1})$ for some $p>1$ then the map conjugating $f$ to $g$ is\nabsolutely continuous.\n \\end{itemize}\n\\end{cor}\n\\medskip\n\nIn particular:\n\\medskip\n\n\\begin{cor} $($Adouani-Marzougui's theorem $($\\cite{am12}, Theorem B$)$$)$\\label{c:16}\nLet $f$ satisfies the assumptions of the main theorem. Then:\n\\begin{itemize}\n \\item [(i)] If $f$ does not have the $(D)$-property then the invariant measure $\\mu_{f}$ is singular with respect\nto the Lebesgue measure $m$.\n \\item [(ii)] If $f$ has the $(D)$-property and $D^{2}f \\in\nL^{p}(S^{1})$ for some $p>1$, then the measure $\\mu_{f}$ is\nequivalent to the Lebesgue measure $m$.\n \\end{itemize}\n\\end{cor}\n\\medskip\n\n\\begin{cor} $($\\cite{am12}, Corollary 1.5$)$ \\label{c:17} Let $f\\in \\mathrm{PL}(S^{1})$\nhave irrational rotation number $\\alpha$ of bounded type. Then the following are equivalent:\n\\begin{itemize}\n \\item [(i)] $f$ has the ($D$)-property\n\n \\item [(ii)] The measure $\\mu_{f}$ is equivalent to the Lebesgue measure $m$.\n\\end{itemize}\n\\end{cor}\n\\medskip\n\n\\textbf{Remark.} The main Theorem and in particular Corollary \\ref{c:17} cannot be extended to rotation number not of bounded type, since very recently,\nTeplinsky \\cite{aT15} constructs an example of a (PL) circle homeomorphism with $4$ non-trivial break points\nlying on different orbits that has invariant measure equivalent to the Lebesgue measure $m$.\nThe rotation number for such example can be chosen either Diophantine or Liouvillean, but not of bounded type.\n\n\\section{\\bf Notations and preliminary results}\n\\bigskip\n\n \\textit{\\bf 2.1. Dynamical partitions}. Let $f$ be a homeomorphism of $S^{1}$ with irrational rotation number $\\alpha = \\rho(f)$.\nWe identify $\\alpha$ to its lift $\\widehat{\\alpha}$ in $]0, 1[$. Let\n$(a_{n})_{n\\in \\mathbb{N}^{\\ast}}$ be the partial quotients of\n$\\alpha$ in the continued fractions expansion. For $n\\in\n\\mathbb{N}^{\\ast}$, the fractions $[a_{1},a_{2},\\dots,a_{n}]$ are\nwritten in the form of irreducible fractions $\\frac{p_{n}}{q_{n}}$.\nThe sequence $\\frac{p_{n}}{q_{n}}$ converges to $\\alpha$ and we say\nthat $\\frac{p_{n}}{q_{n}}$ are \\textit{rational approximations} of\n$\\alpha$. Their denominators $q_{n}$ satisfy the following recursion\nrelation:\n$$q_{n} = a_{n}q_{n-1}+q_{n-2}, ~n\\geq 2, ~q_{0}=1, ~q_{1}=a_{1}.$$\n\nLet $x_{0}\\in S^{1}$ fixed. Denote by:\n\n$$\\Delta_{0}^{(n)}(x_{0}) = \\begin{cases}\n{[x_{0}, f^{q_{n}}(x_{0})]}, &\\text{ if $n$ is even } \\\\\n{[f^{q_{n}}(x_{0}), x_{0}]}, &\\text{ if $n$ is odd }\n\\end{cases}$$\n\\\n\\\\\n$$\\Delta_{i}^{(n)}(x_{0}) : \\ = f^{i}\\left(\\Delta_{0}^{(n)}(x_{0})\\right), \\ i\\in \\mathbb{Z}$$\n\\medskip\n\n\\textit{In all the sequel, we deal with the case $n$ odd} (the case\n$n$ even is obtained by reversing the orientation of $S^{1}$).\n\\smallskip\n\nWe have then:\n\\medskip\n\n\\begin{lem}[See \\cite{yS94}]\\label{l:23} The segments $\\Delta_{i}^{(n-1)}(x_{0}) = f^{i}\\left(\\Delta_{0}^{(n-1)}(x_{0})\\right), \\ 0\\leq\ni0$ such that for any $x_{0}\\in S^{1}$, $n\\geq 1$ and any element\n$\\Delta^{(n)}$ of the dynamical partition $\\xi_{n}(x_{0})$, we have\n\\; $m(\\Delta^{(n)})\\leq C\\lambda^{n}$, where $\\lambda =\n(1+e^{-V})^{-\\frac{1}{2}}< 1$.\n\\end{cor}\n\\medskip\n\nFrom Corollary \\ref{c:24} it follows that every orbit of every $x\\in\nS^{1}$ is dense in $S^{1}$ and this implies the following\ngeneralization of the classical Denjoy theorem:\n\\medskip\n\n\\begin{cor}[Denjoy's theorem: the class P] Let $f\\in \\mathcal{P}(S^{1})$ with irrational rotation number $\\alpha = \\rho(f)$.\nThen $f$ is topologically conjugate to the rotation $R_{\\alpha}$.\n\\end{cor}\n\\medskip\n\nIn the following Lemma we have to compare the lengths of iterates\ndifferent of intervals.\n\\medskip\n\n\\begin{lem}\\label{l:31} Let $f\\in \\mathcal{P}(S^{1})$ with irrational rotation number $\\alpha = \\rho(f)$.\nLet $n\\in \\mathbb{N}^{*}$ and $z_{1}\\in S^{1}$. Set $z_{2} =\nf^{q_{n-1}}(z_{1}), \\ z_{3} = f^{q_{n-1}}(z_{2})$. Then for any\nsegments $K_{1}, \\ K_{2}\\subset [z_{1},z_{3}]$, one has:\\\\\n\n$$e^{-2V}\\dfrac{m(K_{1})}{m(K_{2})}\\leq\n\\dfrac{m(f^{j}(K_{1}))}{m(f^{j}(K_{2}))}\\leq\ne^{2V}\\dfrac{m(K_{1})}{m(K_{2})}$$ for all \\; $j=-q_{n}, \\dots\n,0,\\dots, q_{n}$.\n\\end{lem}\n\\medskip\n\n\\begin{proof} If $j= q_{n}$, Lemma \\ref{l:31} is a consequence of Denjoy's inequality.\nWe suppose that $0\\leq j1$ be a real number and $x_{0}\\in S^{1}$. We say that a triple\n$(z_{1},z_{2},z_{3})$~ of~ $S^{1}$ ($z_{1}\\prec z_{2}\\prec z_{3})$\n satisfies the conditions $(a)$~~\\textrm{and}~~$(b)$ for the point $x_{0}$ and\nthe constant $R$ if:\\\\\n\\medskip\n\n\\begin{itemize}\n \\item [(a)]: $R^{-1}\\leq\\dfrac{m([z_{2},~z_{3}])}{m([z_{1},~z_{2}])}\\leq R$\n\\\n\\\\\n\\item [(b)]: $\\underset{1\\leq i\\leq 3}{\\max}~m([x_{0},z_{i}])\\leq R m([z_{1},z_{2}])$\n\\end{itemize}\n\\end{defn}\n\\medskip\n\nWe call two intervals in $S^{1}$ $R$-\\textit{comparable} if the\nratio of their lenghts is in $[R^{-1}, R]$.\n\\bigskip\n\\medskip\n\n\\textit{\\bf 2.4. Reduction.} In this subsection, we will reduce any\nhomeomorphism $f\\in \\mathcal{P}(S^{1})$ with several break (or\nnon-break)\n points to the one with break points lying on different orbits.\n\\medskip\n\n\\begin{defn} Let $c\\in C(f)$. A \\emph{maximal $f$-connection} of $c$ is a segment\n\\ $[f^{-p}(c),\\dots, f^{q}(c)]:= \\{f^{s}(c): \\ -p \\leq s\\leq q\\}$\nof the orbit $O_{f}(c)$ which contains all the break points of $f$\ncontained on $O_{f}(c)$ and such that $f^{-p}(c)$ (resp.\n$f^{q}(c)$) is the first (resp. last) break point of $f$ on\n$O_{f}(c)$.\n\\end{defn}\n\\medskip\n\\\n\\\\\n We have the following properties:\n\\medskip\n\\\n\\\\\n- Two break points of $f$ are on the same maximal $f$-connection, if\nand only if, they are on the same orbit.\n\n- Two distinct maximal $f$-connections are disjoint.\n\\bigskip\n\nDenote by \\\n\\\\\n\\begin{itemize}\n\n\n \\item [-] $ M_{i}(f) = [c_{i},\\dots, f^{N_{i}}(c_{i})], \\ (N_{i}\\in \\mathbb{N}),$ the maximal $f$-connections of\n$c_{i}\\in C(f)$, ($0\\leq i\\leq p$).\n\n \\item [-] M$(f) = \\coprod _{i=0}^{p}M_{i}(f)$.\n\\end{itemize}\n\\medskip\nSo, we have the decomposition: $C(f) = \\coprod_{i=0}^{p} C_{i}(f)$\nwhere, $C_{i}(f) = C(f)\\cap M_{i}(f), \\ 0\\leq i\\leq p$. We also\nhave $$\\underset{d\\in C_{i}(f)}\\prod \\sigma_{f}(d) = \\underset{d\\in\nM_{i}(f)}\\prod \\sigma_{f}(d).$$\n\\medskip\n\\\n\\\\\n\\begin{prop}[\\cite{am15}, Theorem 2.1]\\label{p:623}\nLet $f\\in \\mathcal{P}(S^{1})$ with irrational rotation number, and\nlet $\\big(k_{0},\\dots,k_{p}\\big)\\in \\mathbb{Z}^{p+1}$. Then there\nexists a piecewise quadratic homeomorphism $K\\in \\mathcal{P}(S^{1})$\nsuch that $F:= K \\circ f \\circ K^{-1}\\in \\mathcal{P}(S^{1})$ with\n$C(F) \\subset \\{K(f^{k_{i}}(c_{i}))= F^{k_{i}}(K(c_{i})); i=0,1,\n\\dots, p\\}$ and such that $\\sigma_{F}(F^{k_{i}}(K(c_{i}))) =\n\\pi_{s,O_{f}(c_{i})}(f),~i=0,1, \\dots, p$.\n\\end{prop}\n\\medskip\n\n\\begin{cor}[\\cite{am15}, Corollary 2.3]\\label{c:21} Let $f\\in \\mathcal{P}(S^{1})$ with irrational rotation number.\nSuppose that $f$ satisfies the (KO) condition. Then, there exists a\npiecewise quadratic homeomorphism $K\\in \\mathcal{P}(S^{1})$ such\nthat: $F = K\\circ f \\circ K^{-1}\\in \\mathcal{P}(S^{1})$ with $C(F)\\subset\\{K(c_{0}),\\dots,K(c_{p})\\}$, where $c_{0},\\dots, c_{p}\\in C(f)$\nare on pairwise distinct orbits.