diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzziirf" "b/data_all_eng_slimpj/shuffled/split2/finalzziirf" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzziirf" @@ -0,0 +1,5 @@ +{"text":"\\section*{Abstract}\nThis article studies the N-vortex problem in the plane with positive vorticities. After an investigation of some properties for normalised relative equilibria of the system, we use symplectic capacity theory to show that, there exist infinitely many normalised relative periodic orbits on a dense subset of all energy levels, which are neither fixed points nor relative equilibria.\n\n\\section{Introduction}\n\\label{sec:1}\n\\subsection{The N-Vortex Problem in the Plane}\n\\label{subsec:1.1}\nThe study of vortex dynamics dates back to Helmholtz's work on\nhydrodynamics in 1858 \\cite{helmholtz}. It has been linked to\nsuperfluids, superconductivity, and stellar\nsystem\\cite{lugt1983vortex}. Known as the Kirchhoff Problem, its\nHamiltonian structure is first explicitly found\nby Kirchhoff \\cite{kirchhoff1876vorlesungen} for $\\mathbb{R}^2$, and later on\ngeneralized by Routh \\cite{routh1880some} and then Lim\n\\cite{lim1943motion} to general domains in the plane. Here\nwe consider the problem in the plane,\n\\begin{equation}\n\\mathbf{\\Gamma} \\dot{\\vz} (t) = \\textit{X}_H(\\vz(t)) = \\mathcal{J}_N\\nabla\n H(\\vz(t)), \\quad \\dot{\\vz}=(z_1,z_2,...,z_N),\\quad z_i=(x_i,y_i)\\in\n \\mathbb{R}^2 \\label{sys:H1} \\tag{H1}\n\\end{equation}\nwhere the Hamiltonian is \n\\begin{equation}\n \\label{eq:Ham}%\n H(z)= -\\frac{1}{4\\pi}\\sum_{1 \\leq i< j\\leq\n N}\\Gamma_{i}\\Gamma_{j}\\log{|z_i -z_j|^2}\n\\end{equation}\nwhile the Poisson matrix $\\mathcal{J}_N$ and the vorticity matrix $\\Gamma$ are\n\\begin{align}\n\\mathcal{J}_{N} = \n\\begin{bmatrix}\n\\mathbb{J} & & \\\\ \n & \\ddots & \\\\\n & & \\mathbb{J}\n\\end{bmatrix}, \\quad \n\\mathbb{J} =\n\\begin{bmatrix}\n0 & 1 \\\\\n-1 & 0\n\\end{bmatrix}\\\\\n\\mathbf{\\Gamma} = \n \\begin{bmatrix}\n \\Gamma_1 &&&&&\\\\\n & \\Gamma_1 &&&&\\\\\n &&\\ddots &&& \\\\\n &&&\\ddots && \\\\ \n &&&&\\Gamma_N & \\\\\n &&&&& \\Gamma_N \n \\end{bmatrix}.\n\\end{align}\nIt is understood that \n\\begin{itemize}\n\\item $N$ is the number of vortices;\n\\item $z_i = (x_i, y_i)$ is the position of the i-th vortex in the plane;\n\\item $\\Gamma_i \\in \\mathbb{R}\\setminus\\{0\\}$ is the vorticity of the i-th vortex;\n\\item $\\textbf{z}= (z_1,z_2,...,z_N)$ is the vortices configuration;\n\\item $\\textit{X}_H$ is the Hamiltonian vector field of $H$;\n\\item $\\mathcal{J}_N$ is a $2N\\times 2N$ block-diagonal matrix;\n obtained by putting $N$ copies of $2\\times 2$ matrice $\\mathbb{J}$\n on the diagonal.\n\\end{itemize}\nSometimes we will need to consider a sequence of vortices\nconfigurations. In that case we will denote this sequence\n$\\{\\textbf{z}^k\\}_{k\\in \\mathbb{N}}$, with upper indices as opposed to\nlower indices refering to particular vorticies. The quantity\n\\begin{align}\n L = \\sum_{1\\leq i< j\\leq N} \\Gamma_i\\Gamma_j\n\\end{align}\nis called the \\textbf{total angular momentum} and will be used\nfrequently; if $V$ is a subset of \\{1,2,...,N\\}, we also define\n$\\displaystyle L_V = \\sum_{i,j \\in V, i< j}\\Gamma_i\\Gamma_j$.\nFinally, throughout this article, if not explicitly emphasized, \\textbf{we\nalways suppose that all vortices are of positive vorticity} :\n\\begin{equation}\n \\label{Hyp:positive}\n \\Gamma_i > 0 \\quad (1\\leq i \\leq N).\n \\tag{Hypo}\n\\end{equation}\nHence $L>0$, and\n$L_V>0$ for all $V$'s.\n\\subsection{Symmetries, First Integrals, and Integrability}\n\\label{subsec:1.2}\nLet $\\mathbb{E}(2)= \\mathbb{T}(2)\\rtimes \\mathbb{O}(2)$ be the Euclidean group, where $\\mathbb{O}(2)$ is the orthogonal group and $\\mathbb{T}(2)$ is the translation group. Consider the action \n\\begin{align*}\n&\\mathbb{E}(2) \\times (\\mathbb{R}^{2})^N \\rightarrow (\\mathbb{R}^{2})^N\\\\\n&(g,\\textbf{z}) \\rightarrow g\\textbf{z} \n\\end{align*}\nwhere\n\\begin{align*}\ng\\textbf{z} = (gz_1,gz_2,...,g z_N), \\forall g\\in \\mathbb{E}(2), z= (z_1,...,z_N) \\in \\mathbb{R}^{2N}\n\\end{align*}\nNote that system \\eqref{sys:H1} is invariant under both translation and\nrotation, thus the corresponding quantities\n\\begin{align}\nP(\\textbf{z}) = \\sum_{1\\leq i\\leq N}\\Gamma_i x_i,\\quad Q(\\textbf{z})= \\sum_{1\\leq i \\leq N} \\Gamma_i y_i,\\quad I(\\textbf{z}) = \\sum_{1\\leq i\\leq N} \\Gamma_i|z_i|^2\n\\end{align}\nare first integrals. These first integrals are \\textbf{not} in\ninvolution in general with respect to the Poisson bracket\n\\begin{align*}\n\\{f,g\\} = \\sum_{1\\leq i\\leq N} \\frac{1}{\\Gamma_i} (\\frac{\\partial f}{\\partial y_i}\\frac{\\partial g}{\\partial x_i}-\\frac{\\partial f}{\\partial x_i}\\frac{\\partial g}{\\partial y_i}).\n\\end{align*}\nActually, \n\\begin{align}\n\\{P,I\\} = -2Q,\\quad \\{Q,I\\} = 2P.\n\\end{align}\nOn the other hand, note that $H, I, P^2+ Q^2$ are independent first\nintegrals in involution. Hence the 3-vortex problem is integrable. For\n$N\\geq 4$, the $N$-vortex problem is in general not integrable\n\\cite{ziglin1980nonintegrability,koiller1989non}. This allows us to\ndraw a parallel between the $N$-vortex problem with $N \\geq 4$ and the\n$N$-body problem with $N \\geq 3$, and thus side with Poincar{\\'e} when he\nfamously described the study of periodic orbits as \\enquote{the only\n opening through which we can try to penetrate in a place which, up\n to now, was supposed to be\n inaccessible}\\cite[section~36]{poincare1892methodes}. In this article, we\nstudy the existence of relative periodic orbits in the $N$-vortex\nsystem, i.e., orbits which are periodic up to rotations. Note that from\nexperimental point of view, it is relative equilibria that have been\nfirst realized for vortices of superfluid $\\ch{^4He}$\n\\cite{yarmchuk1979observation}. Hence from either theoretical or\npractical consideration, relative periodic orbit will be an ideal\ncandidate for our analysis.\n\n\\subsection{Normalised Orbits}\n\\label{subsec:1.3}\nThe closed orbits of N-vortex problem \\eqref{sys:H1} are not\nisolated. Indeed, if $\\textbf{z}(t)$ is an orbit, then so are\n\\begin{itemize}\n\\item $(z_1(t)+c, \\cdots, z_N(t)+c)$, $c\\in \\mathbb{R}^2$\n\\item $\\lambda^{\\frac{1}{2}}\\textbf{z}(\\frac{t}{\\lambda})$, $\n \\lambda \\in \\mathbb{R}^{+}$. \n\\end{itemize}\nWe wish not to distinguish such orbits. To this end, we give the following definition:\n\n\\begin{Def}\n\\label{Def:orbit}\nwe will call an orbit $\\textbf{z}(t)$ of the system \\eqref{sys:H1}\n\\begin{enumerate}\n\\item \\textbf{centred} if it satisfies $P(\\textbf{z}(t)) = Q(\\textbf{z}(t)) = 0$\n\\item \\textbf{normalised} if it is centred and satisfies $I(\\textbf{z}(t)) = 1$\n\\item \\textbf{periodic} if $\\textbf{z}(t) = \\textbf{z}(t+T)$ for some $T>0$\n\\item \\textbf{relatively periodic orbit (RPO)} if $\\textbf{z}(t) = g \\textbf{z}(t+T)$ for some $T>0$ and $g \\in \\mathbb{E}(2)$\n\\end{enumerate}\n\\end{Def}\nThus, the abbreviation \\textbf{NRPO} will stand for a \\textbf{normalised relative periodic\n orbit}. Note that in particular for a NRPO we have $g\\in \\mathbb{O}(2)$ in the above definition.\nA periodic solution of the planar $N$-vortex problem s.t. $\\sum_{i=1}^N\\Gamma_{i} \\neq 0$ is called a\n\\textbf{relative equilibrium}, if it is of the form\n\\begin{align*}\nz_i(t)= e^{\\mathbb{J}\\omega t}(z_i(0)-c)+ c\n\\end{align*}\nwhere $\\displaystyle c = \\frac{\\sum_{i=1}^N \\Gamma_i z_i}{\\sum_{i=1}^N \\Gamma_i }$ is the vorticity center. This is a special configuration where all the vortices rigidly rotate about their center of vorticity $c$. In particular, given a normalised relative equilibrium, i.e., $c = (0,0)$ and $I(\\vz) =1$, it is of course a NRPO. We define\n\\begin{align*}\n&\\mathcal{Z}_{0}(H)= \\{\\textbf{z}| \\text{\\textbf{z} is a normalised orbit of the system } \\eqref{sys:H1} \\}\\\\\n&\\mathcal{Z}_{1}(H) =\\{\\textbf{z}| \\text{\\textbf{z} is a normalised relative equilibrium of the system } \\eqref{sys:H1} \\}\n\\end{align*}\nWe list some properties that will be used frequently later on:\n\\begin{Pro}\n\\label{Pro:NRE}\nThe following are equivalent:\n\\begin{align}\n(1)& \\quad \\textbf{z} \\in \\mathcal{Z}_{1}; \\label{E1} \\\\\n(2)& \\quad \\nabla H(\\textbf{z}) = -\\frac{L}{4\\pi} \\nabla I(\\textbf{z}) \\label{E2}\n\\end{align}\n\\end{Pro}\n\\begin{proof}:\n(1)$\\Rightarrow$ (2) : By definition of relative equilibrium, $\\textbf{z}(t)\\in \\mathcal{Z}_{1}$ implies $\\exists \\omega\\in \\mathbb{R}$ s.t. \n\\begin{align*}\n\\nabla H(\\textbf{z}(t)) = \\frac{\\omega}{2} \\nabla I(\\textbf{z}(t)) \n\\end{align*}\ntaking inner product with $\\textbf{z}(t)$ on both sides. Since $I(\\textbf{z})=1$, one sees that \n\\begin{align*}\n-\\frac{L}{2\\pi} = \\omega I(\\textbf{z}(t)) \\Rightarrow \\frac{\\omega}{2} = -\\frac{L}{4\\pi}\n\\end{align*}\nHence (2) is proved.\\\\\n(2)$\\Rightarrow$ (1) : If $\\textbf{z}$ satisfies that $\\displaystyle \\nabla H(\\textbf{z}) = -\\frac{L}{4\\pi} \\nabla I(\\textbf{z})$, then the flow passing through $\\textbf{z}$ will be a relative equilibrium. We need to show that such a relative equilibrium is normalised. First, by considering $(x,y)\\in \\mathbb{R}^2 $ as a complex number $x+iy \\in \\mathbb{C}$, (\\ref{E2}) implies that \n\\begin{align*}\n-\\frac{1}{2\\pi}\\sum_{j\\neq i } \\Gamma_i\\Gamma_j \\frac{\\bar{z}_i -\\bar{z}_j }{|z_i-z_j|^2} = -\\frac{L}{4\\pi} \\Gamma_i \\bar{z}_i, \\quad \\forall 1\\leq i \\leq N\n\\end{align*}\nIt follows that \n\\begin{align*}\n0 = -\\frac{1}{2\\pi}\\sum_{i=1}^{N}\\sum_{j\\neq i } \\Gamma_j\\Gamma_i \\frac{\\bar{z}_i -\\bar{z}_j }{|z_i-z_j|^2} = -\\sum_{i=1}^{N}\\frac{L}{4\\pi} \\Gamma_i \\bar{z}_i \n\\end{align*}\nThus $\\sum_{i=1}^{N}\\Gamma_i z_i =0 $, and $\\textbf{z}$ is centred. Next, multiply $\\textbf{z}$ on both sides of (\\ref{E2}), so that $ \\displaystyle-\\frac{L}{2\\pi} = \\nabla H(\\textbf{z}) \\textbf{z} =-\\frac{L}{4\\pi} \\nabla I(\\textbf{z})\\vz = -\\frac{L}{2\\pi} I(\\textbf{z})$. Thus $I(\\textbf{z})=1$. \n\\end{proof}\nThe first such configuration, found In 1883 by J.J. Thomson, is the\nso-called Thomson configuration \\cite{thomson1883treatise}, i.e., $N$\nidentical vortices located at the vertices of a N-polygon and rotating\nuniformly around its center of vorticity (which could be fixed to the\norigin).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=40mm,scale=0.5]{fig_1}\n\\captionsetup{justification=centering}\n\\caption{Thomson configuration for 8 vortices which form an octagon}\n\\label{fig:1}\n\\end{center}\n\\end{figure}\nThere has since then been extensive studies on relative equilibria in\nthe planar $N$-vortex problem, see for example\n\\cite{palmore1982relative,o1987stationary,roberts2013stability}. The\nstudy of relative equilibria is a subject in itself. Although their\nnumber and even their finiteness are unknown as functions of $N$ (see \\cite{hampton2009finiteness,o2007relative} for the special case when $N = 4$), one\ndoes not expect that relative equilibria could in general be abundant in the\nphase space. Hence our interest will be on the RPOs that are not relative\nequilibria. \n\n\n\\begin{Def}\n\\label{def:NTNRPO}\nWe say an orbit \\textbf{z}(t) is a \\textbf{non-trivial normalised relative periodic orbit} (NTNRPO) if \\textbf{z}(t) is a normalised relative periodic orbit but \\textbf{not} a normalised relative equilibrium.\n\\end{Def}\n\nNTNRPO's could conjecturally be dense in some open sets of the phase\nspace, similarly to what Poincar{\\'e} conjectured for the $N$-body\nproblem. Correspondingly define\n\\begin{align*}\n \\mathcal{Z}_{2}(H)= \\{\\textbf{z}| \\text{\\textbf{z} is a \n non-trivial normalised relative periodic orbit of the system } \\eqref{sys:H1} \\}.