\n\\end{cor}\n\\medskip\n\n\nIn particular, we have:\n\\medskip\n\n\\begin{cor}\\label{c:22} Let $f$, $K$ and $F$ as in Corollary \\ref{c:21}.\n\\begin{itemize}\n\\item [(i)] If $f$ does not satisfy the ($D$)-property, then there exists\n$0\\leq i\\leq p$ such that $K(c_{i})$ is the unique break point of\n$F$ in its orbit.\n \\item [(ii)] If $f$ satisfies the ($D$)-property then $F$ is a $C^{1}$-diffeomorphism with $\\mathrm{DF}$ absolutely continuous on\n$S^{1}$.\n\\end{itemize}\n\\end{cor}\n\\medskip\n\\bigskip\n\n\\begin{prop}\\label{p:23} Let $f, ~g\\in \\mathcal{P}(S^{1})$ and have the same irrational rotation number such that the break points of\n$f$ (resp. $g$) belong to pairwise distinct $f$-orbits (resp.\n$g$-orbits). Let $h$ be the conjugating homeomorphism from $f$ to\n$g$. Then there exist an integer $q\\geq p$,\n$c_{0},c_{1},\\dots,c_{q}; \\ d_{0},d_{1},\\dots,d_{q}$ belong to pairwise distinct orbits $f$-orbits and $g$-orbits respectively\nsuch that $C(f)\\subset \\{c_{0},c_{1},\\dots,c_{q}\\}$,\n$C(g)\\subset \\{d_{0},d_{1},\\dots,d_{q}\\}$, and a homeomorphism $u$\nof $S^{1}$ such that $G: = u \\circ f \\circ u^{-1}\\in\n\\mathcal{P}(S^{1})$ has break points belong to pairwise distinct\n$G$-orbits and satisfies\n\\\\\n\\begin{itemize}\n\n \\item [(i)] $C(G)\\subset \\{u(c_{i}): ~i=0,1, \\dots, q\\}$.\n \\item [(ii)] $\\sigma_{G}(u(c_{i})) = \\sigma_{g}(d_{i})~,~i=0,1, \\dots, q$.\n \\item [(iii)] $\\pi_{s}(G)=\\pi_{s}(g)$.\n \\item [(iii)] $h$ is singular if and only if so is $u$.\n\\end{itemize}\n\\end{prop}\n\\medskip\n\\\n\\\\\n\\textit{From now on we denote by \\ \\ $B:= \\{c_{0},\\dots, c_{q}\\}$} and $c_{0}=c$.\n\\medskip\n\nIn the sequel, we may assume that $f, ~g\\in \\mathcal{P}(S^{1})$ have\nthe same irrational rotation number $\\alpha$ and satisfy the\nfollowing:\n\\medskip\n\n- The points of $B$ belong to pairwise distinct $f-$ orbits.\n\n- The break points of $g$ belong to pairwise distinct $g$-orbits.\n\n- $C(f)\\subset B$.\n\n- The maps $f, ~g$ satisfy the Katznelson-Ornstein (KO) condition.\n\n- The conjugating $h$ is such that $C(g)\\subset h(B)$.\n\\bigskip\n\\medskip\n\n\\section{\\bf Primary Cells}\n\\medskip\n\\medskip\n\nLet $x_{0}\\in S^{1}$ and $f\\in \\mathcal{P}(S^{1})$\nwith irrational rotation number $\\alpha =\\rho(f)$. Let $c\\in B$\nand $n$ an odd integer. By Lemma \\ref{l:23}, either $c\\in\n\\Delta^{(n-1)}_{i_{n}(c)}(x_{0})$ for some $0\\leq i_{n}(c) < q_{n}$\nor $c\\in\\Delta^{(n)}_{i_{n}(c)}(x_{0})$ for some $0\\leq i_{n}(c)<\nq_{n-1}$. Set\n\n\\[\\ y_{2} = f^{-i_{n}(c)}(c), \\ y_{1} =\nf^{-q_{n-1}}(y_{2}), \\ y_{3} = f^{q_{n-1}}(y_{2}).\\]\n\\smallskip\n\nNotice that $y_{1},y_{2}$ and $y_{3}$ are defined with respect to\n$f,x_{0},c$ and the number $i_{n}(c)$ depends on $x_{0}$.\n\\medskip\n\nLet $\\delta > 0$ and $U_{\\delta}(x_{0})$ a\n$\\delta$-neighbourhood of $x_{0}$.\n\\medskip\n\n\\begin{prop}$($cf. \\cite{am12}, Proposition 3.1$)$ \\label{p:41} Under the notations above, there exists $N = N(x_{0}, \\delta)\\in \\mathbb{N}$ such that for all $n\\geq N$, there is\na triple $(y_{1},y_{2},y_{3})=(y_{1}(n),y_{2}(n),y_{3}(n))_{n\\geq\nN}$ with the following properties:\n\\medskip\n\n \\begin{itemize}\n \\item [(c-0)] $\\left(y_{1},y_{2},y_{3}\\right) \\left(\\mathrm{resp}.\n ~ (f^{q_{n}}\\left(y_{1}),f^{q_{n}}(y_{2}),f^{q_{n}}(y_{3}\\right)\\right))_{n\\geq N}\\subset\nU_{\\delta}(x_{0})$\\\\\n\n\\item [(c-1)] $~y_{2} \\in \\Delta^{(n-1)}_{0}(x_{0})~\\mathrm{ or }~y_{2}\\in \\Delta^{(n)}_{0}(x_{0})$\\\\\n\n \\item [(c-2)] $~m\\left(f^{j}([y_{1},y_{3}])\\right)\\leq K\\lambda^{n}$, for every $0\\leq j0,$ set\n\\medskip\n\n\\textbullet ~ $~l_{n}: = m(\\Delta^{(n)}(y_{1}))$\\\\\n\n\\textbullet ~ $V^{(f)}_{n,\\gamma}(a):=[a-\\gamma ~l_{n-1},~a+\\gamma\n~l_{n-1}]$ \\\\\n\n \\textbullet ~ $k_{n}(d)=\\begin{cases}\n\n i_{n}(d), & \\textrm{if}~d \\in \\{c\\} \\cup Q_{2}(\\mathbb{N}_{0}) \\\\\n j_{n}(d), & \\textrm{if}~d \\in Q_{3}(\\mathbb{N}_{0})\n \\end{cases}$\n\\medskip\n\\medskip\n\n\\begin{prop}\\label{p:47} There exists a positive constant $\\gamma_{0}>0$ such that:\n\\medskip\n\n\\begin{enumerate}\n \\item For every $d\\in E(c_{1},c)$ and every $0<\\gamma< \\gamma_{0}$, there exists\n$n_{\\gamma} \\in \\mathbb{N}_{0}\n $ such that for any $n \\in \\mathbb{N}_{0},~n \\geq n_{\\gamma}$ there exists a unique integer $0 \\leq k_{n}(d)0,~\\exists ~ n_{\\gamma} \\in \\mathbb{N}_{0}\n~\\textrm{such that for every }~n \\geq n_{\\gamma},~ n \\in\n\\mathbb{N}_{0}~:~f^{-i_{n}(d)}(d) \\in V^{(f)}_{n,\\gamma}(y_{2}).\n$$\n$\\bullet$ ~ Similarly, the ratios\n$\\dfrac{m([y_{2},~f^{-j_{n}(d)}(d)])}{m([y_{1},~y_{2}])}$ and\n$\\dfrac{m([y_{2},~f^{-j_{n}(d)}(d)])}{m([y_{2},~f^{i_{n}(d)-j_{n}(d)}(y_{2})])}$\nare comparable, which is comparable to the\nratio\n$\\dfrac{m([f^{j_{n}(d)-i_{n}(d)}(y_{2}),~f^{-i_{n}(d)}(d)])}{m([f^{j_{n}(d)-i_{n}(d)}(y_{2}),~y_{2}])}\n=1-t_{n}(d)\n$ (by the Lemma \\ref{l:31}). So, for $d \\in Q_{3}(\\mathbb{N}_{0}),~\\lim_{n \\to +\\infty}\n~t_{n}(d)=1$, which is equivalent to the assertion:\n$$\\forall ~\\gamma >0,~\\exists ~ n_{\\gamma}\\in \\mathbb{N}_{0}\n~\\textrm{such that for every }~n \\geq n_{\\gamma},~ n \\in\n\\mathbb{N}_{0}:~f^{-j_{n}(d)}(d) \\in V^{(f)}_{n,\\gamma}(y_{2}).\n$$\n\nAssertion (2): Let $d \\in Q_{1}(\\mathbb{N}_{0})$. There exists\n$0<\\gamma (d)<1$ such that for every $n \\in \\mathbb{N}_{0},~t_{n}(d)\n\\geq \\gamma (d) ~ \\textrm{and}~1-t_{n}(d) \\geq \\gamma (d) $. Since\n$Q_{1}(\\mathbb{N}_{0})$ is finite there exists $0<\\gamma_{0}<1$ and\nan infinite subset $M_{2}$ of $\\mathbb{N}_{0}$\n($\\mathbb{N}_{0}\\backslash M_{2}$ is finite) such that the points\n$f^{-i_{n}(d)}(d)$ and $f^{-j_{n}(d)}(d)$ are contained in the set\n$S^{1}\\backslash V^{(f)}_{n,\\gamma_{0}}(y_{2})$ for all $d\\in\nQ_{1}(M_{2})$ and $n \\in M_{2}$. By Lemma \\ref{l:25}, $~ d\\notin f^{i}\n([y_{1},~y_{2}])\\cup f^{j} ([y_{2},~y_{3}])$, for every $0\n\\leq i < q_{n}, ~i \\neq i_{n}(d), ~j = \\varphi_{n}(i)$. Hence, for every $0\n\\leq i 0$. Then the following properties hold:\n\n\\begin{itemize}\n \\item [(1)] $(h(y_{1}),h(y_{2}),h(y_{3}))$ is a\nprimary cell associated to $(g,h(x_{0}),h(c),\\mathbb{N}_{0})$.\n\n\\item [(2)] $(h(z_{1}),h(z_{2}),h(z_{3}))$ is a $(\\frac{\\beta}{2},2\\gamma)$-derived cell associated to\n$(g,h(x_{0}),h(c),\\mathbb{N}_{0})$.\n\n\\item [(3)] There is an integer $n_{\\gamma} \\in \\mathbb{N}_{0}$ such\nthat for every $n \\in \\mathbb{N}_{0},~n \\geq n_{\\gamma}$ :\n$$h \\big(V^{(f)}_{n,\\gamma}(y_{2}) \\big) \\subset V^{(g)}_{n,2\\gamma}\\big(h(y_{2})\\big)\n\\subset h \\big(V^{(f)}_{n,4\\gamma}(y_{2}) \\big)$$\n\\end{itemize}\n\\end{prop}\n\\medskip\n\n\\begin{prop}\\label{p:43} Under the notations of Proposition \\ref{p:46}, for every $n\n\\in \\mathbb{N}_{0},~n \\geq n_{\\beta} $ we have:\n\\medskip\n\\begin{itemize}\n\n \\item [(h-0)] $[h(z_{1}),h(z_{3})]\\subset V^{(g)}_{n,2\\gamma}(h(y_{2}))\\subset\n[h(y_{1}),h(y_{3})]$.\n\n \\item [(h-1)] For any $d \\in E(c_{1},c)$, $k_{n}(d)$ is the unique integer in $ [0,~q_{n}[$ such that\n$g^{-k_{n}(d)}(h(d))\\in V^{(g)}_{n,\\frac{1}{2}\\beta}(h(y_{2}))$.\n\n \\item [(h-2)] $~m\\left(g^{j}([h(z_{1}),h(z_{3})])\\right)\\leq K^{\\prime}(\\lambda^{\\prime})^{n}$, for every $0\\leq\nj0$. By Proposition\n\\ref{p:41} (6), there is an integer $n_{\\gamma}^{\\prime} \\in \\mathbb{N}_{0}$\nsuch that for every $n \\in \\mathbb{N}_{0},~n \\geq n_{\\gamma}^{\\prime}$ :\n$f^{-k_{n}(d)}(d) \\in V^{(f)}(y_{2})$. On the other hand, by\nProposition \\ref{p:46}, (3), there is an integer $n_{\\gamma} \\in\n\\mathbb{N}_{0},~n_{\\gamma}\\geq n_{\\gamma}^{\\prime}$ such that for every $n\n\\in \\mathbb{N}_{0},~n \\geq n_{\\gamma}$ : $ h\n\\big(V^{(f)}_{n,\\frac{\\gamma}{2}}(y_{2}) \\big) \\subset\nV^{(g)}_{n,\\gamma}(h(y_{2}))$. As $g^{-k_{n}(d)}(h(d)) =\nh(f^{-k_{n}(d)}(d))$, it follows that $g^{-k_{n}(d)}(h(d))\\in\nV^{(g)}_{n,\\gamma}(h(y_{2}))$ for every $n \\geq n_{\\gamma},~n \\in\n\\mathbb{N}_{0}$. Since $\\gamma>0$ is arbitrary, so, $$ \\lim_{n \\to\n+\\infty} \\dfrac{m\\big([g^{-k_{n}(d)}(h(d)),h(y_{2}) ] \\big)}{m\n\\big([h(y_{1}),h(y_{2}) ] \\big)}=0\n$$\n$($h-6$)$ is a consequence of Assertions ($h-0$) and ($h-1$).\n\\end{proof}\n\\medskip\n\n\\section{\\bf Control of distortions}\n\\medskip\n\n\\begin{prop}[\\cite{A}, Proposition 5.1]\\label{p:2} Let $\\mathbb{N}_{0}$ and $\\gamma_{0}$ are as in Proposition \\ref{p:47}. Let\n$\\beta,~\\gamma \\in ]0,\\gamma_{0}[~(\\beta <\\gamma)$. Assume that the\nconjugation homeomorphism $h$ from $f$ to $g$ admits at $x_{0}$ a\npositive derivative $\\mathrm{Dh}(x_{0}) = \\omega_{0} >0$. Then there\nexists an integer $n_{\\beta} \\in \\mathbb{N}_{0}$ such that: for\nevery $n\\in \\mathbb{N}_{0}, \\ n\\geq n_{\\beta},$ we have\n\n$$\\left|\n\\dfrac{\\mathrm{Dr}_{g^{q_{n}}}\\big(h(z_{1}),h(z_{2}),h(z_{3})\\big)}{\\mathrm{Dr}_{f^{q_{n}}}(z_{1},z_{2},z_{3})}\n- 1\\right|\\leq \\dfrac{2}{1-\\gamma_{0}}\\beta$$\n\\end{prop}\n\\bigskip\n\\\n\\\\\n\nThe next proposition is an another distortion control which is\nopposite to the one of Proposition \\ref{p:2}, this allows us to\nprove that the conjugation from $f$ to $g$ is singular with respect\nto the Lebesgue measure.\n\n\\begin{prop}\n\\label{p:44} Assume that the irrational rotation number\nof $f$ is of bounded type. Let $c, c_{1}$ in $B(f)$, $c\\neq c_{1}$, $\\mathbb{N}_{0}$ and $\\gamma_{0}$ are as in\nProposition \\ref{p:47}. Then\nfor any $\\varepsilon >0$, there exists $0<\\gamma<\\gamma_{0}$ such\nthat for any $\\beta \\in ]0,\\gamma[$ there exists $n_{\\beta} \\in\n\\mathbb{N}_{0}$ such that for all $n \\geq n_{\\beta},~n \\in\n\\mathbb{N}_{0}$ the ($\\beta,\\gamma$) secondary cell\n$(z_{1},z_{2},z_{3})$ associated to\n$(f,x_{0},c,\\mathbb{N}_{0},\\delta)$ satisfies the following\ninequality\n$$ \\left| \\dfrac{\\textrm{Dr}_{g^{q_{n}}}\\left(h(z_{1}),h(z_{2}),h(z_{3})\\right)}{\\textrm{Dr}_{f^{q_{n}}}(z_{1},z_{2},z_{3})}-\n\\Pi(c_{1},c) \\right|\\leq A \\ \\varepsilon, ~~\\textrm{for some\nconstant} \\; A> 0,$$\n\nwhere\n$$\\Pi(c_{1},c)=\n \\underset{d\\in E (c_{1},c)}\\prod \\dfrac{\\sigma_{g}(h(d))}{\\sigma_{f}(d)}.$$\n\\end{prop}\n\\medskip\n\nThe proof of the proposition \\ref{p:44} is an elaboration of the proof of (\\cite{A}, Proposition 5.3) and so we only\ndescribe the changes that are necessary.\n\\medskip\n\n\\begin{lem}\\label{l:49q}\nAssume that the\nrotation number $\\alpha$ of $f$ is of bounded type. Let $c_{1} \\in C(f)\\backslash \\{c\\}$, $u_{0}$, $\\gamma_{c,c_{1}}$ and\n$\\mathbb{N}_{0}$ as in Proposition \\ref{p:47}. Let $h$ be the conjugating from $f$ to $R_{\\alpha}$:\n$h\\circ f= R_{\\alpha}\\circ h$. Assume that Dh$(x_{0})>0$. Then for any $\\varepsilon >0$,\nthere exists $\\eta >0$ such that for any $u\\in ]0,\\eta[$ there exists $n_{u,r} \\in\n\\mathbb{N}_{c,c_{1}}$ such that for any $n \\in \\mathbb{N}_{c,c_{1}},\\ n\\geq\nn_{u,r}$: the $($r,u$)$-derived cell $(z_{1},z_{2},z_{3})$ associated to\n$(f,x_{0},c)$ satisfies\n\\\n\\\\\n$$\\arrowvert\\textrm{Dcr}_{f^{q_{n}}}(z_{1},z_{2},z_{3})-\n\\Pi_{f}(c,c_{1})\\arrowvert\\leq C_{1}\\varepsilon,$$ where $$\\Pi_{f}(c,c_{1})= \\underset{b\\in E (c_{1},c)}\\prod\n\\sigma_{f}(b)$$ and \\; $C_{1}$ is a positive constant.\n\\end{lem}\n\\bigskip\n\n\n\\begin{proof}\nLet $\\varepsilon>0$. Since $\\textrm{Df}$ is absolutely continuous on\nevery interval $[c_{i},c_{i+1}]~(0 \\leq i \\leq p)$, there exists a\nreal $0<\\eta_{0}0)$, the following properties hold:\\\\\n\n$0\\leq D^{\\ast}_{i}(f)-1 \\leq \\dfrac{1}{m_{1}}\n|\\textrm{Df}(b_{i})-\\textrm{Df}(a_{i})|$.\\\\\n\n$\\sum_{i \\in I} |D^{\\ast}_{i}(f)-1|\\leq \\dfrac{2}{m_{1}}\n\\varepsilon$.\\\\\n\n$ |\\log D_{i}(f)|= \\log D^{\\ast}_{i}(f)\\leq D^{\\ast}_{i}(f)-1$\\\\\n\n$\\dfrac{1}{P^{\\ast}} \\leq P \\leq P^{\\ast}$.\\\\\n\nIt follows that\n\\begin{equation}\\label{(4)}\n e^{-C_{0}\\varepsilon} \\leq P \\leq e^{C_{0}\\varepsilon}\n \\, \\, \\ \\textrm{where} \\, \\, C_{0} = \\dfrac{2}{m_{1}}.\n\\end{equation}\n\\medskip\n\n\\textbf{Step 2}. We consider $D_{k_{n}(d)}(f),~d\\in E(c_{1},c)$. We have $d \\in ~ ]f^{k_{n}(d)}(z_{1}),f^{k_{n}(d)}(z_{3})[$.\\\\\n\n\\textbf{Case 2a}. $d\\in ]f^{k_{n}(d)}(z_{1}),f^{k_{n}(d)}(z_{2})[$\nwith $k_{n}(d)=i_{n}(d)$. By Proposition \\ref{p:42} (5),\n$\\textrm{Df}$ is continuous on the intervals\n$[f^{k_{n}(d)}(z_{1}),d],~[d,f^{k_{n}(d)}(z_{2})]$ and\n$[f^{k_{n}(d)}(z_{2}),f^{k_{n}(d)}(z_{3})]$; by the mean value\ntheorem there exist $t_{1}\\in ]f^{k_{n}(d)}(z_{1}),d[,~t_{2} \\in\n]d,f^{k_{n}(d)}(z_{2})[$ and $t_{3} \\in\n]f^{k_{n}(d)}(z_{2}),f^{k_{n}(d)}(z_{3})[$ such that\n$$D_{k_{n}(d)}(f) = (1-r_{n}(d))\\dfrac{\\textrm{Df}(t_{1})}{\\textrm{Df}(t_{3})}+r_{n}(d)\\dfrac{\\textrm{Df}(t_{2})}{\\textrm{Df}(t_{3})}$$\n~~where~~$$r_{n}(d) =\n\\dfrac{m([d,f^{k_{n}(d)}(z_{2})])}{m([f^{k_{n}(d)}(z_{1}),f^{k_{n}(d)}(z_{2})])}.$$\n\nWe have\n\\medskip\n\n $\\begin{array}{ll}\n |D_{k_{n}(d)}(f)-\\sigma_{f}(d)| & \\leq |\\dfrac{\\textrm{Df}(t_{1})}{\\textrm{Df}(t_{3})}-\\dfrac{D_{-}f(d)}{D_{+}f(d)}|+\nr_{n}(d)\\dfrac{|\\textrm{Df}(t_{1})-\\textrm{Df}(t_{2})|}{\\textrm{Df}(t_{3})}\\end{array}\n$\n\\medskip\n\nSince $$\\begin{array}{ll}\n|\\dfrac{\\textrm{Df}(t_{1})}{\\textrm{Df}(t_{3})}-\\dfrac{D_{-}f(d)}{D_{+}f(d)}|\n\\leq \\dfrac{M_{1}}{m_{1}^{2}} \\big(\n|\\textrm{Df}(t_{1})-D_{-}f(d)|+|\\textrm{Df}(t_{3})-D_{+}f(d)|\\big),\n\\end{array}$$\n\\medskip\n\nand by Lemma \\ref{l:31}\n\\medskip\n\n$$\\begin{array}{ll}\n r_{n}(d) & = \\dfrac{m \\big(f^{k_{n}(d)}([f^{-k_{n}(d)}(d),z_{2}]) \\big)}{m \\big(f^{k_{n}(d)}([z_{1},z_{2}]\n\\big)}\\\\\\\\\n & \\leq e^{2V}\\dfrac{m ([f^{-k_{n}(d)}(d),z_{2}]) }{m ([z_{1},z_{2}])} \\\\\\\\\n& \\leq e^{2V}\\dfrac{m ([y_{1},y_{2}]) }{m ([z_{1},z_{2}])}\\times\n\\dfrac{m ([f^{-k_{n}(d)}(d),z_{2}]) }{m ([y_{1},y_{2}])} \\\\\\\\\n& \\leq \\dfrac{e^{2V}}{\\beta} s_{n}(d)\n \\end{array}$$\n\nwith ~~~~\n$$s_{n}(d):= \\dfrac{m ([f^{-k_{n}(d)}(d),z_{2}]) }{l\n([y_{1},y_{2}])}.$$\n\\medskip\n\nIt follows that:\n\n$$|D_{k_{n}(d)}(f)-\\sigma_{f}(d)| \\leq K_{0} \\big(\n\\dfrac{s_{n}(d)}{\\beta}+|\\textrm{Df}(t_{1})-D_{-}f(d)|+|\\textrm{Df}(t_{3})-D_{+}f(d)|\n \\big)$$\nwhere\n\n$K_{0}=\\max(\\dfrac{M_{1}}{m_{1}}e^{2V},\\dfrac{M_{1}}{m_{1}^{2}})$.