\n\\end{align*}\nSo, in this article, we focus on the search of NTNRPOs of the\n$N$-vortex problem. In 1949, Synge has given a thorough study of\nrelative periodic solutions of three vortices in\n\\cite{synge1949motion} (see also \\cite{aref1979motion}). Since the\n$N$-vortex problem is in general not integrable, it is thus more\ncomplicated to find NTNRPOs when more vortices are presented. Several\ndifficulties occur in the search of periodic solutions for system \\eqref{sys:H1}, for example,\n\\begin{itemize}\n\\item the system is singular around collisions\/infinity\n\\item energy surfaces are not compact\n\\item the Hamiltonian is not convex\n\\end{itemize}\nDue to these difficulties, methods in \\cite{ekeland2012convexity,hofer2012symplectic,carminati2006fixed} cannot be applied\ndirectly. Some NTNRPOs can be found by applying various perturbative\narguments around relative equilibria, as in\n\\cite{borisov2004absolute}; see also \\cite{carvalho2014lyapunov} for\nthe application of the Lyapunov centre theorem, and\n\\cite{bartsch2016periodic,bartsch2017global} for the application of\ndegree theory. We would like to make some contribution for the general\nknowledge on existence of such orbits whose energy might be far from\nthe energy of the relative equilibra with an arbitrary number $N$ of\nvortices and for arbitrary positive vorticity $\\mathbf{\\Gamma}$. To this end\nsome global method is needed. We focus on variational arguments,\ninstead of perturbative methods. The existence of non-relative\nequilibrium solution will rely on the absence of relative equilibrium\n(see \\textbf{Figure} \\ref{fig:2}). Define\n\\begin{align*}\n\\mathcal{H}_{0}& = \\{h \\in \\mathbb{R} | h = H(\\textbf{z}), \\textbf{z}\\in \\mathcal{Z}_{0}(H)\\}\\\\\n\\mathcal{H}_{1}& = \\{h \\in \\mathbb{R} | h = H(\\textbf{z}), \\textbf{z}\\in \\mathcal{Z}_{1}(H)\\}\\\\\n\\mathcal{H}_{2}& = \\{h \\in \\mathbb{R} | h = H(\\textbf{z}), \\textbf{z}\\in \\mathcal{Z}_{2}(H)\\}\\\\\n\\end{align*}\nNote that these are well-defined because the Hamiltonian $H$ is\nautonomous, thus it is constant along its flow. Clearly\n$\\mathcal{H}_{2} \\subset \\mathcal{H}_{0} $. Our main\nresult is that:\n\\begin{Thm}\n\\label{Thm:Main}\nUnder hypothesis \\eqref{Hyp:positive}, $\\mathcal{H}_{2}$ is dense in $\\mathcal{H}_{0}$.\n\\end{Thm}\n\\begin{figure}\n\\label{VariPert}\n\\captionsetup{justification=centering}\n\\begin{center}\n\\includegraphics[width=70mm,scale=0.5]{fig_2}\n\\caption{Perturbative Method(Left) VS Variational Method(Right).}\n\\label{fig:2}\n\\end{center}\n\\end{figure}\nThe rest of the article is organized in the following way:\n\\begin{itemize}\n\\item In chapter \\ref{sec:2}, we study the normalised relative equilibria of $H$. In particular, we show that the they are isolated from the region of singularity (\\textbf{Lemma \\ref{Lem:MinDist}}). Using this fact we show further that the set of critical value is very small(\\textbf{Theorem \\ref{Thm:ClosedNull}}). In case of positive rational vorticity, it even can be shown to be a finite set (\\textbf{Theorem \\ref{Thm:RationalFinite}});\n\\item In chpater \\ref{sec:3}, we show that by applying some symplectic reduction, we can focus on dynamics in the reduced phase space, where the energy level is compact (\\textbf{Lemma \\ref{Lem:compact}}). The capacity theory will then show that there are infinitely many NTNRPOs (\\textbf{Theorem \\ref{Thm:ManyRPOLocal}}). Combining results in chapter \\ref{sec:2} and in chapter \\ref{sec:3}, the main result(\\textbf{Theorem \\ref{Thm:Main}}) is thus proved;\n\\item In chapter \\ref{sec:4}, we add discrete symmetry constraints to get NTNRPOs of special symmetric configuration.\n\\end{itemize}\n\nWe have resumed necessary technical background and some details of proofs in the appendix. \n\\section{Normalised Relative Equilibria of H}\n\\label{sec:2}\n\\subsection{General Positive Vorticities}\n\\label{subsec:2.1}\nBefore we proceed to study NTNRPOs, we first need to have some preparation for properties of the normalised relative equilibria of $H$. In this chapter, we study the normalised relative equilibria of H, with an emphasis on their energy levels. First note that the mutual distances between vortices in a normalised relative equilibrium configuration cannot be too small. More precisely: \n\\begin{Lem}\n\\label{Lem:MinDist}\nFor $\\Gamma_i \\in \\mathbb{R}^{+}$, there exists constant $\\epsilon(\\mathbf{\\Gamma})$ which depends only on the vorticities $\\mathbf{\\Gamma} = (\\Gamma_1,\\Gamma_2,..\\Gamma_N), 1\\leq i\\leq N$, s.t.\n\\begin{align*}\n\\inf_{\\substack{\\textbf{z}\\in \\mathcal{Z}_{1}\\\\ 1\\leq i < j\\leq N}} |z_i-z_j|^2 >\\epsilon > 0\n\\end{align*}\n\\end{Lem}\n\\begin{Rmk}\nAs the relative equilibria are rigid body motions, we have dropped the dependence of time of $\\textbf{z}$ to simplify the discussion. \n\\end{Rmk}\nThis result first appears in the work of O'Neil \\cite{o1987stationary} and has been reproved recently by Roberts \\cite{roberts2017morse} using a renormalisation argument, followed by a detailed discussion on Morse index of relative equilibria. We here give an alternative proof by the observation that for a relative equilibirum, the vorticity center of a given cluster also rotates uniformly. \n\\begin{proof}:\nDenote\n\\begin{align*}\nm(\\textbf{z}) = \\inf_{ 1\\leq i < j\\leq N} |z_i-z_j|^2\n\\end{align*} \nSuppose to the contrary that $\\textbf{z}^k$ is a sequence of relative equilibria whose mutual distances s.t. \n$\\lim_{k\\rightarrow \\infty}m(\\textbf{z}^k) = 0$. Then by consecutively passing to subsequence if necessary, we may suppose that there exists an sub-index set $V\\subset\\{1,2,..,N\\}$ s.t. $z^k_i \\rightarrow z^*, \\forall i\\in V$. Denote $\\textbf{z}_V$ as the vector of vortices with index in V. The Hamiltonian could be separated into two parts, the interactions between vortices in V and otherwise. Let $H(\\textbf{z}) = H_V(\\textbf{z}) + H_{V^c}(\\textbf{z})$, where\n\\begin{align}\nH_{V}(\\textbf{z})&= -\\frac{1}{4\\pi}\\sum_{\\substack{i< j \\\\ i,j\\in V}}\\Gamma_{i}\\Gamma_{j}\\log{|z_i -z_j|^2} \\\\\nH_{V^c}(\\textbf{z})&=-\\frac{1}{4\\pi}\\sum_{\\substack{i< j \\\\ (i,j)\\notin V\\times V}}\\Gamma_{i}\\Gamma_{j}\\log{|z_i -z_j|^2}\n\\end{align}\nIt follows that \n$\\displaystyle \\nabla H(\\textbf{z}^k) \\textbf{z}^k = -\\frac{1}{2\\pi}L, \\text{while } \\nabla H_V(\\textbf{z}_V^k) \\textbf{z}_V^k = -\\frac{1}{2\\pi}L_V$. \nObserve that $c^k_V$, the vorticity centre of $\\textbf{z}^k_V$, also follows a uniform rotation with the vortices. As a result, \n\\begin{align}\n\\dot{c}^k_V &= \\frac{\\sum_{i\\in V}\\Gamma_i\\dot{z}^k_i}{\\sum_{i\\in V} \\Gamma_i} =\\mathbb{J} \\frac{\\omega}{2} c^k_V\\\\\n\\Gamma_i\\dot{z}^k_i &= \\mathbb{J}(\\nabla_{z_i} H_V(\\textbf{z}) + \\nabla_{z_i} H_{V^c}(\\textbf{z}) ) = \\mathbb{J} \\Gamma_i \\frac{\\omega}{2} z_i^k,\\quad i\\in V\n\\end{align}\nSince $\\lim_{k\\rightarrow \\infty}c^k_V = \\lim_{k\\rightarrow \\infty} z^k_i = z^*,\\forall i\\in V $,\nWe see that $\\displaystyle \\lim_{k\\rightarrow \\infty}\\nabla H_V(\\textbf{z}_V^k) = \\textbf{0}$. But we know already that $\\displaystyle \\nabla H_V(\\textbf{z}_V^k) \\textbf{z}_V^k = -\\frac{1}{2\\pi}L_V$. As $|z^i_V|$ is bounded (since $\\vz^k \\in \\mathcal{Z}_1(H)$), this implies that $L_V =0$, which contradicts the fact that $\\Gamma_i > 0,\\forall i\\in V$. As a result, such sequence $\\textbf{z}^k$ does not exist. The lemma is proved.\n\\end{proof}\nLemma \\ref{Lem:MinDist} tells us that the relative equilibria are isolated from the diagonals, where collision happens and singularity rises. With this result in hand, we will study the distribution of energy levels on which normalised relative equilibria exist. For a subeset $\\mathcal{A}\\subset \\mathbb{R}$, we denote by $\\mu(\\mathcal{A})$ its Lebesgue measure. Roughly speaking, we show that $\\mathcal{H}_{1}$ is somehow a small subset of $\\mathbb{R}$. \n\\begin{Thm}\n\\label{Thm:ClosedNull}\nFor $\\Gamma_i \\in \\mathbb{R}^{+},\\forall 1\\leq i \\leq N$, $\\mathcal{H}_{1}$ is a closed set in $\\mathbb{R}$. Moreover $\\mu(\\mathcal{H}_{1}) = 0$.\n\\end{Thm}\n\\begin{proof}:\nSuppose given a sequence of real numbers $h^k \\in \\mathcal{H}_{1}$ s.t. $\\lim_{k\\rightarrow \\infty } h^k \\rightarrow h^{*}\\in \\mathbb{R}$. Then by definition of $\\mathcal{H}_{1}$, there exists a sequence of normalised relative equilibria $\\textbf{z}^k \\in \\mathcal{Z}_{1}$ s.t. \n\\begin{align}\nH(\\textbf{z}^k) = h^k \\rightarrow h^{*}\n\\end{align}\nSince $I(\\textbf{z}^k) = 1$, $\\textbf{z}^k \\in \\mathbb{R}^{2N}$ is a bounded sequence, hence $\\displaystyle \\textbf{z}^k \\xrightarrow{k\\rightarrow \\infty} \\textbf{z}^{*}$. Thanks to lemma \\ref{Lem:MinDist}, we see that points in $\\mathcal{Z}_{1}$ are isolated from collision, hence $H$ is smooth at these points. As a result\n\\begin{align}\n\\nabla H(\\textbf{z}^{*})&= \\lim_{k\\rightarrow\\infty} \\nabla H(\\textbf{z}^k) = \\lim_{k\\rightarrow \\infty} -\\frac{L}{4\\pi} \\nabla I(\\textbf{z}^k(t)) = -\\frac{L}{4\\pi} \\nabla I(\\textbf{z}^{*}) \\\\\nI(\\textbf{z}^*) &= \\lim_{k\\rightarrow \\infty} I(\\textbf{z}^k) = 1,\\\\ H(\\textbf{z}^*) &= \\lim_{k\\rightarrow \\infty} H(\\textbf{z}^k) = \\lim_{k\\rightarrow\\infty} h^k = h^{*}\\end{align}\nIn other words, $z^{*} \\in \\mathcal{Z}_{1}$ and $H(z^*) = h^{*}$. Hence $\\mathcal{H}_{1}$ is a closed set.\\\\\nNext, consider the function \n\\begin{align*}\n&f: \\mathbb{R}^{2N} \\rightarrow \\mathbb{R}\\\\\n&f(\\textbf{z}) = 2H(\\textbf{z}) + \\frac{L}{2\\pi} I(\\textbf{z})\n\\end{align*}\nNow by proposition \\ref{Pro:NRE} $\\nabla f(\\textbf{z}) = 0$ implies that $\\textbf{z}\\in \\mathcal{Z}_{1}$, which is isolated from collision. Hence Sard's theorem applies and $f(\\mathcal{Z}_{1})$ is a null set. But on $\\mathcal{Z}_{1}$, one has $I(\\textbf{z}) = 1$, hence $\\mathcal{H}_1 = H(\\mathcal{Z}_{1})$ is a null set too. The theorem is thus proved.\n\\end{proof}\nOne important consequence of theorem 2 is the following corollary:\n\\begin{Col}\n\\label{Col:OpenDense}\n$\\mathcal{H}_0 \\setminus\t \\mathcal{H}_1$ is an open dense subset of $\\mathcal{H}_0$.\n\\end{Col}\n\\begin{proof}:\nImmediately from theorem \\ref{Thm:ClosedNull}.\n\\end{proof}\n\\subsection{Rational Positive Vorticities And Beyond}\n\\label{subsec:2.2}\nSo far corollary \\ref{Col:OpenDense} is sufficient for our further need. But when vorticities are positive rational numbers we can do even more. Actually, if $\\Gamma_i \\in \\mathbb{Q}^+$, we can even prove that there are only finitely many energy levels on which a normalised equilibrium exists. The proof of theorem \\ref{Thm:RationalFinite} below depends on a transformation of Hamiltonian and some elimination theory in algebraic geometry. We have resumed the detailed proof in Appendix \\ref{appendix:A}. \n\\begin{Thm}\n\\label{Thm:RationalFinite}\nIf $\\text{ }\\Gamma_i \\in \\mathbb{Q}^+, 1\\leq i \\leq N$, then $\\mathcal{H}_{1}$ is a finite set. \n\\end{Thm}\n\\begin{proof}:\nSee appendix \\ref{appendix:A}.\n\\end{proof}\nTheorem \\ref{Thm:RationalFinite} is interesting in its own right, although we still do not know whether the number of normalised relative equilibria configurations are finite or not. Actually, from the proof in appendix \\ref{appendix:A}, we see that $\\Gamma_i \\in \\mathbb{Q}^+$ is sufficient but not necessary. More generally, if $\\displaystyle \\frac{\\Gamma_i}{\\Gamma_j} \\in \\mathbb{Q}^{+},\\forall 1\\leq i< j\\leq N$, the result will hold. In particular, this is case for identical vorticities:\\\\\n\\begin{Col}\n\\label{Col:IdenticalFinite}\nIf $\\text{ }\\Gamma_i = c\\in \\mathbb{R} \\setminus \\{0\\}, 1\\leq i \\leq N$, then $\\mathcal{H}_{1}$ is a finite set. \n\\end{Col}\n\\section{Symplectic Reduction and Relative Periodic Orbits in the Plane}\n\\label{sec:3}\nIn this chapter, we will use standard symplectic reduction to study the Hamiltonian in a reduced phase space. In the first section, we give some properties for the generalized Jacobi variable introduced by Lim \\cite{lim1989canonical}. The main result is the compactness of energy surface of the reduced Hamiltonian in the reduced phase space. We do not give explicit calculation for coordinates transformations in this chpater. Instead, a detailed example of the 5-vortex problem is studied with explicit coordinate transformation in Appendix \\ref{CT}.\n\\subsection{Lim's generalized Jacobi coordinates}\n\\label{subsec:3.1}\nWe would like to fix the center of vorticity to the origin thus study only centred orbits. The reason is that, any non-centred relative equilibrium, when putting into a rotationing framework around the origin, might automatically become a relative periodic solution that is not a relative equilibrium. This situation is illustrated in figure \\ref{fig:3}.\n\\begin{figure}\n\\label{RAnon}\n\\captionsetup{justification=centering}\n\\begin{center}\n\\includegraphics[width=60mm,scale=0.5]{fig_3}\n\\caption{A non trivial relative periodic (left) coming from a non-centred relative equilibrium in the original phase space (right)}\n\\label{fig:3}\n\\end{center}\n\\end{figure}\nHowever, this kind of solution (orbits in red color in the left of figure \\ref{fig:3}) is not the solution that we are searching for. Because it does not give any further insights about our dynamic system. As a result, we should insist on centred orbits, and we need some transformation to fix the vorticity centre to the origin.\\\\\nThe usual tool in celestial mechanics is the so called Jacobi coordinates. However, the usual Jacobi coordinates are not suitable for the $N$-vortex problems. This is because the conjugate variables $(q,p)$ are separated in the Hamiltonian for Newtonian gravitation N-body problem, i.e., \n\\begin{align*}\nH(q,p) = \\frac{|p|^2}{2} + U(q) \\tag{N-Body}\n\\end{align*}\nwhile in $N$-vortex problem the conjudate variables $(x,y)$ are mixed\n\\begin{align*}\nH(x,y)= -\\frac{1}{4\\pi}\\sum_{i,j =1, i< j}^{N}\\Gamma_{i}\\Gamma_{j}\\log{|z_i -z_j|^2} \\tag{N-Vortex}\\end{align*}\nHence if we perform a normal Jacobi transforamtion, we can fix the center of vorticity, but the resulting new Hamiltonian might be no longer invariant under rotation. There has been some study on symplectic transformations adapted to the $N$-vortex problem. For example \\cite{khanin1982quasi,borisov1998dynamics,lim1989canonical} and so on. In particular, Lim's method in \\cite{lim1989canonical} has introduced a canonical transformation for the $N$-vortex Hamiltonian based on graph theory. This transformation works particularly well when all the vorticities are postive, and is quite ideal for our purpose of evaluating the energy surfaces. We hence apply Lim's generalized jacobi coordinates to simplify our $N$-vortex system.\\\\\nFirst, we make the change of variable \n\\begin{align}\nZ_i = (X_i, Y_i) = (\\sqrt{\\Gamma_i}x_i,\\sqrt{\\Gamma_i}y_i )\n\\end{align}\nIt turns out that $\\textbf{Z}=(Z_1,Z_2,...,Z_N)$ follows the usual Hamiltonian system \n\\begin{equation}\n\\dot{\\textbf{Z}}(t) = X_{\\hat{H}}(\\textbf{Z}(t)) = \\mathcal{J}_{N}\\nabla \\hat{H}(\\textbf{Z}(t)) \\quad \\textbf{Z}=(Z_1,Z_2,...,Z_N),\\quad Z_i\\in \\mathbb{R}^{2} \\tag{H2} \\label{sys:H2}\n\\end{equation}\nwhere \n\\begin{align*}\n\\hat{H}(\\textbf{Z}) = -\\frac{1}{4\\pi}\\sum_{i,j =1, i< j}^{N}\\Gamma_{i}\\Gamma_{j}\\log{|\\frac{Z_i}{\\sqrt{\\Gamma_i}} -\\frac{Z_j}{{\\sqrt{\\Gamma_j}}}|^2} \n\\end{align*}\nThen for the new variables,\n\\begin{align*}\n\\hat{P}(\\textbf{Z}(t)) = \\sum_{1\\leq i\\leq N}\\sqrt{\\Gamma_i} X_i(t),\\quad \\hat{Q}(\\textbf{Z}(t))= \\sum_{1\\leq i \\leq N} \\sqrt{\\Gamma_i} Y_i(t),\\quad \\hat{I}(\\textbf{Z}(t)) = \\sum_{1\\leq i\\leq N} |Z_i(t)|^2\n\\end{align*}\nare first integrals.\nWe identify till the end of this section the coordinate in $Z_k = (X_k, Y_k) \\in \\mathbb{R}^2$ to the complex number $Z_k = X_k + iY_k $. A transformation from $\\mathbb{C}^N$ to $\\mathbb{C}^N$ will also be considered as a transformation from $\\mathbb{R}^{2N}$ to $\\mathbb{R}^{2N}$.\n\\begin{Pro}\n\\label{Pro:LimTransform}\n(\\cite[page~263]{lim1989canonical}) There exists a linear transformation for the positive planar N-vortex problem \n\\begin{align*}\n\\phi: \\quad \\mathbb{C}^N &\\rightarrow \\mathbb{C}^N\\\\\nZ= (X,Y) &\\xrightarrow{\\phi} W=(q,p) \n\\end{align*}\ns.t. \n\\begin{enumerate}\n\\item $\\phi$ is unitary; \n\\item In the new coordinate W = (q,p), one has \\\\\n\\begin{align}\n\\begin{cases}\n\\displaystyle q_N = \\frac{\\sum_{1\\leq N}\\sqrt{\\Gamma_i}X_i}{\\sum_{1\\leq i\\leq N} \\Gamma_i}\\\\\n\\displaystyle p_N = \\frac{\\sum_{1\\leq N}\\sqrt{\\Gamma_i}Y_i}{\\sum_{1\\leq i\\leq N} \\Gamma_i}\n\\end{cases}.\n\\end{align}\n\\end{enumerate}\n\\end{Pro}\n\nSince\n$\\mathbb{U}(N) = \\mathbb{O}(2N) \\bigcap \\mathbb{SP}(2N)$, the transformation\n$\\phi$, seen as a transformation $\\mathbb{R}^{2N} \\xrightarrow{\\phi} \\mathbb{R}^{2N}$, is thus a real linear symplectic transformation.\nAs a result, we see that $q_N$ is a first integral and $p_N$ as its conjugate variable is cyclic. We can thus fix $q_N = p_N = 0$, and get a reduced Hamiltonian on $\\mathbb{R}^{2N-2}$: \n\\begin{align}\n\\bar{H}(q_1,p_1, q_2, p_2,..., q_{N-1}, p_{N-1} ; q_N = p_N = 0) = \\bar{H}(\\textbf{W};W_N=0)\n\\end{align}\n\\\\\nConsider the dynamic system\\\\\n\\begin{equation}\n\\dot{\\textbf{W}}(t) = \\textit{X}_{\\bar{H}}(\\textbf{W}(t)) \\tag{H3} \\label{sys:H3}\n\\end{equation}\n\nWe resume some properties of the new Hamiltonian $\\bar{H}$: \n\\begin{Pro}\n\\label{Pro:AfterLimTransform}\nConsider the Hamiltonian system \\eqref{sys:H3} and the original Hamiltonian system \\eqref{sys:H1} and \\eqref{sys:H2} . Then:\n\\begin{enumerate}\n\\item Any orbit of $\\bar{H}$ is a centred orbit of $H$;\\\\\n\\item The system \\eqref{sys:H3} is invariant under rotation;\n\\item Define \n\\begin{align}\n\\bar{I}(W) = \\sum_{1\\leq i\\leq N-1} (p_i^2+ q_i^2)\n\\end{align}\nThen $\\bar{I}(\\textbf{W}) = \\hat{I}(\\textbf{Z})$.\n\\end{enumerate}\n\\end{Pro}\n\\begin{proof}:\nThese propositions are direct consequences of the special symplectic transformation $\\phi$.\\\\\n1. $(q_N,p_N)$ corresponds to the vorticity centre in the original Hamiltonian and they are fixed at 0. Hence all the orbits of $\\bar{H}$ are centred orbit of $H$.\\\\\n2. $\\phi$ is a linear transformation $\\mathbb{C}^N \\xrightarrow{\\phi} \\mathbb{C}^N$. The term $\\displaystyle \\log|\\frac{Z_i}{\\Gamma_i}-\\frac{Z_j}{\\Gamma_j}|^2$\nunder the transformation $\\phi$ now becomes \n\\begin{equation}\n\\displaystyle \\log|\\frac{Z_i}{\\sqrt{\\Gamma_i}}-\\frac{Z_j}{\\sqrt{\\Gamma_j}}|^2= \\log|\\frac{\\sum_{1\\leq k\\leq N-1} c_{ki} W_i}{\\sqrt{\\Gamma_i}}-\\frac{\\sum_{1\\leq k\\leq N-1} c_{kj} W_j}{\\sqrt{\\Gamma_j}}|^2 \\label{eq:quatratic}\n\\end{equation}\nwhere the coefficients $c_{ki}$ and $c_{kj}$ are decided by $\\phi$. It is clearly still invariant under rotation. \\\\\n3. We know that $I(\\textbf{z})$ is a first integral for system \\eqref{sys:H1}, hence $\\hat{I}(\\textbf{Z}) = \\sum_{1\\leq i \\leq N} |Z_i|^2$ is a first integral for system \\eqref{sys:H2}. Now that $\\phi$ is orthogonal, we have $\\sum_{1\\leq i \\leq N} |Z_i|^2 = \\sum_{1\\leq i \\leq N} |W_i|^2$, while\n$W_N = (q_N,p_N) = 0$, we see that actually $\\sum_{1\\leq i \\leq N} |W_i|^2= \\sum_{1\\leq i \\leq N-1} |W_i|^2$. In other words,\n$\\bar{I}(\\textbf{W}) = \\hat{I}(\\textbf{Z})$.\n\\end{proof}\nRecall we are interested in normalised orbits of the original Hamiltonian system \\eqref{sys:H1}. According to results in the previous proposition, they can be characterized by the new coordinates, i.e.:\n\\begin{Pro}\n\\label{Pro:NewNormalOrbit}\n\\label{equivalence}\nThe orbits of system (\\ref{sys:H3}) which satisfies $\\bar{I}(\\textbf{W}) = 1$ are the normalised orbits of the system \\eqref{sys:H1}.\n\\end{Pro}\n\\subsection{Energy Surface in Reduced Phase Space} \n\\label{subsec:3.2}\n The Hamiltonian system \\eqref{sys:H3} with $\\bar{H}(\\textbf{W};W_N=0) : \\mathbb{R}^{2N-2} \\rightarrow \\mathbb{R}$ is invariant under rotation, and $\\bar{I}(\\textbf{W})$ is the first integral.\nBy the theory of the standard symplectic reduction, we can fix $\\bar{I} =1 $ and apply Hopf-fibration, it turns out that \\eqref{sys:H3} canonically induces a Hamiltonian system \n\\begin{equation}\n\\dot{\\tilde{\\textbf{W}}} = \\textit{X}_{\\tilde{H}}(\\tilde{\\textbf{W}} ) =\\mathcal{\\tilde{J}}(\\tilde{\\textbf{W}})\\nabla \\tilde{H}(\\tilde{\\textbf{W}}) \\tag{H4} \\label{sys:H4}\n\\end{equation}\n on $\\mathbb{CP}^{N-2}$ \\cite{abraham1978foundations}. Each point in $\\mathbb{CP}^{N-2}$ represents a equivalent class of configurations up to the translation (by fixing $q_N= p_N=0$) the rotation (by taking quotient of $\\mathbb{SO}(2)$), and the homothety(by fixing $\\bar{I}(\\textbf{W})= 1$, thus $\\nabla \\bar{I}(\\textbf{W}) \\neq 0$). By Proposition \\ref{equivalence}, each orbit on $\\mathbb{CP}^{N-2}$ stands for a relative normalised orbit of system \\eqref{sys:H1}. We resumed the whole reduction process in the following diagram:\\\\\n\\begin{tikzcd}\n&&&& & \\mathbb{S}^1 \\arrow[hookrightarrow]{d} \\\\\n&&& \\mathbb{R}^{2N} \\arrow[hookleftarrow]{r}{q_N=p_N=0}&\\mathbb{R}^{2N-2} \\arrow[hookleftarrow]{r}{\\bar{I}= 1} & \\mathbb{S}^{2N-3} \\arrow [twoheadrightarrow]{d} {\/ \\mathbb{SO}(2)} \\\\\n&&&& & \\mathbb{CP}^{N-2} \\\\ \n\\end{tikzcd}\n\\\\\nAlthough the energy surfaces for original Hamiltonian is not even bounded, due to the invariance under translation and the opposited singularities in logarithm function, the energy surface of the reduced Hamiltonian is indeed compact.\n\\begin{Lem}\n\\label{Lem:compact}\nLet $c \\in \\mathbb{R}$. Consider the hypersurface $S_c = \\tilde{H}^{-1}(c) \\subset \\mathbb{CP}^{N-2} $. If $S_c \\neq \\emptyset$, then $S_c$ is compact. \n\\end{Lem}\n\\begin{proof}:\nConsider the set $\\bar{S}_c=\\bar{H}^{-1}(c)\\cap \\bar{I}^{-1}(1)$, which is the lifted set of $S_c$ from $\\mathbb{CP}^{N-2} $ to $\\mathbb{S}^{2N-3}$. If $\\bar{S}_c$ is compact, then $S_c$ will be compact by quotient topology. First, $\\mathbb{S}^{2N-3}$ is a bounded manifold, hence the boundedness of $\\bar{S}_c$. Next, recall that $\\bar{I}(\\textbf{W})= 1$ for all points in $\\bar{S}_c$, which implies that all the mutual distances are bounded from above, since each squared mutual distance is a quadratic functions of $\\textbf{W}$, as is shown in \\eqref{eq:quatratic}. In other word, by the fact that $\\bar{H}$ and $\\bar{I}$ are preserved by the lifted flow of $\\phi_{\\bar{H}}$, the mutual distances cannot be too small. As a result, the energy surface $\\bar{S}_c$ is isolated from singularity. But then the preimage of a closed set must be closed, hence $\\bar{S}_c$ is closed. Hence $\\bar{S}_c$ is compact. So is $S_c$.\\\\\n\\end{proof}\n\\subsection{Symplectic Capacity and Existence of Normalised Non-Trivial Relative Periodic Orbits}\n\\label{subsec:3.3}\nWe are now ready to prove the theorem concerning the existence of NTNRPOs of system \\eqref{sys:H1}. Our main tool is the so called symplectic capacity, in particular the Hofer-Zehnder capacity $c_0$\\cite{hofer2012symplectic}, which links periodic solution of Hamiltonian system to symplectic invariant. It is closely related to the searching of periodic orbits on a prescribed energy surface, initially studied by Rabinowitz \\cite{rabinowitz1978periodic} and Weinstein \\cite{weinstein1978periodic}. For general introduction to symplectic capacity theory one could turn to \\cite{viterbo31capacites,hofer2012symplectic} and the references therein. \n\n\\begin{Thm}\n\\label{Thm:ManyRPOLocal}\nSuppose that $S_c = \\tilde{H}^{-1}(c)$ is a non-empty regular hypersurface, then there exists a non-constant sequence $\\lambda_k \\rightarrow c$ and a sequence of normalised non-trivial relative periodic orbits $\\vz^k(t)$ of system \\eqref{sys:H1} s.t. $H(\\vz^k) = \\lambda_k$.\n\\end{Thm}\n\\begin{proof}:\nSince the hypersurface $S_c$ is regular, and by Lemma \\ref{Lem:compact} it is compact. In other words, the vector field $\\dot{\\tilde{W}} = \\frac{\\nabla \\tilde H(\\tilde{W})}{|\\nabla \\tilde H(\\tilde{W})|^2 }$ is locally well defined. By consequence we can almost surely extend $S_c$ to a 1-parameter family of regular energy surfaces $S(\\delta)$, with $-\\epsilon< \\delta < \\epsilon$ and $S(\\delta) = S_{c+\\delta}$. \nDefine\n\\begin{align*} \nU_{\\epsilon} = \\bigcup_{\\delta\\in (-\\epsilon, \\epsilon)} S(\\delta)\n\\end{align*} \nLet $c_0(\\mathbb{CP}^{N-2},\\omega)$ be the symplectic capacity, where $\\omega= Im(g)$ and $g$ is the induced K{\\\"a}hler metric by the standard Hermitian, then $c_0(\\mathbb{CP}^{N-2},\\omega) = \\pi <\\infty$ (\\cite[Corollary~1.5]{hofer1992weinstein}), thus \\textit{a fortiori}, $c_0(U,\\omega)< \\infty$. Classical result of almost existence (\\cite[Theorem~4.1]{hofer2012symplectic}) now implies the existence of infinitely many \\textbf{non-constant} periodic solutions $\\{\\tilde{\\textbf{W}}^k\\}_{k\\in \\mathbb{N}}$ of the Hamiltonian system (\\ref{sys:H4}) and a corresponding non-constant sequence $\\{\\lambda_k\\}_{k\\in \\mathbb{N}}$, which satisfy that \n$\\displaystyle \\tilde{H}(\\tilde{\\textbf{W}}^k) = \\lambda_k \\rightarrow c$.\\\\\nNow given a non-constant periodic orbit $\\tilde{\\textbf{W}}^k(t) = \\phi_{\\tilde{H}}(t) \\subset \\mathbb{CP}^{N-2}$ of system \\eqref{sys:H4}, its lifted orbit $\\textbf{z}^k =\\phi_{H}(t) \\subset \\mathbb{R}^{2N}$ is a normalised relative periodic solution of the original Hamiltonian system \\eqref{sys:H1}. We show that $\\textbf{z}^k$ is not a relative equilibrium. \nRecall that by our construction of the reduced phase space, the vortex center of $\\vz^k(t)$ is fixed at 0. If $\\vz^k(t)$ is a relative equilibrium, then $\\tilde{\\textbf{W}}^k(t)$ is a fixed point in the reduced space, which contradicts the fact that $\\tilde{\\textbf{W}}^k(t)$ is a non-constant periodic solution. The theorem is thus proved.\n\\end{proof}\n\\begin{Rmk}\nStrictly speaking the reduced dynamics is only defined on $\\mathbb{CP}^{N-2} \\setminus \\tilde{\\Delta}$.