\n\\medskip\n\nSince $\\underset{n\\to +\\infty}\\lim s_{n}(d)=0$ (Corollary\n\\ref{c:37}), there exists $n_{d,\\beta} \\in \\mathbb{N}_{0}$ such that\nfor any $n \\in \\mathbb{N}_{0},~ n \\geq n_{d,\\beta}$, one has\n$s_{n}(d)< \\beta\\varepsilon $. The intervals\n$(]t_{1},d[;~]d,t_{3}[)_{d\\in E(c_{1},c)}$ are disjoint intervals of\ncontinuity of $\\textrm{Df}$, they satisfy by Lemma \\ref{l:31}:\n\\bigskip\n\n $\n\\begin{array}{ll}\n m([t_{1},d])+m([d,t_{3}]) & \\leq 2 m([f^{k_{n}(d)}(z_{1}),f^{k_{n}(d)}(z_{2})])+m([f^{k_{n}(d)}(z_{2}),f^{k_{n}(d)}(z_{3})])\\\\\\\\\n\n & \\leq 3 e^{2V}\\gamma m([f^{k_{n}(d)}(y_{1}),f^{k_{n}(d)}(y_{2})])\\\\\\\\\n\n & \\leq 3 e^{2V} \\gamma < \\eta_{0}\n\\end{array}\n$\n\n So, $$ |\\textrm{Df}(t_{1})-D_{-}f(d)|+|\\textrm{Df}(t_{3})-D_{+}f(d)|< \\varepsilon $$\n\\medskip\n\nHence, there exists $n_{d,\\beta} \\in \\mathbb{N}_{0}$ such that for\nevery $n \\in \\mathbb{N}_{0}$, $n \\geq n_{d,\\beta}$,\n\n\\begin{equation}\\label{(5)}\n |D_{k_{n}(d)}(f)-\\sigma_{f}(d)|\\leq 2 K_{0}\\varepsilon\n\\end{equation}\n\\smallskip\n\n\\textbf{Case 2b}. $d\\in ]f^{k_{n}(d)}(z_{2}),f^{k_{n}(d)}(z_{3})[$\nwith $k_{n}(d)=j_{n}(d)$. By Proposition \\ref{p:42} (b-5),\n$\\textrm{Df}$ is continuous on the intervals\n$[f^{k_{n}(d)}(z_{1}),f^{k_{n}(d)}(z_{2})],~[f^{k_{n}(d)}(z_{2}),d]$\nand $[d,f^{k_{n}(d)}(z_{3})]$. By the mean value theorem, there\nexist $s_{1} \\in ]d, f^{k_{n}(d)}(z_{3})[,~s_{2} \\in\n]f^{k_{n}(d)}(z_{2}),d[$ and $s_{3} \\in\n]f^{k_{n}(d)}(z_{1}),f^{k_{n}(d)}(z_{2})[$ such that\n$$D_{k_{n}(d)}^{-1}(f)=(1-r^{\\prime}_{n}(d))\\dfrac{\\textrm{Df}(s_{1})}{\\textrm{Df}(s_{3})}+r^{\\prime}_{n}(d)\\dfrac{\\textrm{Df}(s_{2})}{\\textrm{Df}(s_{3})}$$\nwhere\n$$r^{\\prime}_{n}(d) = \\dfrac{m[f^{k_{n}(d)}(z_{2}),d])}{m[f^{k_{n}(d)}(z_{2}),f^{k_{n}(d)}(z_{3})])}.\n$$\n\\medskip\n\nAs in Case 2a, we have\n\\medskip\n\n $\\begin{array}{ll}\n |D_{k_{n}(d)}(f)^{-1}-\\sigma_{f}(d)^{-1}| & \\leq |\\dfrac{\\textrm{Df}(s_{1})}{\\textrm{Df}(s_{3})}-\\dfrac{D_{+}f(d)}{D_{-}f(d)}|+\nr^{\\prime}_{n}(d)\n\\dfrac{|\\textrm{Df}(s_{1})-\\textrm{Df}(s_{2})|}{\\textrm{Df}(s_{3})}\\end{array}\n$\n\\bigskip\n\nSince $\\begin{array}{ll}\n|\\dfrac{\\textrm{Df}(s_{1})}{\\textrm{Df}(s_{3})}-\\dfrac{D_{+}f(d)}{D_{-}f(d)}|\\leq\n\\dfrac{M_{1}}{m_{1}^{2}}\n\\big(|\\textrm{Df}(s_{1})-D_{+}f(d)|+|\\textrm{Df}(s_{3})-D_{-}f(d)|\n\\big),\\end{array} $ \\\n\\\\\n\\medskip\n\nand by Lemma \\ref{l:31}\n\\medskip\n\n$$\\begin{array}{ll}\n r^{\\prime}_{n}(d) & = \\dfrac{m \\big(f^{k_{n}(d)}([z_{2},f^{-k_{n}(d)}(d)]) \\big)}{m \\big(f^{k_{n}(d)}([z_{2},z_{3}]\n\\big)}\\\\\\\\\n & \\leq e^{2V}\\dfrac{m ([z_{2},f^{-k_{n}(d)}(d)]) }{m ([z_{2},z_{3}])} \\\\\\\\\n& \\leq e^{2V}\\dfrac{m ([y_{1},y_{2}]) }{m([z_{2},z_{3}])}\\times\n\\dfrac{m ([z_{2},f^{-k_{n}(d)}(d)]) }{m ([y_{1},y_{2}])} \\\\\\\\\n& \\leq \\dfrac{e^{2V}}{\\beta} s_{n}^{\\prime}(d),\\end{array}\n$$ where $$s_{n}^{\\prime}(d):= \\dfrac{m ([z_{2},f^{-k_{n}(d)}(d)]) }{m\n([y_{1},y_{2}])}.$$ \\\n\\\\\n\nIt follows that $$|D_{k_{n}(d)}(f)^{-1}-\\sigma_{f}(d)^{-1}| \\leq\nK_{0} \\big(\n\\dfrac{s_{n}^{\\prime}(d)}{\\beta}+|\\textrm{Df}(s_{1})-D_{+}f(d)|+|\\textrm{Df}(s_{3})-D_{-}f(d)|\n \\big)$$\nwhere $$K_{0} =\n\\max(\\dfrac{M_{1}}{m_{1}}e^{2V},\\dfrac{M_{1}}{m_{1}^{2}}).$$ By\nCorollary \\ref{c:37} : $\\underset{n\\to +\\infty}\\lim\ns_{n}^{\\prime}(d)=0$. Hence, there exists $n_{d,\\beta} \\in\n\\mathbb{N}_{0}$ such that for every $n\\in \\mathbb{N}_{0}, \\ n \\geq\nn_{d,\\beta}$,~~ $s_{n}^{\\prime}(d)< \\beta\\varepsilon $. The\nintervals $(]d,s_{1}[;~]s_{3},d[)_{d\\in E(c_{1},c)}$ are disjoint\nintervals of\ncontinuity of $\\textrm{Df}$, they satisfy, by Lemma \\ref{l:31}, \\\\\n\\medskip\n\n $\\begin{array}{ll}\n m([d,s_{1}])+m([s_{3},d]) & \\leq m([f^{k_{n}(d)}(z_{1}),f^{k_{n}(d)}(z_{2})])+ 2m([f^{k_{n}(d)}(z_{2}),f^{k_{n}(d)}(z_{3})])\\\\\\\\\n\n & \\leq 3e^{2V}\\gamma m([f^{k_{n}(d)}(y_{1}),f^{k_{n}(d)}(y_{2})])\\\\\\\\\n\n & \\leq 3e^{2V}\\gamma < \\eta_{0}\n\\end{array}\n$\n\n So, $$|\\textrm{Df}(s_{1})-D_{+}f(d)|+|\\textrm{Df}(s_{3})-D_{-}f(d)|< \\varepsilon $$\n\\medskip\n\nHence, there exists $n_{d,\\beta}\\in \\mathbb{N}_{0}$ such that for\nevery $n \\in \\mathbb{N}_{0}, n\\geq n_{d,\\beta}$,\n\n\\begin{equation}\\label{(6)}\n |D_{k_{n}(d)}(f)^{-1}-\\sigma_{f}(d)^{-1}| \\leq 2 K_{0}\\varepsilon\n\\end{equation}\n\\medskip\n\nTherefore from the cases 2a and 2b, we conclude that there exists\n$n_{d,\\beta} \\in \\mathbb{N}_{0}$ such that for every $n \\in\n\\mathbb{N}_{0},~n \\geq n_{d,\\beta}$,\n\\medskip\n\n\\begin{equation}\\label{(7)}\n |D_{k_{n}(d)}(f)-\\sigma_{f}(d)| \\leq K_{0}\\varepsilon\n\\end{equation}\n\\medskip\n\nSince $E(c_{1},c)$ is finite, there exists $n_{\\beta} \\in\n\\mathbb{N}_{0}$ such that for every $n \\in \\mathbb{N}_{0}, ~n \\geq\nn_{\\beta}$,\n\n\\begin{equation}\\label{(8)}\n |\\prod_{d \\in E(c_{1},c)}~D_{k_{n}(d)}(f)-\\prod_{d \\in\nE(c_{1},c)}~\\sigma_{f}(d)| \\leq \\varepsilon\n\\end{equation}\n\\medskip\n\nHence, (\\ref{(4)}) and (\\ref{(8)}) imply that: there exist $n_{\\beta} \\in \\mathbb{N}_{0}$ such that for every $n \\in\n\\mathbb{N}_{0},~n \\geq n_{\\beta}$,\n\n\\begin{equation}\\label{(9)}\n |Dcr_{f^{q_{n}}}(z_{1},z_{2},z_{3})-\\nu(c) | \\leq C_{1}\\varepsilon\n\\end{equation}\n\\medskip\n\nwhere $\\nu(c) = \\prod_{d\\in E(c_{1},c)}~\\sigma_{f}(d)$ and $C_{1}$ is a\npositive constant.\n\\end{proof}\n\\bigskip\n\n\\begin{lem}\n\\label{l:49} Under the hypothesis of Proposition \\ref{p:44}, for any $\\varepsilon >0$, there exists\n$0<\\gamma<\\gamma_{0}$, such that for any $\\beta \\in ]0,\\gamma[$,\nthere exist $n_{\\beta}\\in \\mathbb{M}_{0}$ such that for any $n\\in\n\\mathbb{M}_{0}, ~n\\geq n_{\\beta}$, the ($\\beta,\\gamma$)-secondary\ncell $(z_{1},z_{2},z_{3})$ associated to $(f,x_{0},b,\\mathbb{M}_{0}, \\delta)$ satisfies\n\n$$\\arrowvert\\textrm{Dr}_{g^{q_{n}(b)}}(h(z_{1}),h(z_{2}),h(z_{3}))-\n\\Pi_{g}(h(c), h(c_{1}))\\arrowvert\\leq C_{2}\\varepsilon$$\n\\medskip\n\nwhere \\ $\\Pi_{g}(h(c), h(c_{1})) = \\underset{d\\in E (c_{1},c)}\\prod \\sigma_{g}(h(d))$ and \\; $C_{2}$ is a\npositive constant.\n\\end{lem}\n\\medskip\n\n\\begin{proof} The proof is a consequence of Lemma \\ref{l:49q} and (Proposition \\ref{p:43}, (h-5)) applied to\nthe ($\\frac{\\beta}{2},\\gamma$)-derived cell\n$(h(z_{1}),h(z_{2}),h(z_{3}))$ associated to $(g,h(x_{0}),h(c),\n\\mathbb{N}_{0})$ instead of the ($\\beta,\\gamma$)-derived cell\n$(z_{1},z_{2},z_{3})$ associated to $(f,x_{0},c,\\mathbb{N}_{0})$.\n \\end{proof}\n\\medskip\n\\\n\\\\\n{\\it Proof of Proposition \\ref{p:44}}. The proof results from the\nLemmas \\ref{l:49q} and \\ref{l:49}. \\qed\n\\bigskip\n\n \\section{\\bf Proof of Main Theorem }\n \\medskip\n\nLet $f, ~g\\in \\mathcal{P}(S^{1})$ with the same irrational rotation number $\\alpha$ of bounded type.\nSuppose that $f$ and $g$ satisfy the (KO) condition. By Corollary \\ref{c:21}, there exist two\n piecewise quadratic homeomorphisms $K, L\\in \\mathcal{P}(S^{1})$ such that $F= L \\circ f \\circ L^{-1}$ and\n $G = K \\circ g \\circ K^{-1}$ have the following properties:\n\n \\begin{enumerate}\n \\item $F, G\\in \\mathcal{P}(S^{1})$ and have the same irrational rotation number $\\alpha$,\n\n \\item The break points of $F$ (resp. $G$) are on \\textit{pairwise distinct} $F$-orbits (resp. $G$-orbits),\n\n \\item $F$ and $G$ satisfy the (KO) condition,\n\n \\item Let $h$ the conjugating map between $f$ and $g$ ~ i.e. $h \\circ f = g \\circ h$ and set $v = K \\circ h \\circ L^{-1}$. Then\n\n \\subitem - $v$ is an absolutely continuous (resp. a singular) function if and only if so is $h$.\n\n \\subitem - $v\\circ F= G\\circ v$.