\nHere $\\tilde{\\Delta}$ is projection of the generalized diagonal $\\Delta$ where collision ( of two or multiple vortices) happens, i.e.,\n\\begin{align*}\n\\Delta = \\{\\vz \\in \\mathbb{R}^{2N} |\\quad z_i = z_j \\text{ for some } 1\\leq i0$. At time 0, we put each group $M_l$ into a $C_N$ symmetric configuration, i.e., $\\forall 1\\leq i \\leq N, 1\\leq l\\leq M$\n\\begin{align}\nz_{li} &= e^{\\mathbb{J}\\frac{2\\pi(i-1)}{N}}z_{l1} \n\\end{align}\nThen by symmetry of the Hamiltonian, we will have an orbit s.t. each vortices in each group $M_i, 1\\leq i \\leq M$ follow a $C_N$ symmetric orbit. We only need to study the Hamiltonian taking the $C_N$ symmetry into account. Denote $\\displaystyle w_l = z_{l1}, 1\\leq l \\leq M$ for short, which serves as a representative of the $N$ vortices in the $l$-th group $M_l$. We then consider the simplified Hamiltonian system \n\\begin{equation}\n\\label{sys:Hsym} \n\\Gamma \\dot{\\textbf{w}}(t) = \\textit{X}_{H^{sym}}(\\textbf{w}(t)) = \\mathcal{J}_M\\nabla H^{sym}(\\textbf{w}(t)) \\quad \\textbf{w}=(w_1,w_2,...,w_M),\\quad w_i\\in \\mathbb{R}^2 \\tag{H-Sym}\n\\end{equation}\nwhere \n\\begin{align*}\nH^{sym}(\\textbf{w})&= -\\frac{1}{4\\pi}\\sum_{\\substack{1\\leq p,q\\leq M\\\\ 1\\leq i,j\\leq N \\\\(p,i) \\neq (q,j)}}\\Gamma_p \\Gamma_q \\log{|e^{\\mathbb{J}\\frac{2\\pi i}{N}}w_p-e^{\\mathbb{J}\\frac{2\\pi j}{N}}w_q |^2} \n\\end{align*}\nClearly each periodic solution of the system \\eqref{sys:Hsym} will imply a $C_N$ symmetric periodic solution of the original $M\\times N$-vortex problem as in system \\eqref{sys:H1}. If we further more require that $I(\\textbf{w}) = \\frac{1}{N}$, then it corresponds to a normalised $C_N$-symmetric periodic solution of the original $M\\times N$-vortex problem as in system \\eqref{sys:H1}.\n\n\\begin{figure}\n\\captionsetup{justification=centering}\n\\begin{center}\n\\includegraphics[width=40mm,scale=0.5]{fig_4}\n\\caption{An example of a $M\\times N$-vortex configuration that is $C_N$ symmetric, with M=3, N=4}\n\\label{fig:4}\n\\end{center}\n\\end{figure}\nWe resume the above discussion in the following proposition:\n\\begin{Thm}\nConsider the above symmetric $M\\times N$-vortex problem with positive vorticities s.t. $\\Gamma_{li} =\\Gamma_{lj} ,1\\leq l \\leq M, 1\\leq i 1$, the saddle is called \\emph{area-expanding}. \n\nThe saddle $p$ is called \\emph{sectionally dissipative} if it has a unique expanding eigenvalue $\\lambda_1: |\\lambda_1|>1,$ and for any two eigenvalues $\\lambda_i, \\lambda_j$ ($i\\ne j$) one has $|\\lambda_i\\cdot\\lambda_j|<1.$ \n\\end{defin}\n\n\\begin{rem}\\label{rem:volume}\nThe saddle $p$ is sectionally dissipative iff in some coordinates $dF^{\\operatorname{per}(p)}$ contracts two-dimensional euclidean volumes, i.e., its restriction to every (not necessarily invariant) two-dimensional plane is volume contracting (hence the name `sectionally dissipative').\n\\end{rem}\n\n\\begin{defin}\nAn $F$-invariant set $\\Lambda$ is called \\emph{locally maximal} if it coincides with the intersection of images of some its neighborhood $U$ under positive and negative iterates of the map: $\\Lambda = \\bigcap_{n\\in\\mathbb Z}F^n(U(\\Lambda)).$ A topologically transitive locally maximal (closed) hyperbolic invariant set is called \\emph {a basic set}. \n\\end{defin}\n\nBasic sets survive small perturbations of the diffeomorphism, and we will often denote the hyperbolic continuation of a basic set by the same symbol that we use for the original basic set. Periodic points are always dense in basic sets.\n\n\\begin{rem}\\label{rem:fiber_density}\n Suppose $\\Lambda$ is a basic set of saddle type and $p\\in\\Lambda$ is a periodic saddle. Denote by $O(p)$ the orbit of $p$. Then $W^u(O(p))$ is dense in $W^u(\\Lambda).$ Indeed, consider a point $x\\in\\Lambda$ with a dense forward orbit\\footnote{Our set $\\Lambda$ is a complete separable metric space, and in this case topological transitivity implies the existence of points with dense forward and backward orbits.}. We may assume that $x$ is close to $p$. Then $W^u(p)\\pitchfork W^s(x)\\ne\\emptyset$ because of the local product structure. But for any point $y$ in this intersection $\\dist(F^n(y), F^n(x))\\to 0$ as $n\\to\\infty$. Then the density of the forward orbit of $x$ implies that $W^u_{loc}(\\Lambda) \\subset \\overline{W^u(O(p))}$, and hence $W^u(\\Lambda) \\subset \\overline{W^u(O(p))}$. An analogous statement is true for $W^s(O(p))$ and $W^s(\\Lambda)$. \n\\end{rem}\n\n\nTwo periodic saddles $p$ and $q$ are called \\emph{heteroclinically related} if $W^u(O(p))\\pitchfork W^s(O(q))\\ne\\emptyset$ and $W^s(O(p))\\pitchfork W^u(O(q))\\ne\\emptyset.$ It follows from the $\\lambda$-lemma (also known as the inclination lemma) that this is an equivalence relation. The closure of the set of all saddles heteroclinically related to $p$ is called \\emph{the homoclinic class} of $p.$ We will denote the homoclinic class of a saddle $p$ by $H(p, F)$, where $F$ stands for the mapping. Whenever we use this notation, we assume that $p$ is a hyperbolic saddle for $F$. \n\n\n\\begin{defin}\nWe will say that there is a homoclinic tangency associated with a hyperbolic periodic saddle $p$ if $W^u(O(p))$ and $W^s(O(p))$ have a point (and therefore an orbit) of non-transverse intersection.\n\\end{defin} \n\nIn what follows in dimension greater than two we will consider only homoclinic tangencies that appear for sectionally dissipative saddles.\n\n\\begin{defin}\\label{def:per_tan}\nSuppose that for each diffeomorphism in an open domain $U\\subset {\\rm Diff}^r(M)$ there is a hyperbolic basic set $\\Lambda(F)$ of saddle type that depends continuously on the diffeomorphism and for each $F \\in U$ there are two points $p_1, p_2\\in\\Lambda (F)$ such that $W^s(p_1)$ has a point of tangency with $W^u(p_2).$ Then we say that there is \\emph{a $C^r$-persistent tangency} in $U$ \\emph{for a hyperbolic set} $\\Lambda(F)$, or that $\\Lambda(F)$ \\emph{exhibits a $C^r$-persistent tangency}. \n\\end{defin}\n\nWhen we have a diffeomorphism with a basic set such that there is a persistent tangency in the neighborhood of this diffeomorphism associated with the continuation of this basic set, we will informally say that this diffeomorphism and this basic set exhibit a persistent tangency.\n\nThe detailed proof of the following proposition can be found in \\cite{CIME} (see Lemma 8.4).\\footnote{Though the primary focus is on the two-dimensional case there, the proof itself is valid in any dimension.}\n\\begin{prop}\\label{prop:dense}\nSuppose that $\\Lambda$ is a hyperbolic basic set for a diffeomorphism $F\\in \\operatorname{Diff}^r(M)$, $p\\in\\Lambda$ is a periodic saddle, and there are two points $p_1, p_2\\in\\Lambda$ such that $W^u(p_1)$ and $W^s(p_2)$ have an orbit of tangency. Then a homoclinic tangency associated with (the continuation of) $p$ can be obtained by a $C^r$-small perturbation of $F.$ \n\\end{prop}\n\nProposition~\\ref{prop:dense} implies that whenever we have a domain $U\\subset{\\rm Diff^1}(M)$ with a persistent tangency for a hyperbolic basic set $\\Lambda(F)$, the diffeomorphisms with a homoclinic tangency associated with a given saddle $p(F)\\in\\Lambda(F)$ are $C^r$-dense in $U$ for any $r.$ This inspires the definition of a persistent tangency related to a given saddle.\n\n\\begin{defin}\\label{def:tan_p}\nWe will say that an open set $U\\subset\\operatorname{Diff}^r(M)$ exhibits \\emph{a $C^r$-persistent tangency associated with the saddle} $p$ of the diffeomorphism $F\\in U$ if for any $G\\in U$ the continuation of $p$ is defined and diffeomorphisms with a homoclinic tangency related to the continuation of $p$ are dense in $U$ in $C^r$-topology. \n\\end{defin}\n\n\\begin{defin}\nSuppose that $\\Lambda$ is an $F$-invariant subset of $M$ and $TM|_\\Lambda = E \\oplus G$ is a $dF$-invariant splitting of $TM$ over $\\Lambda$ such that the dimensions of the fibers of $E$ and $G$ are constant. Then the splitting $E \\oplus G$ is called \\emph{dominated} if for some $n\\in\\mathbb N$ for any $x\\in\\Lambda$ and any $u\\in E(x),\\; v\\in G(x)$ one has\n$$ \\;\\;\\frac{\\|dF^n(x)u\\|}{\\|u\\|} \\le \\frac{1}{2}\\cdot\\frac{\\|dF^{n}(x)v\\|}{\\|v\\|}.$$ \n\\end{defin}\n\nOne may regard the existence of a dominated splitting as a very weak form of hyperbolic-like behavior. If we add to this definition the assumption that $E$ is purely contractive (or $G$ is dilative), we will obtain the definition of partial hyperbolicity; and if we, moreover, assume that $G$ is purely dilative, we will obtain the definition of a hyperbolic set.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{A sufficient condition for the Lyapunov instability of the Milnor attractor}\n\n\\begin{prop} \\label{prop:suff} Suppose that a diffeomorphism $F\\in\\operatorname{Diff}^1(M)$ satisfies the following conditions:\n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n\\item $F$ has a hyperbolic saddle $p$ whose unstable manifold $W^u(p)$ intersects the basin of attraction of some sink $\\gamma$. \n\\item $F$ has a sequence of periodic sinks $\\gamma_j, \\ j\\in\\mathbb N,$ that accumulate to $p$, i.e.,\n $$\n\\mbox{\\rm dist }(\\gamma_j, p) \\to 0 \\mbox{ as } j \\to \\infty.\n $$\n\\end{itemize}\nThen $A_M(F)$ is Lyapunov unstable.\n\\end{prop}\n\n\\begin{proof} \n\nFirst note that the Milnor attractor $A_M(F)$ lies in the non-wandering set $\\Omega(F).$ Indeed, since a wandering point has a neighborhood free of $\\omega$-limit points, it can not belong to the attractor, for otherwise one could subtract the neighborhood of this wandering point from the attractor and obtain a smaller closed set that attracts almost every positive orbit, which is a contradiction.\n\nFurther note that every sink $\\gamma_j$ belongs to the Milnor attractor since the basin of attraction of $\\gamma_j$ contains a neighborhood of this sink and therefore has positive measure.\nAs these sinks accumulate to the saddle $p$ and the Milnor attractor is, by definition, closed, we get: $p \\in A_M(F)$.\n\nNow consider the sink $\\gamma$ from the first assumption of the proposition. The whole basin of this sink, except for the sink itself, consists of wandering points.\nThe first assumption says that there is a point of $W^u(p)$ in that basin. Since such a point is wandering, it is separated from the attractor. But the preimages of this point under the iterates of $F$ come arbitrarily close to the saddle $p\\in A_M(F)$, which implies the Lyapunov instability of the Milnor attractor.\n\\end{proof}\n\n\\begin{rem}\nAll statements about Milnor attractor which we prove below are reduced to Proposition~\\ref{prop:suff}. Note that the proof of Proposition~\\ref{prop:suff} uses only the following three properties of the Milnor attractor: the attractor is closed, contains every sink and lies in the non-wandering set. Therefore, analogous statements are true for any other definition of attractor provided that the definition under consideration implies these three properties.\nFurther note that Proposition~\\ref{prop:suff} does not really need the assumption that $F$ is a diffeomorphism: this map may as well be just a local diffeomorphism.\n\\end{rem}\n\n\\begin{rem}\nPreviously we defined Lyapunov stability in purely topological terms. However, one can consider another definition that is very much alike but not equivalent. Namely, let us call an invariant set $K$ \\emph{metrically Lyapunov stable} if for any $\\varepsilon >0$ there is $\\delta>0$ such that any positive semi-orbit that starts $\\delta$-close to $K$ never leaves the $\\varepsilon $-neighborhood of~$K$. Then an analogue of Proposition~\\ref{prop:suff} with Lyapunov instability replaced by metric Lyapunov instability is true and the proof does not utilize the closeness of the attractor. Thus, if we confine ourselves with metric Lyapunov stability, we can restate every result of this paper for any other definition of attractor if this definition implies that the attractor contains every sink and lies in the non-wandering set. \n\\end{rem}\n\n\n\n\n\n\n\n\n\n\\section{The Newhouse phenomena and the instability of attractors}\n\nIn this section we will prove that every domain with a persistent tangency for a sectionally dissipative saddle has a residual subset of diffeomorphisms with Lyapunov unstable Milnor attractors.\n\n\n\n\n\n\\subsection{The Newhouse phenomena}\n\nIn the 70th S. Newhouse proved a number of results on persistent homoclinic tangencies and coexistence of infinitely many sinks or sources.\n\n\\begin{thm}[Newhouse, \\cite{N70, N74}]\\label{thm:N}\nFor any manifold $M$ of dimension greater than one and for any $r\\ge 2$ there is an open set $U\\subset\\operatorname{Diff}^r(M)$ such that any diffeomorphism $G\\in U$ exhibits a persistent tangency for a basic set $\\Lambda(G)$ and a topologically generic diffeomorphism in $U$ has infinitely many periodic sinks.\n\\end{thm}\n\n \n\\begin{thm}[Newhouse, \\cite{N79}]\\label{thm:N2}\nLet $F\\in{\\rm Diff^2}(M),\\; \\dim M = 2,$ have a periodic saddle $p$ with a homoclinic tangency. Then arbitrarily close to $F$ there exists an open set $U$ of $C^2$-diffeomorphisms with a persistent tangency for some basic set $\\Lambda(G)$ that continuously depends on $G\\in U$.\n\\end{thm}\n\n\nJ. Palis and M. Viana in \\cite{PV} generalized the second result for higher dimensions assuming that the saddle $p$ was sectionally dissipative. Although they did not construct a single basic set with persistent tangency, they obtained a persistent heteroclinic tangency for a couple of basic sets and deduced from it the persistent tangency associated with the continuation of some sectionally dissipative saddle. The latter persistent tangency implied a locally generic coexistence of infinitely many sinks. The argument in \\cite{PV} generalized the new proof of the Newhouse phenomena presented in the book \\cite{PT}. \n\n\\begin{thm}[Palis, Viana, \\cite{PV}]\\label{thm:ptpv}\nLet $F\\in{\\rm Diff^2}(M)$ have a sectionally dissipative periodic saddle $p$ with a homoclinic tangency. Then arbitrarily close to $F$ there exists an open set $U$ of $C^2$-diffeomorphisms with a persistent tangency associated with the continuation of some sectionally dissipative saddle $p_1$ and topologically generic diffeomorphisms in $U$ have infinitely many sinks.\\footnote{Actually, in \\cite{PT, PV} only tangencies between invariant manifold of one and the same saddle are considered, but Theorem~\\ref{thm:ptpv} is true as stated here; see Proposition~\\ref{prop:goodtan3} below.}\n\\end{thm}\n\n \n\n\n\n\n\n\\subsection{Instability of attractors}\nThis subsection is devoted to the applications of Theorem \\ref{thm:NI}. Let us first repeat this theorem for convenience.\n\n\\begin{repthm1}\nSuppose that in an open set $U\\subset{\\rm Diff}^r(M),\\; r\\ge 1,$ there is a persistent tangency associated with a sectionally dissipative periodic saddle $p$. Then a topologically generic diffeomorphism in $U$ has a Lyapunov unstable Milnor attractor.\n\\end{repthm1}\n\n\n\n\\subsubsection{Homoclinic tangencies and instability}\n\n\n\\begin{cor}\\label{cor:mger}\nSuppose that in an open domain $U\\subset\\operatorname{Diff}^r(M),\\; r\\ge 1,$ there is a persistent tangency associated with a basic set $\\Lambda(F),\\; F\\in U,$ with a sectionally dissipative periodic saddle $p(F)$ that continuously depends on $F\\in U$. Then a topologically $C^m$-generic ($m\\ge r$) diffeomorphism in $U$ has a Lyapunov unstable Milnor attractor. \n\\end{cor}\n\\begin{proof}\nProposition \\ref{prop:dense} implies that diffeomorphisms with a tangency associated with the saddle $p(F)$ are $C^m$-dense in $U\\cap\\operatorname{Diff}^m(M)$. Then we can apply Theorem~\\ref{thm:NI} to $U\\cap\\operatorname{Diff}^m(M)$ and conclude the proof.\n\\end{proof}\n\n\\begin{cor}\\label{cor:generic2}\nFor any smooth compact two-dimensional manifold $M$ there exists a $C^2$-open set $U\\subset\\operatorname{Diff}^2(M)$ such that any $C^r$-generic ($r\\ge2$) diffeomorphism $F\\in U$ has a Lyapunov unstable Milnor attractor.\n\\end{cor}\n\\begin{proof}\nConsider an open set $U$ given by Theorem \\ref{thm:N}. Take any $F\\in U$ and note that after a small perturbation and, perhaps, time inversion, we may assume that in a $C^2$-neighborhood $V$ of $F$ the basic set $\\Lambda$ that exhibits the persistent tangency also has a dissipative saddle. Then Corollary~\\ref{cor:mger} applied to $V$ finishes the proof.\n\\end{proof}\n\n\\begin{cor}\\label{cor:generic3}\nFor every smooth compact manifold $M$ of dimension $k\\ge 3$ there exists a $C^1$-open domain $U\\subset{\\rm Diff^1}(M)$ such that any $C^r$-generic ($r\\ge1$) diffeomorphism in $U$ has a Lyapunov unstable Milnor attractor.\n\\end{cor}\n\\begin{proof}\nThis is verified by applying Corollary~\\ref{cor:mger} to the following theorem\\footnote{It is more convenient to give a reference to the work of M.~Asaoka here, though the construction that underlies this result can be found in \\cite{CIME}.}.\n\n\\begin{thm}[Asaoka, \\cite{Masa}]\\label{MA}\nFor any smooth manifold of dimension at least three there exists a $C^{\\infty}$-diffeomorphism such that it admits a hyperbolic basic set that contains a sectionally dissipative saddle and exhibits a $C^1$-persistent tangency.\n\\end{thm}\n\n\\end{proof}\n\n\n\\begin{rem}\\label{rem:easy}\nActually, one does not need Theorem \\ref{thm:NI} to prove either of Corollaries~\\ref{cor:generic2} and~\\ref{cor:generic3}.\n\nIn the two-dimensional case, as it was observed independently by Yu.~S.~Ilyashenko and A.~Okunev, we may take $F\\in{\\rm Diff^2}(M)$ as in Theorem~\\ref{thm:ptpv}, i.e. with a homoclinic tangency associated with a dissipative saddle $p$, and additionally require that $W^u(O(p))$ intersect the basin of some sink.\\footnote{Both conditions can be achieved by isotopically changing a linear mapping with a saddle at the origin, and this procedure may be adapted to any two-dimensional manifold, so the required map exists.} The latter property persists under small perturbations. Therefore we may assume that any diffeomorphism in the domain $U$ given by Theorem~\\ref{thm:ptpv} possesses this property. The next step would be to modify the proof of the two-dimensional version of Theorem~\\ref{thm:ptpv} given in \\cite{PT} in order to show that one may take the saddle $p_1$ that is homoclinically related to the continuation of $p$. This would imply, with the help of the inclination lemma, that for the maps in $U$ the unstable manifold $W^u(p_1(G))$ intersects the basin of the aforementioned sink as well. It is easy to show (and we will show it below) that for a $C^r$-generic diffeomorphism in $U$ sinks accumulate to (the continuation of) $p_1$. Then a $C^r$-generic diffeomorphism in $U$ satisfies both assumptions of Proposition~\\ref{prop:suff} with respect to the continuation of $p_1$ and therefore has an unstable Milnor attractor.\n\nMoreover, it turns out peculiarly that the construction presented by M.~Asaoka in \\cite{Masa} also ensures that generically both requirements of Proposition~\\ref{prop:suff} are satisfied. He utilizes a normally hyperbolic repelling disk such that the restriction of the mapping to this disk is a Plykin map (see the original paper \\cite{Ply} by Plykin and also \\cite{GUCK} for a slightly modified and somewhat simpler example of such map) with an area expanding saddle that plays a key role in constructing the persistent tangency. One may observe that the stable manifold of this saddle intersects the repelling basin of a source of the Plykin map. After the inversion of time one gets a sectionally dissipative saddle $p$ with $W^u(p)$ intersecting a basin of the sink that was the aforementioned source prior to time inversion. Then a variant of the Newhouse argument (or Baire argument, if you like) yields that sources accumulate to that saddle locally generically. \n\\end{rem}\n\n\\begin{cor}\\label{cor:approx}\nAny $C^2$-diffeomorphism $F$ exhibiting a homoclinic tangency associated with a sectionally dissipative periodic saddle $p$ belongs to the closure of a $C^2$-open set $U$ such that a generic diffeomorphism in $U$ has an unstable Milnor attractor. \n\\end{cor}\n\\begin{proof}\nTheorem \\ref{thm:ptpv} states that arbitrarily close to $F$ there is a domain with a persistent tangency associated with the continuation of some sectionally dissipative saddle $p_1$. Consider a sequence of such domains that approaches $F$ (saddles $p_1$ may be different for different domains). In each domain we apply Theorem~\\ref{thm:NI} to the persistent tangency associated with~$p_1$, which yields the genericity of maps with unstable attractors. Thus we can take the union of these domains as $U$. \n\\end{proof}\n\n\n\n\\subsubsection{Infinitely many sinks and instability}\n\n It must be mentioned that some authors define the \\emph{Newhouse phenomenon} as a generic coexistence of infinitely many sinks, not assuming any persistent tangency. Persistent tangencies for sectionally dissipative saddles imply generic coexistence of infinitely many sinks, but it is not known whether the converse is true. S.~Crovisier, E.~Pujals and M.~Sambarino have announced (see~\\cite[Cor. 4.5]{Crovisier14}) that, at least in $C^1$, locally generic coexistence of infinitely many sinks implies the density of homoclinic tangencies, which seems to be pretty close to having a persistent tangency.\n\n\\begin{thm}[Crovisier, Pujals, Sambarino]\\label{thm:CPS}\nFor any open set $V\\subset\\operatorname{Diff}^1(M)$, $M$ being compact, the following properties are equivalent:\n \\begin{itemize}\n \\item{Baire-generic diffeomorphisms in $V$ have infinitely many sinks,}\n \\item{densely in $V$ there exist diffeomorphisms exhibiting homoclinic tangencies associated with sectionally dissipative periodic points.}\n \\end{itemize}\n\\end{thm}\n\nSince homoclinic tangencies in this theorem appear, in general, for different sectionally dissipative saddles (we do not know whether those are heteroclinically related or not), Theorem~\\ref{thm:NI} does not yield the local genericity of diffeomorphisms with unstable Milnor attractors. However, since a $C^1$-diffeomorphism with a homoclinic tangency can be approximated by a $C^2$-diffeomorphism with a tangency related to the continuation of the same saddle, a straightforward application of Corollary~\\ref{cor:approx} provides the following statement.\n\n\\begin{cor}\\label{cor:dense_instability}\nSuppose $M$ is a smooth compact manifold and there is an open set $V\\subset\\operatorname{Diff}^1(M)$ such that a topologically generic diffeomorphism in $V$ has infinitely many sinks. Then densely in $V$ diffeomorphisms have Lyapunov unstable Milnor attractors. \n\\end{cor}\n\nTo sum up, the positive answer to the following problem seems rather plausible.\n\\begin{prob}\nIs it true that on compact manifolds the local Baire-generic coexistence of infinitely many sinks is always accompanied by the generic instability of Milnor attractors? \n\\end{prob}\n\nRecall that the subset of the phase space is called \\emph{asymptotically stable} if it is Lyapunov stable and attracts every point in some its neighborhood. The following argument due to A.~Okunev shows that coexistence of infinitely many sinks implies, at least, that the attractor lacks asymptotic stability. \n\n\\begin{thm}[A.~Okunev] If the diffeomorphism $F$ has infinitely many sinks, then its Milnor attractor is not asymptotically stable.\n\\end{thm}\n\\begin{proof}\nConsider a sink $\\gamma$ of $F$. Denote by $B$ the basin of attraction of $O(\\gamma)$. Its boundary $\\partial{B}$ is a closed invariant set. If it has a nonempty intersection with the attractor, then the attractor is Lyapunov unstable. Indeed, arbitrarily close to a point $a\\in \\partial{B}\\cap A_M$ one can take a point $b\\in B$ whose forward orbit inevitably passes through a wandering domain $U\\setminus \\overline{F(U)}$, \\ $U$ being some fixed dissipative neighborhood of $O(\\gamma)$.\n\nSuppose now that for every sink of the diffeomorphism $F$ the boundary of the basin of this sink is separated from the attractor. Note then that no point in this boundary is attracted to $A_M$.\nTake a sequence $(\\gamma_j)_{j\\in\\mathbb N}$ of sinks that converges to some point $z\\in M$ (as always, we assume $M$ to be compact). Since the attractor is closed, $z$ belongs to $A_M$. For every $\\gamma_j$ denote by $B_j$ the basin of $O(\\gamma_j)$. There is also a sequence of points $b_j\\in\\partial{B_j}$ that converges to the same point $z$. Indeed, for $\\gamma_j$ close to $z$ one can take a segment that connects $z$ with $\\gamma_j$ and find a point of $\\partial{B_j}$ inside this segment. Thus, we have obtained a sequence $(b_j)$ that converges to $z\\in A_M$ but consists of points that are not attracted to $A_M$, which is in contradiction with asymptotic stability. \n\\end{proof}\n\n\n\n\n\\subsection{Reduction of Theorem \\ref{thm:NI} to the capture lemma}\\label{sec:thmNI}\n\n\n\n\\begin{lem}[capture lemma]\\label{lem:capture}\nLet $F\\in{\\rm Diff}^r(M), \\; r\\ge 1,$ have a sectionally dissipative periodic hyperbolic saddle $p = p(F)$ with a homoclinic tangency. Then arbitrary $C^r$-close to $F$ there is a diffeomorphism $G$ for which $W^u(p(G))$ intersects the basin of a periodic sink. \n\\end{lem} \n\n\\begin{rem}\\label{rem:newhouse_gap}\nA two-dimensional version of this lemma can be found in the paper by S.~Newhouse \\cite[Lemma 2.2]{N_new} but there is a gap in the proof. Namely, that proof would work only if the eigenvalues $|\\lambda|<1<|\\sigma|$ of the saddle $p$ satisfied the inequality $|\\lambda\\sigma^2|<1.$ \n\\end{rem} \n\n\\begin{rem}\nAs J.~C.~Tatjer kindly informed the author, the two-dimensional version of the capture lemma is implied by the main result of the work \\cite{TS}(see Thm 5.8 there). Although it seems that the argument of \\cite{TS} can be generalized to the general case, below we use a different approach based on the paper \\cite{PV} by J.~Palis and M.~Viana. \n\\end{rem}\n\nNow we will reduce Theorem~\\ref{thm:NI} to the capture lemma, which we will prove in the next section.\n\nAssume as in Theorem~\\ref{thm:NI} that there is an open domain $U\\subset\\operatorname{Diff}^r(M), \\; r\\ge 1,$ with a persistent tangency associated with a sectionally dissipative periodic saddle $p$ (see Definition~\\ref{def:tan_p}) and take a diffeomorphism $F\\in U$ with a tangency related to $p(F)$. According to Prop.~1 of~\\cite{N74}, when unfolding a homoclinic tangency by a $C^r$-small perturbation, we can create a hyperbolic periodic sink that passes arbitrarily close to the point where the tangency was.\\footnote{J.~Palis and M.