\n \\end{enumerate}\n\n Therefore, we may assume that all break points of $f$ (resp. $g$) are on \\textit{pairwise distinct} $f$-orbits\n (resp. $g$-orbits). Now by Proposition \\ref{p:23}, there exists a\n homeomorphism $u$ of $S^{1}$ such that $G= u\\circ f\\circ u^{-1}\\in \\mathcal{P}(S^{1})$ with the following properties:\n\n \\\n \\\\\n \\textbullet ~ $C(f)\\subset B:= \\{c_{i}: ~i=0,1, \\dots, q\\}$, ($q\\geq p$). \\\\\n \\textbullet ~ $C(G)\\subset u(B):=\\{u(c_{i}): ~i=0,1, \\dots, q\\}$. \\\\\n \\textbullet ~ $\\sigma_{G}(u(c_{i})) = \\sigma_{g}(d_{i})~,~i=0,1, \\dots, q$.\\\\\n \\textbullet ~ $\\pi_{s}(G)=\\pi_{s}(g)$.\\\\\n \\textbullet ~ $h$ is singular if and only if so is $u$.\n \\medskip\n\n So we may assume that $u=h$ and $G=g$.\n\\\n\\\\\n To show that the conjugation homeomorphism $h$ from $f$ to $g$ is singular with respect to the Lebesgue measure $m$, it\n suffices to prove that its derivative $\\textrm{Dh}$ is zero on\n a set of Lebesgue total measure. Assume on the contrary that $h$ admits at a point $x_{0}$ a positive derivative $Dh(x_{0})>0$.\n\\bigskip\n\nLet $\\varepsilon >0, ~c \\in B$. For $0<\\beta<\\gamma <\\gamma_{0}$,\nwrite:\n\n$$D_{n}(\\beta, \\gamma):= \\dfrac{\\textrm{Dr}_{g^{q_{n}}}\\big(h(z_{1}),h(z_{2}), h(z_{3}) \\big)}{\\textrm{Dr}_{f^{q_{n}}}\\big(z_{1},z_{2}, z_{3} \\big)} $$\n\\medskip\n\nBy Proposition \\ref{p:2}, there exists $n_{\\beta} \\in\n\\mathbb{N}_{0}$ such that for every $n\\in \\mathbb{N}_{0},~n\\geq\nn_{\\beta}$:\n\n\\begin{equation}\\label{(10)}\n |D_{n}(\\beta, \\gamma)-1| \\leq \\dfrac{2}{1-\\gamma_{0}}\\beta\n\\end{equation}\n\\medskip\n\nBy Proposition \\ref{p:44}, there exists\n$\\gamma_{\\varepsilon}<\\gamma_{0}$ such that for every\n$0<\\beta<\\gamma_{\\varepsilon}$ there exists\n$n_{\\beta}(\\varepsilon)\\in \\mathbb{N}_{0}$ such that for every $n\\in\n\\mathbb{N}_{0},~~n\\geq n_{\\beta}(\\varepsilon)$, the $(\\beta,\\gamma)$\nsecondary cell $(z_{1},z_{2},z_{3})\\ (\\gamma<\\gamma_{\\varepsilon})$-\nassociated to $(f,x_{0},c,\\mathbb{N}_{0},\\delta)$ satisfies:\n\n\\begin{equation}\\label{(11)}\n |D_{n}(\\beta, \\gamma)-\\Pi(c_{1},c)| \\leq A \\varepsilon,\n\\end{equation}\n\\\n\\\\\nwhere $$\\Pi(c_{1},c):= \\underset{d \\in\nE(c_{1},c)}\\prod~\\dfrac{\\sigma_{g}(h(d))}{\\sigma_{f}(d)}$$ and $A$\nis a positive\nconstant.\\\\\n\nLet $\\beta < \\gamma <\\gamma_{\\varepsilon}$ and $n \\in\n\\mathbb{N}_{0}, ~n \\geq n_{\\beta}(\\varepsilon)$. Then (\\ref{(10)})\nand (\\ref{(11)}) imply that\n\\bigskip\n\n$ \\begin{array}{lll}\n |\\Pi(c_{1},c)-1| & \\leq |D_{n}(\\beta, \\gamma)-1|+|D_{n}(\\beta,\n\\gamma)-\\Pi(c_{1},c)| & \\leq \\dfrac{2\\beta}{1-\\gamma_{0}}+A\n\\varepsilon\n\n \\end{array}$\n\\bigskip\n\nSince $\\varepsilon$ and $ \\beta $ are arbitrary, it follows that $\\Pi(c_{1},c)=1$. i.e.\n\\begin{equation}\\label{(12)}\n \\prod_{d\\in E(c_{1},c)} ~\\dfrac{\\sigma_{g}(h(d))}{\\sigma_{f}(d)}=1\n\\end{equation}\nSince $c_{1}, c \\in B$,~~$c_{1}\\neq c~$ are arbitrary, so for every\n$(i,k)\\in \\{0,1, \\dots,~q \\}^{2}~~(i \\neq k),$ one has\n$$\\underset{d\\in E(c_{k}, c_{i})}\\prod~\\dfrac{\\sigma_{g}(h(d))}{\\sigma_{f}(d)}=1$$\n\nAs by Proposition \\ref{p:461}, $c_{k} \\notin E(c_{k},c_{i})$, so we have $$\\underset{0 \\leq j \\leq q\n}\\prod~\\big (\\dfrac{\\sigma_{g}(h(c_{j}))}{\\sigma_{f}(c_{j})}\n\\big)^{e(i,k,j)}=1,$$\n\nwhere $e(i,k,j) \\in \\{0,1\\}$ with\n$$\\varepsilon(i,k,i)=1 \\textrm{and } \\varepsilon(i,k,k)=0.$$\nIt follows that for every $(i,k)\\in \\{0,1, \\dots,~q \\}^{2}, \\ i\\neq\nk$, we have\n\n$$\\underset{0 \\leq j \\leq q}\\sum~ e (i,k,j)~\\log\\left(\\dfrac{\\sigma_{g}(h(c_{j}))}{\\sigma_{f}(c_{j})}\\right)=0.$$\n\\medskip\n \\\n\\\\\n\nFor $(i,j,k) \\in \\{0, \\dots, q\\}^{3}$, set\n$$\\varepsilon_{k+i(q+1),j}\n =\\left\\{\n \\begin{array}{ll}\n e (i,k,j) & \\textrm{if} ~~i \\neq k,~~j \\neq i ~~\\textrm{and}~~j\\neq\nk,\n\\\\\\\\\n 1 & \\textrm{if} ~~i \\neq k ~~\\textrm{and}~~j=\ni,\n\\\\\\\\\n 0 & \\textrm{if} ~~i \\neq k\n~~\\textrm{and}~~j=k,\n\\\\\\\\\n0 & \\textrm{if} ~~i = k\n \\end{array}\n \\right.\n$$\n\nIt follows that for every $(i,k) \\in \\{0, \\dots,\nq\\}^{2}$,~~$$\\sum_{j=0}^{q}\\varepsilon_{k+i(q+1),j}\\log\\left(\\frac{\\sigma_{g}(h(c_{j}))}{\\sigma_{f}(c_{j})}\\right)=0\n$$\n\nSet $A_{q+1}=(a_{l,j})_{0\\leq l<(q+1)^{2},~0 \\leq j1$. By\nKatznelson-Ornstein's theorem, $\\mu_{F}$ is equivalent to the\nLebesgue measure $m$ and so is $\\mu_{f}$. Hence there exists an\nabsolutely continuous map $\\varphi$ such that $\\varphi \\circ f=\nR_{\\rho(f)} \\circ \\varphi$. Similarly, there exists an absolutely\ncontinuous map $\\psi$ such that $\\psi \\circ g = R_{\\rho(f)} \\circ\n\\psi$. In addition, $\\psi^{-1}$ is absolutely continuous. It\nfollows that $(\\psi \\circ h\\circ\\varphi^{-1}) \\circ\nR_{\\rho(f)}=R_{\\rho(f)}\\circ (\\psi\\circ h\\circ \\varphi^{-1})$. As\n$\\rho(f)$ is irational then $\\psi\\circ h\\circ\n\\varphi^{-1}=R_{\\beta}$ a rotation, for some $\\beta \\in S^{1}$.\nTherefore $h= \\psi^{-1}\\circ R_{\\beta}\\circ \\varphi$, which is an\nabsolutely continuous map. This completes the proof. \\qed\n\\\n\\\\\n\n\\textit{Proof of Corollary \\ref{c:16}}. The proof follows easily from\nCorollary \\ref{c:15} by taking $g$ the rotation $R_{\\rho(f)}$. \\qed\n\n\n\n\\bigskip\n\n\\bibliographystyle{amsplain}\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nIn recent years, the atmospheric~\\cite{sk}, solar~\\cite{solar}, \nreactor~\\cite{kamland}, and accelerator~\\cite{k2k, minos_first} experiments \nhave provided convincing evidence of neutrino oscillations and \ntherefore have demonstrated that neutrinos have non--zero masses. This phenomenon \nis the first clear example of new physics beyond the Standard Model which \nassumes neutrinos are massless particles. Three \ngeneration neutrino oscillations are described by six independent parameters: \nthree mixing angles ${\\rm sin^2\\theta}_{12}, \n{\\rm sin^2\\theta}_{23}, {\\rm sin^2\\theta}_{13}$, two mass-squared \ndifferences $\\Delta m^2_{21} = m^2_2 - m^2_1$ and \n$\\Delta m^2_{23} = m^2_3 - m^2_2$, and one complex phase $\\delta$. \nBoth mass differences and two mixing \nangles ($\\theta_{12}$ and $\\theta_{23}$) are measured. The mixing angle \n$\\theta_{13}$ was found to be small and only an upper limit was \nobtained~\\cite{chooz,k2k_theta_13}. Presently nothing is known \nabout the CP violating Dirac phase $\\delta$. The near--future neutrino\noscillation experiments will be focused on the measurements of the unknown \nneutrino parameters: $\\theta_{13}$, mass hierarchy, and \n$\\delta$. Another important goal of these experiments is to \nmeasure the known mixing parameters more precisely.\n\n\\section{Principles of T2K}\n\\label{sec:t2k}\nThe T2K (Tokai--to--Kamioka) experiment~\\cite{t2k} will use a \nhigh intensity off--axis neutrino \nbeam generated by a 50 GeV (initially 30 GeV) proton beam at JPARC \n(Japan Proton Accelerator Research Complex), SuperKamiokande as a far \nneutrino \ndetector, and a set of dedicated neutrino detectors located \nat a distance of 280 m from the pion production target to\nmeasure the properties of the unoscillated neutrino beam. The schematic view \nof the T2K setup is shown in Fig.~\\ref{fig:t2k_setup}.\n\\begin{figure}[h]\n\\centering\\includegraphics[width=14cm,angle=0]{t2k_setup.eps}\n \\caption{General layout of the T2K experiment. The basic elements are \n the neutrino beam line, muon monitor, near neutrino detector at 280 meters \n from the pion production target, and the far neutrino detector \n SuperKamiokande. Possible future 2km near detector is also shown.}\n\\label{fig:t2k_setup}\n\\end{figure}\n The first phase of \n T2K has two main goals: a sensitive \nmeasurement of ${\\rm\\theta}_{13}$ and a more accurate determination of the \nparameters ${\\rm sin^22\\theta}_{23}$ and $\\Delta m^2_{23}$ than any previous\nexperiment. \n\nThe probability of $\\nu_{\\mu}$ transition to $\\nu_e$ can be approximately \ngiven by\n\\begin{equation}\nP(\\nu_{\\mu}\\to \\nu_e) \\approx 4{\\rm cos}^2\\theta_{13}{\\rm sin}^2\\theta_{13}\n{\\rm sin}^2\\theta_{23}{\\rm sin}^2\\Bigl\n(\\frac{1.27\\Delta m^2_{13}({\\rm eV}^2)L({\\rm km})}{E_{\\nu}({\\rm GeV})}\\Bigr),\n\\label{eq:p(numu-nue)}\n\\end{equation}\nwhere $L$ is the $\\nu$ flight distance, and $E_{\\nu}$ is the neutrino energy.\nIt follows from this expression, the maximum sensitivity to the\n$\\nu_{\\mu}\\to \\nu_e$ transition is expected around the oscillation maximum \nfor $\\Delta m_{13} \\simeq \\Delta m_{23} = \\Delta m_{atm} \\simeq 2.5\\times\n10^{-3}$. Based on this value, the neutrino peak energy in T2K should be tuned \nto $\\leq 1$ GeV to\nmaximize the sensitivity for muon neutrino \noscillations for a baseline of 295 km. \n\nT2K will adopt an \noff--axis beam configuration in which neutrino energy is almost independent of \npion energy and quasi-monochromatic neutrino spectrum can be achieved. \nThe neutrino beam is produced \nfrom pion decays in a 94 m decay tunnel filled with 1 atm He gas at an\nangle of \n$2.5^{\\circ}$ with respect to the proton beam axis, providing a narrow neutrino \nspectrum with mean neutrino energies from 0.7 to 0.9 GeV, as shown in \nFig.~\\ref{fig:t2k_nu_beam}. \n\\begin{figure}[h]\n\\centering\\includegraphics[width=10cm,angle=0]{nu_beam.eps}\n\\caption{Neutrino energy spectra at $0^{\\circ}$ and different \noff-axis angles.}\n\\label{fig:t2k_nu_beam}\n\\end{figure}\nThe high energy tail is considerably reduced at $2.5^{\\circ}$ in \ncomparison with\nthe standard on-axis wide-band beam. This minimizes the neutral \ncurrent $\\pi^0$ background in the $\\nu_e$ appearance search.\nMoreover, the intrinsic contamination of $\\nu_e$'s from muon and\nkaon decays is expected to be about 0.4\\% around the peak energy.\n\nTo achieve T2K goals, precise measurements \nof the neutrino flux, spectrum and \ninteraction cross sections are needed. For these purposes, the near detector \ncomplex (ND280)~\\cite{nd280} will \nbe built at a distance of 280 m from the target along the line defined by the \naverage pion decay point and SK (see Fig.~\\ref{fig:t2k_setup}). \nThis complex has two detectors: an on-axis detector (neutrino beam\nmonitor) and an off--axis detector. Physics \nrequirements for ND280 \ncan be briefly summarized as \nfollows: the absolute energy scale of the neutrino spectrum must\nbe calibrated with 2\\% precision, and the neutrino flux monitored \nwith \nbetter than 5\\% accuracy. The momentum resolution of muons from the \ncharged current quasi-elastic interactions~(CCQE) should be less than 10\\%, and the \nthreshold for \ndetection of recoil protons is required to be about 200~MeV\/c. The \n$\\nu_e$ fraction should be measured with an uncertainty of $\\leq 10$\\%.\nA measurement of the neutrino beam direction, with a precision much better \nthan 1 mrad, is required from the on-axis detector. \n\n\\section{Near neutrino detectors}\n\\label{sec:nd280}\n\\subsection{On-axis neutrino monitor}\nThe role of the on-axis neutrino detector~(INGRID) is to monitor the neutrino \nbeam direction and profile on a day to day basis. It consists of $7 + 7$ \nidentical modules, arranged to form a cross-configuration, and 2 diagonal modules, \n as shown in Fig.~\\ref{fig:ingrid}.\n \\begin{figure}[h]\n\\centering\\includegraphics[width=13cm,angle=0]{ingrid_combined.eps} \n\\caption{(a) schematic view of INGRID; (b) segmented iron-scintillator \nsandwich module; (c) a charged current neutrino interaction with muon \ntrack.}\n\\label{fig:ingrid}\n\\end{figure}\nINGRID samples the neutrino beam profile with an area of $9\\times 9$ m$^2$.\nEach iron-scintillator sandwich module covers an area of $125 \\times 125$ cm$^2$ \nand weighs 10 tons. The module consists of ten 6.5 cm-thick \niron layers and 11\nscintillator tracking planes, and is surrounded by four veto counters. Each\ntracking plane has one vertical and one horizontal scintillator layer \ncomposed of $5\\times 1\\times\n121$ cm$^3$ scintillator slabs.\nEach scintillator has a central hole to insert a wavelength shifting~(WLS) \nfiber for light \ncollection and routing to a photosensor. \n The total mass of INGRID is $\\sim 160$ tons.\nA typical event rate detected by the center module every spill is expected \nto be about 0.5 per ton, i.e. the whole INGRID will detect more than $10^5$\nneutrino events\/day. In order to minimize the systematic from the uncertainty \nof the off-axis angle, the neutrino beam direction will be monitored by \nINGRID with a precision of $<< 1$ mrad each day at designed intensity.\n \n\\subsection{Off-axis near detector}\nThe off-axis detector (Fig.~\\ref{fig:nd280})\n\\begin{figure}[h]\n\\centering\\includegraphics[width=10cm,angle=0]{nd2.eps} \n\\caption{The cutaway view of the T2K near detector.}\n\\label{fig:nd280}\n\\end{figure}\n includes the UA1 \nmagnet operating with a magnetic field of 0.2 T, \na Pi-Zero detector (POD), a tracking detector which includes time projection \nchambers (TPC's) and fine grained scintillator detectors (FGD's), an \nelectromagnetic calorimeter\n(Ecal), and a side muon range detector~(SMRD). \n\n \\subsubsection{Photosensors}\n Wavelength \nshifting fibers will be widely used for readout of all \nscintillator detectors which are the main active element of the ND280 \ndetector. \n A magnetic field environment and limited space inside the \nUA1 magnet put serious constraints for the usage of standard photodetectors \nsuch as traditional multi-anode photomultipliers. Since the ND280 has about \n60k individual readout channels, the cost of \nphotosensors is also very important. \n After studying several candidates, \na multi-pixel avalanche photo-diode operating in the \nlimited Geiger multiplication mode was selected as the baseline detector. These \n novel devices \nare compact, well matched to spectral emission of WLS fibers, and insensitive \nto magnetic fields~\\cite{gm1,gm2,andreev}. The required parameters for these\nphotosensors from all ND280 subdetectors can be summarized as follows: \nan active area diameter of $\\sim 1$ mm, photon detection efficiency for green \nlight\n$\\geq 20$\\% , pixels number $> 400$, and a dark rate at operating \nconditions $\\leq 1$ \nMHz. The gain should be $(0.5-1.0)\\times 10^6$, \nthe cross$-$talk $\\sim10$\\%, and \npulse width should be less than 100 ns to meet the spill structure of the \nJPARC proton beam. For \ncalibration and control purposes, it is very desirable to obtain well separated \nsingle photoelectron peaks in amplitude spectra at operating temperature.\n\nAfter a R\\&D study of 3 years, a Hamamatsu MPPC was chosen as the \nphotosensor for ND280. The description of this device\nand its parameters can be found in Ref.~\\cite{hamamatsu}. \nThe final T2K version is a 667 pixel MPPC with a sensitive area \nof $1.3\\times 1.3$ mm$^2$ (Fig.