~Viana prove this for a generic unfolding of a tangency in~\\cite{PV} (see Remark~6.1), and we use a similar but a little simpler argument in the proof of the capture lemma, see Section~\\ref{sect:gcfeds}.} Since finite segments of orbits depend continuously on the initial point and the mapping, this proposition yields that for any $\\delta$ we can create a sink passing $\\delta$-close to the hyperbolic continuation of the saddle~$p$. Indeed, before the perturbation the point of tangency $q$ was in the stable manifold of $p(F)$, and therefore there were a neighborhood $V$ of $q$ and an integer $n$ such that $F^n(V)$ lied in the (open) $\\delta$-neighborhood of $p(F)$. The same is true for any diffeomorphism $G$ sufficiently close to $F$: $p(G)$ is close to $p(F)$, $G^n(V)$ is close to $F^n(V)$, and therefore $G^n(V)$ is in the $\\delta$-neighborhood of $p(G)$. So, if $G$ has a sink in $V$, this sink passes $\\delta$-close to $p(G)$. \n\nAt this point we can use a standard argument due to S. Newhouse. Namely, for each $n\\in\\mathbb N$ denote by $U_n$ the subset of $U$ that consists of diffeomorphisms $F$ with a sink at a distance less than $\\frac{1}{n}$ from the saddle $p(F)$. These sets possess the following properties.\n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n\\item{Each $U_n$ is $C^r$-open, because hyperbolic sinks have hyperbolic continuations.}\n\n\\item{Each $U_n$ is $C^r$-dense in $U$.\n\nIndeed, we can take any $F\\in U$, slightly perturb it to obtain a homoclinic tangency, then make another $C^r$-perturbation to unfold this tangency and create a sink that passes $\\frac 1 n$-close to the saddle $p(\\cdot)$, which yields a diffeomorphism in~$U_n$.} \n\\end{itemize}\n\nSince the sets $U_n$ are $C^r$-open and $C^r$-dense in $U\\subset\\operatorname{Diff}^r(M)$, the set $R = \\bigcap_n U_n$ is a residual subset of $U$. For every $F\\in R$ one can find a sink in any neighborhood of $p(F)$, which implies that there is a sequence of sinks that accumulates to this saddle.\n\nSuppose that Lemma~\\ref{lem:capture} is true. Then the diffeomorphisms $F$ for which $W^u(p(F))$ intersects a basin of some sink form a $C^r$-dense subset $C$ in $U$. Moreover, this subset is open due to the continuous dependence of local stable and unstable manifolds on the mapping. \n\nThe subset $R\\cap C$ is residual in $U$, and any diffeomorphism in this subset satisfies both assumptions of Proposition~\\ref{prop:suff}: it has sinks accumulating to a saddle whose unstable manifold intersects the basin of one of the sinks. Therefore, any such diffeomorphism has an unstable Milnor attractor. The proof of Theorem~\\ref{thm:NI} modulo Lemma~\\ref{lem:capture} is complete. \n\n\n\n\n\n\\section{Proof of the capture lemma}\n\n\\subsection{Model example}\nIn this section we discuss the simplest two-dimensional example of unfolding of a homoclinic tangency. This example shows, without introducing technical difficulties of the general case, how the sink emerges near the point where the tangency was and how a part of the unstable manifold of the saddle is captured by that sink. \n\n\\subsubsection*{Description of the family}\nTake two numbers $\\lambda, \\sigma \\in \\mathbb R$ such that $0<\\lambda<1<\\sigma $ and $\\lambda\\sigma <1$ and consider a plane $\\mathbb R^2$ with coordinates $x, y$ and a one-parameter family $\\{F_\\mu\\}_{\\mu\\in[-\\varepsilon , \\varepsilon ]}$ of $C^\\infty$-diffeomorphisms of that plane such that\n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n\\item{for any $\\mu$, the restriction of $F_\\mu$ to the rectangle $R_0 = [-2,2]^2$ has the form\n\\begin{equation}\\label{eq:fn}\nF_\\mu|_{R_0}\\colon (x, y)\\mapsto(\\lambda x, \\sigma y);\n\\end{equation}\n}\n\\item{there is a small rectangle $R_1$ centered at the point $r = (0, 1)$ and an integer $N\\in \\mathbb N$ such that for any $\\mu$ the restriction $\\left.F^N_\\mu\\right|_{R_1}$ has the form\n\\begin{equation}\\label{eq:fN}\n\\left.F^N_\\mu\\right|_{R_1}\\colon (x, y)\\mapsto \\left(y, \\;\\mu - x + (y-1)^2\\right).\n\\end{equation}\n}\n\\end{itemize} \n\nObviously, for any $F_\\mu$ there is a fixed dissipative saddle at the origin. We will denote this saddle $p$. The segments $[-2, 2]\\times \\{0\\}$ and $\\{0\\}\\times[-2, 2]$ are local stable and unstable manifolds of $p$ respectively. According to the second condition on the family, $F^N_\\mu$ maps the segment $R_1\\cap Oy \\subset W^u_{loc}(p, F_\\mu)$ into an arc $\\Gamma_\\mu$ of a parabola which shifts along the $y$-axis when we change the parameter, and when $\\mu = 0$, there is a homoclinic tangency associated with $p$ at the point $q = (1, 0)$ which is the vertex of $\\Gamma_0$ (Figure~\\ref{pic}).\n\n\\begin{figure}\n \\begin{center}\n\n \\begin{tikzpicture}[scale=5]\n \n \\coordinate (x) at (1.5, 0);\n \\coordinate (y) at (0, 1.35);\n \n \\draw (y) node[above] {$y$} -- (0,0) coordinate (p) node[below left] {$p$} -- (0,-0.35);\n \\filldraw [black] (p) circle (0.3pt);\n \\draw (-0.5, 0) -- (x) node[right] {$x$};\n \n \n \\filldraw [black] (1,0) coordinate (q) node[below, align = center] {$q$\\\\{\\footnotesize $(1,0)$}} circle (0.3pt);\n \\filldraw [black] (0,1) coordinate (r) node[left, align = center] {{\\scriptsize $(0, 1)$} $r$} circle (0.3pt);\n\n \n \\draw[>-<] (-0.25, 0) -- (0.25, 0);\n \\draw[<->] (0, -0.25) -- (0, 0.25);\n \\node[below] at (0.25, 0) {\\footnotesize $W^s_{loc}(p)$};\n \\node[left] at (0, 0.25) {\\footnotesize $W^u_{loc}(p)$};\n \n \n \\begin{scope}[yshift = 1cm, scale = 0.25]\n \\coordinate (s) at (0, 0);\n \\coordinate (a) at (-1, 1);\n \\coordinate (b) at (1, 1);\n \\coordinate (c) at (1, -1);\n \\coordinate (d) at (-1, -1);\n \n \\path[draw, dotted] (a) -- (b) -- (c) -- (d) -- cycle; \n \\node[below right] at (-1, 1) {$R_1$};\n \\end{scope}\n\n\n \n \\begin{scope}[xshift = 1cm, yshift = 0.15cm, scale = 0.2]\n \\coordinate (s) at (0, 0);\n \\coordinate (a) at (-1, 0.5);\n \\coordinate (b) at (1, 0.5);\n \\coordinate (c) at (1, -0.5);\n \\coordinate (d) at (-1,-0.5);\n \\filldraw [black] (s) node[right] {$s_n$} circle (0.5pt);\n \\path[draw, thick] (a) -- (b) -- (c) -- (d) -- cycle; \n \\node[right] at (1, 0) {$D_n$};\n \\end{scope}\n\n \\filldraw [black] (1,0.15) circle (0.3pt);\n\n \n \\begin{scope}[xshift = 0.10cm, yshift = 1cm, xscale = 0.05, yscale = 0.3]\n \\coordinate (s) at (0, 0);\n \\coordinate (a) at (-1, 0.5);\n \\coordinate (b) at (1, 0.5);\n \\coordinate (c) at (1, -0.5);\n \\coordinate (d) at (-1,-0.5);\n \\path[draw, thick, color = blue] (a) -- (b) -- (c) -- (d) -- cycle; \n \\end{scope}\n\n \n \\draw[thick, ->, color = blue] (0.7, 0.15) to [out = 180, in = -90] (0.10, 0.7);\n \\node[above right] at (0.3, 0.33) {$F^n_\\mu$};\n\n \n \\begin{scope}[xshift = 0.05cm, yshift = 0.03cm]\n \\draw[thick, ->, color = green] (0.25, 1.1) {[rounded corners] to [out = 10, in = 85] (1.05, 0.35)};\n \\node[above right] at (0.85, 0.95) {$F^N_\\mu$};\n \\end{scope}\n\n \n \\begin{scope}[xshift = 1cm, yshift = 0.185cm, xscale = 0.15, yscale = 0.1]\n \n \\coordinate (a) at (-1, 0.5);\n \\coordinate (b) at (1, 0.5);\n \\coordinate (c) at (1, -0.5);\n \\coordinate (d) at (-1,-0.5);\n \n \\draw[thick, color = green] (a) parabola[bend pos = 0.5] bend +(0,-0.4) (b) -- (c) parabola [bend pos = 0.5] bend +(0,-0.4) (d) -- cycle;\n \\draw[thick, color = magenta] (-2, 2) parabola[bend pos = 0.5] bend +(0,-1.6) (2, 2) node[above, right, color = black] {$\\Gamma_\\mu$}; \n \\end{scope}\n\n \\end{tikzpicture}\n \\caption{Model example.}\\label{pic}\n \\end{center}\n\\end{figure}\n\n\n\nWhen $\\mu$ is small and positive, $\\Gamma_\\mu$ is shifted upwards and does not intersect the $x$-axis. Take a small rectangle $R$ (with sides parallel to the axes) that lies between the point $q$ and $\\Gamma_\\mu$. Under the iterates of $F_\\mu$ this rectangle will be contracted along $x$-axis and stretched along $y$-axis. If the center of $R$ is at $(1, \\sigma ^{-n})$, then the $n$-th image of this rectangle is a thin and long rectangle near the point~$(0, 1)$. The latter rectangle, if it is contained in $R_1$, is mapped by $F^N_\\mu$ into a curvilinear figure that lies below $\\Gamma_\\mu$ and may intersect $R$. The idea is that by adjusting $\\mu$ and the size of the rectangle we can assure that $F^{n+N}_\\mu$ maps it into its interior and therefore there is a periodic point inside the rectangle $R$.\n\n\\subsubsection*{The sink}\nFirstly, we will show how to obtain a sink ``manually''. This will give some motivation to the renormalization technique below.\n\nSuppose that we are looking for a periodic point $s = (x_s, y_s)$ of $F_\\mu$ such that $s$ lies in the vicinity of $q$ and goes to the vicinity of the point $r$ after $n$ iterations, and then returns to its original position after another $N$ iterations. If it exists, then $F^n_\\mu(x_s, y_s) = (\\lambda^nx_s, \\sigma ^nx_s)$ lies in $R_1$ and substituting this into (\\ref{eq:fN}) yields the analytic expression for $F^{n+N}_\\mu(\\cdot) = \\left(F^N_\\mu\\circ F^n_\\mu\\right)(\\cdot)$ in a neighborhood of $s$:\n\\begin{equation}\\label{eq:fnN}\nF^{n+N}_\\mu(x, y) = (\\sigma ^ny, \\; \\mu - \\lambda^nx + (\\sigma ^ny-1)^2).\n\\end{equation}\nThen we can try to find $s$ by solving the equation\n$$(x_s, y_s) = (\\sigma ^ny_s, \\; \\mu - \\lambda^nx_s + (\\sigma ^ny_s-1)^2).$$\n\nLet us take $x_s = 1$. Then $y_s = \\sigma ^{-n}$, and we have to set $\\mu = \\mu_n = \\sigma ^{-n}+\\lambda^n$. We have obtained the solution $(x_s, y_s, \\mu) = (1,\\; \\sigma ^{-n},\\; \\sigma ^{-n}+\\lambda^n)$. Note that $s \\to q$ and $\\mu_n\\to 0$ as $n\\to\\infty$. Now it is easy to verify that for sufficiently large values of $n$ we have $F^n_{\\mu_n}(x_s, y_s)\\in R_1$, which implies that $F^{n+N}_{\\mu_n}$ restricted to the neighborhood of $s$ is given by expression~(\\ref{eq:fnN}). Thus, the point $s$ is indeed a periodic point for the given value of $\\mu$.\n\nUsing (\\ref{eq:fnN}) we can write the Jacoby matrix of $F^{n+N}_{\\mu_n}$ in the neighborhood of $s$ explicitly:\n\\begin{equation}\\label{eq:jacoby}\ndF^{n+N}_{\\mu_n}=\n\\left(\n \\begin{array}{cc}\n 0 & \\sigma ^n \\\\\n -\\lambda^n & 2\\sigma ^n(\\sigma ^ny - 1)\\\\\n \\end{array}\n\\right)\n\\end{equation}\n\nThe matrix of $dF^{n+N}_{\\mu_n}(s)$ can be obtained by a substitution $y = y_s = \\sigma ^{-n}$. The eigenvalues of this matrix are $\\pm i(\\lambda\\sigma )^{n\/2}$. Since $\\lambda\\sigma < 1$, the moduli of eigenvalues are smaller than one and the point $s$ is a periodic sink.\n\nTo sum up, for any $n$ we have obtained a parameter value $\\mu_n$ such that the map $F^{n+N}_{\\mu_n}$ has a sink $s_n$:\n$$ s_n = (1, \\sigma ^{-n}), \\ \\mu_n = \\lambda^n + \\sigma ^{-n}.$$\n\n\n\\subsubsection*{Renormalization}\n Now we can use an argument utilizing a renormalization technique as in \\cite[\\S 3.4]{PT}. Consider the following $n$-dependent affine change of coordinates:\n\\begin{equation}\\label{eq:newcoords}\n H_n\\colon (x, y)\\mapsto (X, Y) = (\\sigma^n(x - 1),\\ \\sigma^{2n}(y - \\sigma^{-n})).\n\\end{equation}\nThe origin of the new $X,Y$-coordinates is at the point $(1, \\sigma ^{-n})$ in the $x,y$-coordinates, and the square $K = [-1,1]^2$ in new coordinates corresponds to the following rectangle $D_n$:\n$$\nD_n = H_n^{-1}(K) = \\{ (x,y)\\mid |x - 1| \\le \\sigma ^{-n}, \\ |y - \\sigma ^{-n}| \\le \\sigma ^{-2n}\\}.\n$$\nFrom now on we will consider only large values of $n$ and assume that $F^n_\\mu(D_n)\\subset R_1.$\n\nLet us calculate the expression for $\\mathcal F_{n,\\mu} = H_n \\circ F_{\\mu}^{n+N} \\circ H_n^{-1}$ at $K$, i.e., the expression for $F_\\mu^{n+N}$ at $D_n$ in our new coordinates:\n$$\n(X,Y) \\stackrel{H_n^{-1}}{\\mapsto }(x, y) = (\\sigma ^{-n}X + 1, \\ \\sigma ^{-2n}Y + \\sigma ^{-n})) \\stackrel{F_{\\mu}^{n+N}}{\\mapsto }\n$$\n$$\n\\stackrel{F_{\\mu}^{n+N}}{\\mapsto }(\\sigma ^{-n}Y + 1, \\ \\sigma ^{-2n}Y^2 - \\lambda^n(\\sigma ^{-n}X + 1) + \\mu ) \\stackrel{H_n}{\\mapsto }\n$$\n$$\n\\stackrel{H_n}{\\mapsto } (Y, \\ Y^2 - \\lambda^n\\sigma ^nX + \\sigma ^{2n}(\\mu - (\\sigma ^{-n} + \\lambda^n))) =\n\\mathcal F_{n,\\mu }(X,Y).\n$$ \n\nConsider the following $n$-dependent reparametrization defined in a neighborhood of $\\mu_n = \\sigma ^{-n} + \\lambda^n$:\n\\begin{equation}\\label{eq:reparam}\n\\nu = \\sigma ^{2n}(\\mu - (\\sigma ^{-n} + \\lambda^n)).\n\\end{equation}\nThen $|\\nu| \\le 1$ corresponds to $|\\mu - (\\sigma ^{-n} + \\lambda^n)| \\le \\sigma ^{-2n}$, and we can suppose that the reparametrization is defined for $\\mu$ in a segment of radius $\\sigma ^{-2n}$ centered at $\\mu_n = \\sigma ^{-n} + \\lambda^n$ and that $\\nu\\in [-1, 1]$. \n\nLet $\\mathcal G_{n, \\nu(\\mu)} := \\mathcal F_{n,\\mu}.$ Since \n$$\\mathcal G_{n, \\nu}(X,Y) = (Y, \\ Y^2 - (\\lambda^n\\sigma ^n)X + \\nu),$$ \nand $(\\lambda\\sigma )^n \\to 0$ as $n\\to \\infty,$ we have\n\\begin{equation}\\label{eq:conv}\n\\mathcal G_{n,\\nu }(X,Y) \\rightrightarrows (Y, Y^2 + \\nu )=: \\mathcal G_\\nu(X, Y) \\mbox{ for } (X, Y)\\in K, \\text{ and } |\\nu|\\le 1 \\ (n\\to\\infty).