~\\ref{fig:mppc}). \n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{mppc_new_a.eps}\n\\includegraphics[width=6cm,angle=0]{mppc_new_b.eps}\n\\end{center}\n\\caption{(a) The photograph of a 667 pixel MPPC: a magnified face view of \nan MPPC with a sensitive area of $1.3\\times 1.3$ mm$^2$ (left),\nthe package of this MPPC (right); (b) ADC spectrum from an LED \nsignal. Clearly separated peaks at equal intervals correspond to 0, 1, 2, \n3... fired pixels.}\n\\label{fig:mppc}\n\\end{figure}\n These devices\ndemonstrated good performance at room temperature: a low cross-talk value of about 10\\%, \na photon detection \nefficiency for green light of $\\geq 25$\\%, a low dark rate of $\\sim 0.3$ \nMHz at the operating voltage, a high gain of about \n $0.7\\times 10^6$, and a pulse width of less than 50 ns. \n \n\\subsubsection{POD}\nThe POD is optimized for measurement of the inclusive $\\pi^0$ production by \n$\\nu_{\\mu}$ interactions on oxygen and will be installed in the upstream end of the \nmagnet. The cross section measurements on an oxygen target will be achieved by \nusing the following POD geometry: the upstream and downstream regions are\nconfigured as electromagnetic calorimeters providing energy containment and\nactive veto, and the central region of the POD provides the fiducial mass for \nthe $\\pi^0$ measurements (Fig.~\\ref{fig:pod}(a)).\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=6.2cm,angle=0]{pod_a.eps}\n\\includegraphics[width=7.5cm,angle=0]{pod_b_c.eps}\n\\end{center}\n\\caption{(a) POD schematic view: the central region is constructed of\nalternating water target and scintillator tracking layers; (b) one layer of \nthe target region; (c) a neutral current\n$\\pi^0$ event in POD. The horizontal and vertical axes are in centimeters. }\n\\label{fig:pod}\n\\end{figure}\n The schematic view of a plane of the POD target is shown in \n Fig.~\\ref{fig:pod}(b).\nIt consists of alternating water target layers of about 3 cm-thick and \ntracking layers composed of X-Y extruded triangular shaped scintillator bars of \n17 mm in height and 32.5 mm at the base and a central hole for a WLS fiber. \n A thin sheet of brass\n($\\sim 1.6$ mm-thick) is sandwiched in the 26 X-Y tracking layers of the target\nregion. The upstream and downstream regions have 7 X-Y scintillator layers \nwith 4 mm-thick lead radiators between them.\n The POD has a total mass \nof approximately 17 tons with a fiducial mass of about 3 tons of water and \n8 tons of other materials. \n The tests of light yield of the POD\nscintillators with MPPC's connected to one end of a Y11 WLS fiber \n showed good results. The light yields for a minimum \nionizing particles~(MIP) are 19.8 p.e.\/MeV and 8.7 p.e.\/MeV at 25 and 205 cm \nfrom the MPPC, respectively. With the mirrored far end of the WLS fiber, \nthe light yield \nof 15.7 p.e.\/MeV is much greater than the 5 p.e.\/MeV required for \nefficient reconstruction of electromagnetic showers.\n\nOxygen cross\nsection measurements will be made by comparing the interaction rate of events \ncollected with water in the target region versus similar running \nperiods with water removed from the target region. A typical neutral current \nevent with a $\\pi^0$ is shown in Fig.~\\ref{fig:pod}(c) in which a neutral pion is\naccompanied by a neutron.\nThe energy resolution for events fully contained in the active\ntarget is expected to be about 10\\% + 3.5\\%\/$\\sqrt{{\\rm GeV}}$ and \nthe reconstruction efficiency of a $\\pi^0$ with a momentum $\\geq 200$ MeV\/c\nis expected to be approximately 33\\%.\n\n\\subsubsection{ND280 tracker}\nThe ND280 tracker consists of three TPC's and two FGD's, as shown in \nFig.~~\\ref{fig:nd280}. Its main function is \nto measure the muon and electron neutrino beam fluxes and energy spectra, \nplus\nvarious \ncharged current cross sections. The tracker is designed to accurately \nmeasure CCQE events, the main \nprocess at the T2K peak neutrino energy, \n\\begin{equation}\n\\nu_{\\mu} + n \\to \\mu^- + p.\n\\label{eq:ccqe}\n\\end{equation}\n In order to measure this process the reconstruction of both\n proton and muon is useful. The proton will be primarily identified and measured \n by the FGD while the muon will be measured by the TPC. The initial neutrino energy \n will be reconstructed from the muon momentum. The measurements of CCQE events \n will be used for flux normalization in the oscillation analysis. \\\\\n \n{\\it FGD.} The ND280 will \ncontain two FGD's, each \nwith dimensions $1.84\\times 1.84\\times 0.3$ m$^3$ resulting in a total target \nmass of about 2.0 tons. \nThe first FGD will be an active scintillator detector, similar to the \nSciBar detector~\\cite{scibar} of the K2K experiment. \nIt consists of thirty scintillator layers of 192 \n$0.96\\times 0.96 \\times 184$ cm$^3$ extruded \nscintillator bars which are arranged \nin alternating vertical and horizontal layers perpendicular to the beam\ndirection. The second FGD is composed of seven X-Y \nsandwiches of scintillator \nlayers alternating with six 3-cm thick layers of water. The weight of the \nscintillator is 0.56 ton and of water, 0.44 ton. The readout of each\nscintillator bar is provided by an MPPC connected to one end of a WLS fiber \ninserted in \na central hole. \nA beam test of scintillator bars performed at TRIUMF showed that the light yield\nfor 120 MeV\/c pions, muons, electrons will be more than 10 p.e. at the far end \nfrom a photosensor without mirroring the far end of the fiber. The \nmirroring increases the light yield by $\\geq 80$\\%,\nguaranteeing a detection efficiency of more \nthan 99\\% for minimum ionizing particles. The FGD allows a clear\nseparation between protons and pions using dE\/dx information and tagging \nMichel electrons from the decay of short-ranged pions.\nComparing the interaction rates in both FGD's permits separate measurement \nof neutrino cross sections on carbon and on water.\nAbout $4\\times 10^5$ \nneutrino interactions are expected in both FGD modules for a one year exposure \nwith $10^{21}$ protons on target.\n\n{\\it TPC.} The primary purpose of the TPC is \nto measure the 3-momenta of muons produced in CCQE interactions in the FGD. \n The TPC will use a \nlow diffusion gas to obtain the \nmomentum resolution of $\\leq 10$\\% for particles below 1 GeV\/c. A 700 $\\mu$m \nspace point resolution per ``row'' of pads is required to achieve this momentum resolution. \nThe absolute energy scale will be checked at the 2\\% level using the invariant \nmass of neutral kaons produced in DIS neutrino interactions and decaying in the\nTPC volume. A good dE\/dx resolution of $< 10$\\% \nis expected for 72 cm long tracks which will provide better than 5$\\sigma$ \nseparation between muon and electron tracks \nin the momenta range 0.3-1.0 GeV\/c. \n\nThe three TPC modules are \nrectangular boxes with outer \ndimensions of approximately $2.5\\times 2.5$ m$^2$ in the plane perpendicular \nto the neutrino beam direction, and 0.9 m along the beam direction. A simplified\ndrawing of the TPC is shown in Fig.~\\ref{fig:tpc}.\n\\begin{figure}[h]\n\\centering\\includegraphics[width=8cm,angle=0]{tpc3d.eps} \n\\caption{The layout of TPC showing the inner and outer boxes and the \ncentral cathode.}\n\\label{fig:tpc}\n\\end{figure}\nThe TPC modules are operated at an electric field of 200 V\/cm. \nThe central cathode, which divides the drift space into two halves to limit the \nmaximum drift distance to $\\sim 1$ m, will be at a potential of -25 kV. The\nbaseline gas choice is Ar(95\\%)--CF$_4$(3\\%)--iC$_4$H$_{10}$(2\\%).