\n\\end{equation}\nNote that if we take any $k>1$, then for $(X, Y)\\in k\\cdot K = [-k, k]^2$ and $\\nu\\in[-k, k]$ the aforementioned reparametrization (\\ref{eq:reparam}) and coordinate change (\\ref{eq:newcoords}) (more precisely, the inverse ones) will be well-defined for sufficiently large values of $n$ and, moreover, the same uniform convergence will take place. Of course, in this case $H_n^{-1}(k\\cdot K)$ will be a rectangle $D_{n,k}$ with the same center as $D_n$ but $k$ times larger, and the original parameter $\\mu$ will take on values in the segment \n$\\left[\\mu_n - k\\sigma ^{-2n}, \\; \\mu_n + k\\sigma ^{-2n}\\right]$.\n\nThe map $\\mathcal G_\\nu$ (viewed as a map from $\\mathbb R^2$ to $\\mathbb R^2$) ``forgets'' the $X$-coordinate, maps the square $K$ into an arc of parabola, and is semi-conjugated with the map $Y\\mapsto Y^2+\\nu$ (from $\\mathbb R$ to $\\mathbb R$). For $\\nu$ small the latter map has exactly two fixed points: a sink at $a_\\nu$ close to $\\nu$ and a source at $r_\\nu$ close to $1 - \\nu$, and the orbit of any point $y \\in [-r_\\nu, r_\\nu]$ is attracted to the sink. Thus, for any $r < r_\\nu$ the rectangle\n$$\nB: |Y| \\le r, \\ |X| \\le 1\n$$\nis attracted to the sink $(a_\\nu, a_\\nu)$ of $\\mathcal G_\\nu$, and the distance between the $j$-th image of a point and the sink decreases uniformly on $B$ as $j$ increases.\n\nSince the convergence $\\mathcal G_{n,\\nu }|_K\\to \\mathcal G_\\nu|_K$ also holds in $C^1$ (in fact, it holds even in $C^\\infty$), for sufficiently large $n$ and small $\\nu$ the map $\\mathcal G_{n,\\nu}$ has a sink $s_{n,\\nu}$ close to $(a_\\nu, a_\\nu)$, and it is not difficult to show that $B$ is in its basin of attraction: it is enough to notice that it takes a uniformly limited number of iterations for $B$ to plunge into a dissipative domain where all maps $C^1$-close to $\\mathcal G_n$ are contracting in a suitable metric. \n\n \n\n\n\n\\subsubsection*{The capture}\n\nSuppose additionally that\n\\begin{equation}\\label{cond:super_dissipative}\n\\lambda\\sigma ^2 < 1.\n\\end{equation}\nWe will call saddles that satisfy such inequality \\emph{extremely dissipative}\\footnote{\\emph{Strongly dissipative} sounds better, but since some authors use this name for sectionally dissipative saddles, we would rather introduce extremely dissipative saddles to prevent any ambiguity.}.\n\nLet us show that for the diffeomorphism $F_\\mu$ with any $\\mu$ sufficiently close to $\\mu_n = \\sigma ^{-n} + \\lambda^n$ the unstable manifold of the saddle $p$ intersects the attraction basin of the continuation of the sink $H_n^{-1}(s_{n,\\nu(\\mu_n)})$.\n\nRecall that in the original $x,y$-coordinates we had $r = (0,1) \\in W^u(p, F_\\mu)$ and $F^N_\\mu(r) = (1,\\mu)\\in W^u(p, F_\\mu)$.\nCondition~\\ref{cond:super_dissipative} implies that $\\lambda^n\\sigma ^{2n}\\to 0$ as $n\\to \\infty$, and hence we have:\n$$H_n(1,\\mu ) = (0,\\ \\sigma ^{2n}\\mu - \\sigma ^{n} ) = (0,\\ \\nu + \\lambda^n\\sigma ^{2n})\\to (0, \\nu).$$\nObviously, when $n$ is large and $\\nu$ is small, $H_n(1,\\mu ) \\in B,$ that is, the capture happens. \n\n\\bigskip\nCondition (\\ref{cond:super_dissipative}) is important for the capture, because when it is violated, the point $(1,\\mu)$ may be outside the basin of attraction of the sink $H_n^{-1}(s_{n,\\nu(\\mu)})$. For example, take $\\lambda, \\sigma $ such that $\\sigma >10$ and $\\lambda^2\\sigma ^3 = 1$ and consider again $\\mu = \\mu_n = \\sigma ^{-n} + \\lambda^n$. Then it is easy to check by a direct calculation using expression (\\ref{eq:fnN}) that $F^{2n+N}_{\\mu_n}(1, \\mu_n) = (\\lambda^n+\\lambda^{2n}\\sigma ^n, \\; 2)\\notin R_1$ and $F^{2n+N-1}_{\\mu_n}(1, \\mu_n) = (\\lambda^{n-1}+\\lambda^{2n-1}\\sigma ^n, \\; 2\\sigma ^{-n})\\notin R_1$. Then the image of $(1,\\mu_n)$ missed the rectangle $R_1$ on the second ``wind'', and one can add some additional conditions on the family to assure that this orbit never returns to the vicinity of our sink: for example, one can require that, for every map in the family, some neighborhood of the point $(0, 2)$ be attracted to another sink.\n\n However, due to the complexity of dynamics introduced by homoclinic tangencies and intersections, it is difficult to see whether the global unstable manifold of $p$ intersects the basin of the sink $H_n^{-1}(s_{n,\\nu(\\mu)})$ when $p$ is not extremely dissipative. In the general case we will use the following trick: a certain perturbation of the map with a homoclinic tangency associated with the saddle $p$ yields a diffeomorphism with an extremely dissipative saddle heteroclinically related to the original one and with a new homoclinic tangency associated with this new saddle. \n\n\n\n\n\n\n\\subsection{General case for extremely dissipative saddles}\n\\label{sect:gcfeds}\n\nLet us begin with a general definition of an extremely dissipative saddle in any dimension. Consider a periodic hyperbolic sectionally dissipative saddle $p$ of a map $F$. Let $\\sigma $ be the (unique) expanding eigenvalue of $dF^{{\\rm per\\,}(p)}(p)$ and $\\lambda$ be the restriction of $dF^{{\\rm per\\,}(p)}(p)$ to the hyperplane $E^s_p$ of contracting eigenvectors. Recall that for any norm on $E^s_p$ the corresponding operator norm of $\\lambda$ is defined. \n\n\\begin{defin}\nA periodic hyperbolic sectionally dissipative saddle $p$ of a map $F$ is called \\emph{extremely dissipative} if there is a norm on $E^s_p$ such that the following inequality holds:\n\\begin{equation}\\label{eq:exdis}\n\\|\\lambda\\|\\cdot \\sigma ^2 < 1.\n\\end{equation}\n\\end{defin}\n\nIn this subsection we will prove the capture lemma for extremely dissipative saddles.\n\n\\medskip\n\n\\subsubsection*{Preliminary perturbations}\n\nSuppose $F$ is a $C^r$-smooth ($r\\ge 1$) diffeomorphism of $M$ with a homoclinic tangency for an extremely dissipative saddle $p$. Then $F$ can be $C^r$-approximated by a $C^\\infty$-diffeomorphism $\\hat F$ with the following properties:\n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n\\item $\\hat F$ has a non-degenerate homoclinic tangency associated with the con\\-tinuation of~$p$;\n\\item the saddle $p$ is non-resonant for $\\hat F$, and all eigenvalues of this saddle are different.\n\\end{itemize}\nThus, without loss of generality we can assume from the very beginning that $F$ is $C^\\infty$-smooth and possesses these properties. To prove the capture lemma for extremely dissipative saddles, it suffices to show that such a diffeomorphism can be $C^\\infty$-approximated by a diffeomorphism for which the unstable manifold of the continuation of $p$ intersects a basin of a sink.\n\n\\subsubsection*{Linearizing coordinates}\n\nSince $p$ is now non-resonant, by Sternberg's theorem there is a neighborhood of $p$ where $F^{{\\rm per\\,}(p)}$ can be linearized by a suitable $C^2$-smooth change of coordinates. We will assume for simplicity that $p$ is a fixed saddle, but the argument is almost literally the same for a periodic saddle.\n\nLet us denote by $x, y$ the linearizing coordinates in the neighborhood of $p$: $x\\in\\mathbb R^{m-1}, \\ y\\in\\mathbb R, \\ m = \\dim M.$ For the sake of simplicity we identify the points in the domain of the linearizing chart with their images in the codomain. We may assume that in our linearizing coordinates $p$ is the origin, $W^s_{loc}(p)\\subset\\{y = 0\\}$, and $W^u_{loc}(p)\\subset\\{x = 0\\}.$ Then, after a linear change of coordinates that preserves the $y$-axis and the $x$-hyperplane, we can also assume that\n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n\\item[-]{the point $r = (0, 1)\\in W^u_{loc}(p)$ belongs to the orbit of tangency and $F^N$ takes $r$ to the point $q = (e, 0)\\in W^s_{loc}(p)$, the euclidean norm of $e$ in $\\mathbb R^{m-1}$ being smaller than 1;} \n\\item[-]{the linear map $\\lambda$\n contracts the unit ball $B$ of the $x$-plane;}\n\\item[-]{our coordinates linearize $F$ in a neighborhood $R_0$ of the cylinder $B\\times [-1, 1]$.}\n\\end{itemize}\nLet $R_1$ be a small neighborhood of the point $r$. If we write $F^N|_{R_1}$ in the following form:\n$$\\left.F^N\\right|_{R_1}: (x, y) \\mapsto (\\mathcal A(x, y), \\mathcal B(x,y)),$$ \nthen, since there is a quadratic tangency at $q$, the maps $\\mathcal A\\colon \\mathbb R^m\\to \\mathbb R^{m-1}$ and $\\mathcal B\\colon \\mathbb R^m\\to\\mathbb R$ will satisfy:\n\\begin{equation}\\label{eq:cond0}\n\\mathcal A(0,1) = e, \\ \\mathcal B(0, 1) = \\partial_y\\mathcal B(0, 1) = 0, \\;\\; \\partial_{yy}\\mathcal B(0, 1)\\ne 0.\n\\end{equation}\n\n\\subsubsection*{A special family}\n\nNow let us choose a one-parameter $C^\\infty$-family $(F_\\mu)$ that passes through our diffeomorphism~$F$. The required perturbation will then be obtained by an arbitrary small shift along the parameter, as we did in the model case.\n\n When choosing the family we should keep in mind that, first, we want $F^N_\\mu$ to have the simplest possible form in the neighborhood $R_1$ of $r$, second, we do not really want the map to change inside $R_0$ when we change the parameter (in particular, we want our saddle to stay at the same place and have the same eigenvalues), and third, we want the tangency to be unfolded non-degenerately. To satisfy these requirements we will consider a family $(F_\\mu)_{\\mu\\in[-\\varepsilon , \\varepsilon ]}$ such that\n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n \\item[-]{$F_0 = F$ and any $F_\\mu$ coincides with $F$ outside a small neighborhood of the point $z = F^{-1}(q) = F^{N-1}(r)\\notin R_0$,}\n \\item[-]{when we change the parameter, the $F_\\mu$-image of every point in some even smaller neighborhood of $z$ shifts in the $y$-direction with the same speed equal to 1.}\n\\end{itemize}\nIn fact, keeping in mind restrictions (\\ref{eq:cond0}) we can make $F^N_\\mu$ have the following form in the neighborhood $R_1$ of $r$:\n\\begin{equation}\\label{eq:fNgeneral}\n\\left.F^N_\\mu\\right|_{R_1}\\colon (x, y)\\mapsto \\left(e + a\\cdot(y-1) + \\gamma x + \\rho_1(x,y), \\;\\mu - cx + b(y-1)^2 + \\rho_2(x, y)\\right),\n\\end{equation}\nwhere $a\\in\\mathbb R^{m-1}, \\ b\\in\\mathbb R,$ $c\\colon \\mathbb R^{m-1}\\to \\mathbb R$ and $\\gamma\\colon \\mathbb R^{m-1}\\to \\mathbb R^{m-1}$ are linear maps, and $\\rho_{1},\\rho_2$ are remainder terms which satisfy the following conditions: \n\n\\begin{equation}\\label{rho_conditions}\nj^1_{(0,1)}\\rho_1 = 0, \\ j^1_{(0,1)}\\rho_2 = 0,\\;\\; \\partial_{yy}\\rho_2(0,1) = 0.\n\\end{equation}\nHere $j^1_t$ stands for the 1-jet of the map at the point $t$. Note that by choosing a proper family and a sufficiently small neighborhood $R_1$ we made the remainder terms independent of $\\mu$.\n \nExpression (\\ref{eq:fNgeneral}) resembles expression~(\\ref{eq:fN}) from the model example: the only difference is that several new ``coefficients'' and these two remainder terms have appeared.\n\nAs we have already mentioned, in our special family the restriction $F_\\mu|_{R_0}$ does not depend on $\\mu$:\n\\begin{equation}\\label{eq:fngeneral}\nF_\\mu|_{R_0}\\colon (x, y)\\mapsto(\\lambda x, \\sigma y).\n\\end{equation}\n\n\n\\subsubsection*{Renormalization}\nNow we can use the renormalization technique as in the model case. We follow \\cite{PT, PV}, as before.\n\n\nConsider the following $n$-dependent affine change of coordinates:\n\\begin{equation}\\label{eq:newcoordsgeneral}\nH_n: X = \\sigma^n(x - e), \\ Y = b\\sigma^{2n}(y - \\sigma^{-n}).\n\\end{equation}\nIn what follows we will consider even $n$ only, so $\\sigma ^n$ will always be positive.\nThe origin of the new coordinates is at the point $(e, \\sigma ^{-n})$ in the $x,y$-coordinates and the cube $K = [-k, k]^m, \\; k\\ge 2,$ in the new coordinates corresponds to the following parallelotope $D_{n, k}$ in the original coordinates:\n$$\nD_{n,k} = H_n^{-1}(K) = \\{ (x,y)\\mid |y - \\sigma ^{-n}| \\le k\\sigma ^{-2n}|b^{-1}|, \\ |(x - e)_j| \\le k\\sigma ^{-n}, \\; j = 1,\\dots, m-1 \\}.\n$$\n\nAs in the model case, we calculate the expression for $\\mathcal F_{n,\\mu} = H_n \\circ F_{\\mu}^{n+N} \\circ H_n^{-1}$ at $K$ assuming that $n$ is sufficiently large:\n$$\n(X,Y) \\stackrel{H_n^{-1}}{\\mapsto }(x, y) = (\\sigma ^{-n}X + e, \\ b^{-1}\\sigma ^{-2n}Y + \\sigma ^{-n})) \\stackrel{F_{\\mu}^{n+N}}{\\mapsto }\n$$\n$$\n\\stackrel{F_{\\mu}^{n+N}}{\\mapsto }\\left(e + a\\cdot \\frac{\\sigma ^{-n}}{b}Y + \\gamma(\\lambda^n(\\sigma ^{-n}X + e)) + \\hat\\rho_1, \\ \\mu - c\\lambda^n(\\sigma ^{-n}X + e) + b(b^{-1}\\sigma ^{-n}Y)^2 + \\hat{\\rho}_2\\right ) \\stackrel{H_n}{\\mapsto }\n$$\n$$\n \\left(ab^{-1}Y + \\gamma(\\lambda^n(X + \\sigma ^{n}e)) + \\sigma ^n\\hat{\\rho}_1, \\ Y^2 - cb\\sigma ^n\\lambda^n(X) + b\\sigma ^{2n}(\\mu - (\\sigma ^{-n} + c\\lambda^n(e))) + b\\sigma ^{2n}\\hat{\\rho}_2\\right)\n$$\n\nHere \n$$\\hat\\rho_i(X, Y) = \\rho_i(\\lambda^n(\\sigma ^{-n}X + e), 1+b^{-1}\\sigma ^{-n}Y), \\; i = 1,2.