\n\n \nThe `bulk' Micromegas detectors will be used to instrument the TPC readout\nplane. The active surface area of the \nMicromegas \nmodule is $359\\times 342$ mm$^2$ with 1726 active pads of $9.7\\times 6.9$ \nmm$^2$. 12 Micromegas modules will be used for each\nreadout plane of the TPC.\n In total, the 3 TPC's will consist of 72 modules with $\\sim 124000$ readout \nchannels. \n\nThe first prototypes of Micromegas detectors have been tested with cosmic \nmuons in the former HARP \nfield cage setup with a magnetic field~\\cite{micromegas} and demonstrated good\nmomentum resolution of 8.3\\% at 1 GeV\/c. A dE\/dx resolution of \nabout 12\\% for track lengthes of about 40 cm and a good \nuniformity of $\\sigma = 3.4$\\% for the gain $\\sim 1000$ have been obtained. \n\nA typical CCQE event for neutrino interaction in FGD1 is shown in \nFig.~\\ref{fig:ccqe_track}.\n\\begin{figure}[h]\n\\centering\\includegraphics[width=9cm,angle=0]{ccqe_track_new.eps} \n\\caption{Typical CCQE event in the tracker.}\n\\label{fig:ccqe_track}\n\\end{figure}\nThe reconstruction efficiency of CCQE \nevents produced in the FGD with a track in the TPC is estimated to be about \n50\\% at a $E_{\\nu} \\sim 0.7$ GeV.\n\n\\subsubsection{Electromagnetic calorimeter}\nThe Electromagnetic calorimeter (Ecal) shown in Fig.~\\ref{fig:ecal} \n\\begin{figure}[h]\n\\centering\\includegraphics[width=7cm,angle=0]{ecal.eps} \n\\caption{Basic structure of the electromagnetic calorimeter.}\n\\label{fig:ecal}\n\\end{figure}\nconsists of two sections. \nOne surrounds the POD (POD Ecal) for detectioning photons and muons escaping the \nPOD, and the second section, surrounding the FGD's and TPC's (TEcal), detects \nparticles leaving the tracking volume. \n\nTEcal modules are made of 4 cm-wide, 1 cm-thick plastic\nscintillator bars arranged in 32 layers and separated by 31 layers of 1.75\nmm-thick lead sheets. The orientation of the bars alternates between layers \nso that the bars in any layer are perpendicular to the bars in the two adjacent \nones. This bar width allows good $\\pi^0$ reconstruction efficiency and provides \nthe spatial resolution required for reconstruction of the direction of detected\nphotons. The active length of the TEcal along the neutrino beam is 384 cm and \nthe total depth is 50 cm corresponding to 10.5$X_0$. TEcal has two side\nmodules, one on each side of the UA1 iron yokes, one top and one \nbottom modules, each is split into two (left and right) so that they can move \nwith the magnet yoke when the magnet opens.\nAll scintillator \nbars have a hole in the center with a 1 mm WLS fiber inserted in it. All \nlong bars running along the neutrino beam are readout by an MPPC at each end\n(double-end readout), while all shorter bars (perpendicular to the neutrino\nbeam) are mirrored at one end and readout by an MPPC at the other (single-end\nreadout). The downstream Ecal is a single module with the same granularity \nas TEcal modules with an effective depth of 11$X_0$. It is located at the \ndownstream end of the magnet and covers \nan active surface area of $2\\times 2$ m$^2$. All bars have double-ended readout. \nThe total weight of the TEcal and downstream Ecal is 28.3 tons. \n\n The POD Ecal has modules with coarser segmentation and less total $X_0$ and \n does not provide good energy and spatial resolution required for $\\pi^0$ \n reconstruction. These modules consist of 6 scintillator layers separated by \n 5 layers of 5 mm-thick lead converters resulting in an effective depth of \n $4.5X_0$. \n \n The energy resolution of TEcal, dominated by sampling fluctuations, is\n estimated to be about 7.5\\%\/$\\sqrt{E(\\rm GeV)}$ for energies up to \n 5 GeV. TEcal is expected to provide good electron\/pion separation. An\n efficiency of 90\\% for electrons is expected with 95\\% pion rejection. \n \n\\subsubsection{Side muon range detector}\nMuons which escape at large angles with respect to the neutrino \nbeam can not be measured by the TPC's. However, they will intersect in the iron yoke \nand therefore a muon's momentum can be obtained from its range by \ninstrumenting the iron at various depths. About 40\\% of muons from \nCCQE reactions and about 15\\% of muons from \ncharge current \nnon-quasi-elastic reactions are expected to enter the SMRD. In addition, \nthe SMRD will be used to veto events from \nneutrino interactions in the magnet and in walls of the ND280 pit and will \nprovide a cosmic trigger \nfor calibration of inner detectors. \n\nThe UA1 iron yoke \nconsists of 16 C-shaped elements made of sixteen 5 cm thick iron plates, with \n1.7 cm air gap between the plates and is segmented in 12 azimuthal sections. \nThe active component of the SMRD will consist of 0.7 cm thick scintillator slabs\nsandwiched between the iron plates of the magnet yokes. \nDetails of the extrusion of the scintillator slabs and the method of \netching the plastic surface by a chemical agent can be found \nin Ref.~\\cite{extrusion}. For the readout, we employ a \nsingle WLS fiber embedded in a serpentine shaped (S--shape)\ngroove, as shown in Fig.~\\ref{fig:smrd_counter}.\n\\begin{figure}[h]\n\\centering\\includegraphics[width=12cm,angle=0]{smrd_counter_1.eps} \n\\caption{The SMRD counter with embedded Kuraray Y11 WLS fiber.}\n\\label{fig:smrd_counter}\n\\end{figure}\n Such a \nshape allows the fiber to collect the scintillation light over the whole \nsurface of a scintillator slab~\\cite{smrd_nim}. Two MPPC's are coupled to \nboth ends of a \nWLS fiber glued into the S--shape \ngroove. The detector performance has been tested using cosmic muons. Typical \nADC spectra for \nMIP's obtained with $1.0\\times 1.0$ mm$^2$ MPPC's are \nshown in Fig.~\\ref{fig:adc_smrd_spectra}. \n\\begin{figure}[h]\n\\centering\\includegraphics[width=11cm,angle=0]{good_spectra.ps} \n\\caption{The ADC spectra of the SMRD counter for minimum ionizing \nparticles measured at 22$^{\\circ}$C. The light yield (sum of both ends) is \nequal to 58 p.e.}\n\\label{fig:adc_smrd_spectra}\n\\end{figure}\nThe SMRD counters tests resulted in a high detection efficiency measurement \nof \ngreater than 99\\%, a time resolution of about 1 ns, and a spatial resolution \nalong the slab of about 8 cm for minimum ionizing particles.\n\\section{Conclusion}\nThe T2K experiment has a rich physics potential and provides an excellent \nopportunity to greatly extend our understanding of neutrino properties.\nTo achieve the physics goals of T2K, the complex of near neutrino detectors \nneeded for measurement of the unoscillated neutrino beam properties \nis under construction. The on-axis detector will be ready to accept the first \nneutrino beam in April 2009, the installation and commissioning of the whole \noff-axis detector will be finished during 2009. The T2K experiment is \nexpected to start data taking in 2009. \n \n{\\bf Acknowledgments.} I thank M.~Gonin, A.~Grant, D.~Karlen, T.~Nakaya, \nV.~Paolone and M.~Yokoyama for providing material for the talk and useful \ncomments on the manuscript. This work was supported in part by \nthe ``Neutrino Physics'' Program \n of the Russian Academy of Sciences and by the RFBR (Russia)\/JSPS (Japan) \n grant \\#08-02-91206.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}