$$\n Note that, since the saddle $p$ is dissipative, we have $\\|\\lambda^n\\|\\sigma ^n \\to 0$ as $n\\to \\infty$ for any operator norm of $\\lambda$. Then, obviously, $\\|\\gamma(\\lambda^n(\\sigma ^{n}e))\\|\\to 0$, $\\|\\gamma\\circ\\lambda^n\\|\\to 0$ and $\\|cb\\sigma ^n\\cdot\\lambda^n\\|\\to 0$ as $n\\to\\infty$. Furthermore, it is easy to check that conditions (\\ref{rho_conditions}) yield that\n$$\\sigma ^n\\hat\\rho_1|_{K} \\to 0, \\ \\sigma ^{2n}\\hat\\rho_2|_{K} \\to 0 \\text{ in } C^1.$$\n \nNow let us make, for $n$ sufficiently large, an $n$-dependent reparametrization\n\\begin{equation}\\label{eq:reparamgeneral}\n\\nu = b\\sigma ^{2n}(\\mu - (\\sigma ^{-n} + c\\circ\\lambda^n(e)))\n\\end{equation}\ndefined in a closed $k\\frac{\\sigma ^{-2n}}{|b|}$-neighborhood of $\\mu_n = \\sigma ^{-n} + c\\circ\\lambda^n(e)$. Denote this neighborhood by $I_n$. Then $I_n$ corresponds to the interval $[-k, k]$ in the new parameter space. \n\n\n\nTake $A = a\\cdot b^{-1}\\in \\mathbb R^{m-1}$ and consider the family of maps\n $$\\mathcal G_\\nu\\colon \\mathbb R^{m-1}\\times\\mathbb R\\to\\mathbb R^{m-1}\\times\\mathbb R\\colon (X, Y) \\mapsto (AY, \\; Y^2+\\nu).$$ \nWe can conclude now that for $\\mathcal G_{n, \\nu} := \\mathcal F_{n,\\mu(\\nu)}$ restricted to $K$ the following convergence takes place:\n\n\\begin{equation}\\label{eq:convgeneral}\n\\mathcal G_{n,\\nu }|_K \\xrightarrow{C^1} \\mathcal G_\\nu|_K \\text{ uniformly for } |\\nu|\\le 1 \\text{ as } n\\to\\infty.\n\\end{equation}\n\n\n\n\\subsubsection*{The sink and the capture}\n\nAnalogously to the model example, for $\\nu$ close to zero the map $\\mathcal G_\\nu$ has a sink $(Aa_\\nu, a_\\nu)$, and there is a number $\\delta > 0$ independent of $\\nu$ such that the cube $B$:\n$$\nB: |Y| \\le \\delta, \\ |X_i| \\le \\delta, \\ i = 1,\\dots, m-1,\n$$\nis in the basin of this sink. \n\nThen the convergence (\\ref{eq:convgeneral}) implies, for $n$ sufficiently large and $\\nu$ sufficiently small, that the map $\\mathcal G_{n,\\nu}$ has a sink $s_{n,\\nu}$ close to $(Aa_\\nu, a_\\nu)$ and that $B$ is in the basin of attraction of this sink.\n\nFinally, we calculate the $X,Y$-coordinates of the point $F^N_\\mu(r)\\in W^u(p, F_\\mu)$. Since this point has $x,y$-coordinates equal to $(e, \\mu)$, we have\n\n$$H_n(e, \\mu) = (0, \\nu + bc(\\sigma ^{2n}\\lambda^n(e)))\\to (0, \\nu) \\text { as } n\\to\\infty.$$\nThis convergence holds because the saddle is extremely dissipative. Note that though the definition of extreme dissipativity requires condition (\\ref{eq:exdis}) to hold in some suitable norm, it nevertheless implies the convergence $\\|\\sigma ^{2n}\\lambda^n\\|\\to 0$ (as $n\\to \\infty$) in any norm. \nWe see that when $n$ is large and $\\nu$ is small, $H_n(e, \\mu)$ is in $B$. We have obtained a diffeomorphism $F_\\mu$ for which the unstable manifold of the saddle $p$ intersects the basin of the sink. Recall that the reparametrization was defined for $\\mu\\in I_n$ and the intervals $I_n$ are close to zero when $n$ is large, therefore such $F_\\mu$ can be taken arbitrarily close to $F$, which finishes the proof. \n\n\\medskip\n\n\\begin{rem}\nWe presented the argument for a fixed saddle but, as we have already mentioned, essentially the same argument works for a periodic saddle even if the tangency involves the invariant manifolds of different points of the orbit. The only difference is that for a given $n$ we should consider $F^{n\\cdot\\operatorname{per}(p)+N}_\\mu$ instead of $F^{n+N}_\\mu$.\n\\end{rem}\n\n\n\\begin{rem}\\label{rem:genfamily}\nThe argument of this subsection can be modified in order to show that the capture happens when a quadratic homoclinic tangency associated with an extremely dissipative saddle is unfolded in a generic family of $C^r$-diffeomorphisms, and then one can also get rid of the extreme dissipativity assumption essentially in the same way as it is done in the next subsection. However, we do not need this stronger version of the capture lemma in this paper.\n\\end{rem}\n\n\n\n\\subsection{Extreme dissipativity requirement eliminated}\n\nNow we proceed to the case when the saddle $p$ is not extremely dissipative. As in Section~\\ref{sect:gcfeds}, we can assume that our map $F$ is $C^\\infty$-smooth, the linearization theorem is applicable in the neighborhood of $p$, and the tangency is quadratic. Every perturbation discussed in this section is supposed to be $C^\\infty$-small.\n\n\\subsubsection*{Finding another saddle}\n\nImagine that after another small perturbation we find an extremely dissipative saddle $p'$ heteroclinically related to the (continuation of the) original saddle and, more than that, there is a homoclinic tangency associated with $p'$. Then we can apply our capture lemma to extremely dissipative saddle $p'$ and conclude that after another perturbation $W^u(p')$ intersects the basin of a sink. We preserve the original notation $p, p'$ for the continuations of our saddles. \n\nSince two saddles are heteroclinically related, $\\lambda$-lemma implies that $W^u(O(p))$ accumulates on $W^u(O(p'))$. But then $W^u(O(p))$ intersects the basin of the same sink, which proves our lemma in the general case.\n\nThe renormalization technique provides a natural candidate for such a saddle $p'$. \nConsider the map $\\mathcal G_\\nu$ (introduced in the previous subsection) for $\\nu = -2$:\n$$(X, Y) \\mapsto (AY, Y^2 - 2).$$\nIt has a fixed saddle point $(2A, 2)$ with eigenvalues 4 and 0, the latter being of multiplicity $m-1$. \nSince the maps $\\mathcal G_{n, -2}$ approximate the map $\\mathcal G_{-2}$ in $C^1$ as $n\\to\\infty$, for sufficiently large values of $n$ and for $\\nu\\in [-2-\\varepsilon , -2+\\varepsilon ]$ the maps $\\mathcal G_{n, \\nu}$ have extremely dissipative fixed saddles $p_{n, \\nu}$ close to $(2A, 2)$. Indeed, for $n$ large, $m-1$ eigenvalues at the saddle $p_{n, \\nu}$ are close to zero and the last eigenvalue is close to 4, thus this saddle has to be extremely dissipative. Then the point $\\tilde{p}_{n, \\mu(\\nu)} = H_n^{-1}(p_{n, \\nu})$ is an extremely dissipative periodic point of the corresponding diffeomorphism $F_{\\mu(\\nu)}$. Such a saddle is a natural candidate for the role of the saddle $p'$ because, as the following proposition shows, for some $\\nu_0$ close to $-2$ there is a homoclinic tangency associated with the saddle $\\tilde{p}_{n, \\mu(\\nu_0)}$, provided that $n$ is sufficiently large. \n\n\n\\begin{prop}\\label{prop:renormsaddletan}\nLet $\\varepsilon >0$ be fixed. Then, for $n$ sufficiently large, there is $\\nu_0\\in {[-2-\\varepsilon , -2+\\varepsilon ]}$ such that the map $\\mathcal G_{n, \\nu_0}$ has a homoclinic tangency associated with the saddle~$p_{n, \\nu_0}$.\n\\end{prop}\nThe proof for the two-dimensional case can be found in \\cite{PT}(\\S 6.3, Prop. 2), and the generalization of this proof for higher dimensions is straightforward.\n\\begin{comment}\nConsider the map $\\mathcal G_{\\nu}$ for $\\nu$ close to $-2$ and denote by $p_{\\nu} = (x_\\nu, y_\\nu)$ the continuation of the saddle $(2A, 2)$ of $\\mathcal G_{-2}$. It is not difficult to show that when $\\nu > -2$, the unstable separatrix of $p_{\\nu}$ \\todo{define} that goes down relative to the $y$-axis does not intersect the hyperplane $H_\\nu = \\{(x, y)\\mid y = y_\\nu\\}\\subset W^s(p_\\nu)$, but when $\\nu < -2$ this separatrix intersects this plane transversely. Now consider the map $\\mathcal G_{n, -2-\\varepsilon }$ for a large $n$. Recall that compact disks of the invariant manifolds of hyperbolic saddles depend continuously on the map and that the map $\\mathcal G_{n, -2-\\varepsilon }$ is close to $\\mathcal G_{-2-\\varepsilon }$ in $K = [-k, k]^m$. Then there are a disk in $W^s(p_{n, -2-\\varepsilon })$ and a segment in $W^u(p_{n, -2-\\varepsilon })$ which intersect transversely at some point far from the boundaries of both the disk and the segment while for the map $\\mathcal G_{n, -2+\\varepsilon }$ the continuations of these disk and segment do not intersect. Then for some $\\nu_0\\in[-2-\\varepsilon , -2+\\varepsilon ]$ there is a homoclinic tangency associated with the saddle $p_{n, \\nu_0}$ of $\\mathcal G_{n, \\nu_0}$.\n\\end{comment} \nApplication of $H_n^{-1}$ yields the corresponding tangency for the saddle $H_n^{-1}(p_{n, \\nu_0})$ of $F_{\\mu(\\nu_0)}$. \n\n\n\n\\subsubsection*{Heteroclinic relations} \n\n\nJ.~Palis and M.~Viana study the saddles $\\tilde{p}_{n, \\mu(\\nu)} = H_n^{-1}(p_{n, \\nu}), \\ \\nu\\in[-2-\\varepsilon , -2+\\varepsilon ],$ in~\\cite{PV} to construct basic sets with large stable thickness. They prove that, under some assumptions on the original saddle $p$ and on the tangency unfolded to create the saddle $\\tilde{p}_{n, \\mu}$, this latter saddle is heteroclinically related to the continuation of $p$. In particular, they consider only tangencies between invariant manifolds of one and the same periodic saddle while we allow tangencies between invariant manifolds of different points of a periodic orbit. \n\nSince we want to prove capture lemma for any sectionally dissipative saddle and any tangency, some additional work has to be done in order to use their results. Our plan is to replace, if necessary, the original saddle, or rather its continuation, with a heteroclinically related saddle, simultaneously obtaining a tangency associated with this new saddle, and then to replace this tangency by another one so that both the new saddle and the new tangency satisfy the aforementioned assumptions of~\\cite{PV}. Then the unfolding of this last tangency will yield an extremely dissipative saddle heteroclinically related to the continuations of both previous saddles. \n\n\\bigskip\n\n \n\n\n At this point we assume that $F\\in\\operatorname{Diff}^\\infty(M)$ has a sectionally dissipative periodic saddle~$p$, that $F^{\\operatorname{per} p}$ is linearizable in the neighborhood of $p$, and that the tangency associated with the saddle $p$ is quadratic. We will use the notation introduced in the subsection of Section~\\ref{sect:gcfeds} devoted to linearizing coordinates, i.e., we will suppose that these coordinates are defined in a neighborhood $R_0$ of $p$, that $W^u_{loc}({p})$ lies in the $y$-axis and $W^s_{loc}({p})$ lies in the $x$-hyperplane, that there are points $r, q: F^N(r) = q$ in the orbit of tangency, etc. Yet again we do not distinguish the points in $R_0$ and their images under the linearizing chart. When we resort to geometric intuition, we regard the $y$-axis as ``vertical'' and the $x$-plane as ``horizontal''.\n\nThe saddle $p$ of $F$ may have several weakest contracting eigenvalues, i.e., contracting eigenvalues with maximal absolute value, but in a generic situation, which we assume to be the case, there is either one real weakest contracting eigenvalue or a pair of conjugated non-real ones. \n\nDenote by $w$ the number of these weakest contracting eigenvectors.\nLet $E^{uw}_{p}$ be the subspace of $T_pM$ spanned by the expanding and the weakest contracting eigenvectors of $p$ and $E^{ss}_{p}$ be the subspace spanned by the rest contracting eigenvectors, which we will call strong. The saddle $p$ has an $(m-w-1)$-dimensional strong stable manifold $W^{ss}(p)$ tangent to $E^{ss}_{p}$. If $M$ is two-dimensional, we take $E^{ss}_p = \\{0\\}$ and $W^{ss}(p) = \\{p\\}$. After another small perturbation we can assume that $q\\notin W^{ss}(p)$. We will always assume that in what follows.\n\nLinearizing coordinates allow us to define a $dF^{\\operatorname{per}(p)}$-invariant splitting $E^{uw} \\oplus E^{ss}$ of $T_{R_0}M$: in the linearizing coordinates the fibers $E^{uw}_z, z\\in R_0,$ are simply parallel to $E^{uw}_p$, and the fibers $E^{ss}_z$ are parallel to $E^{ss}_p$. \n\n \nDenote by $\\pi$ the projection onto $E^{uw}$ along $E^{ss}$. Consider the map \n$$\\Phi \\colon E^{uw}_r\\to E^{uw}_q, \\ \\ \\Phi = \\pi \\circ dF^{N}(r)|_{E^{uw}_r}.$$\nThe linearizing coordinates allow us to identify $E^{uw}_r$ and $E^{uw}_q$ and regard $\\Phi$ as a linear map from $\\mathbb R^{w+1}$ to itself. Then we can consider the following condition on the tangency at $q$:\n\\begin{equation}\\label{eq:pvcond}\n\\det(\\pi \\circ dF^{N}(r)|_{E^{uw}_r}) > 0.\n\\end{equation}\n\nIn Section 6 of \\cite{PV} J.~Palis and M.~Viana consider the case when $q\\in W^s_{loc}(p)\\cap W^u(p)$ and the saddle $p$ has a unique weakest contracting eigenvalue. \nThey prove that if $\\Phi$ defined above is an isomorphism, the saddle $\\tilde{p}_{n, \\mu(\\nu)}$, obtained trough the renormalization technique when unfolding the tangency at $q$, has a unique weakest contracting eigenvalue too (for $n$ large). Moreover, if this eigenvalue is positive and there exist transverse homoclinic orbits that involve the same connected components of $W^u(p)\\setminus\\{p\\}$ and $W^s(p)\\setminus W^{ss}(p)$ as the former tangency at $q$, then $\\tilde{p}_{n, \\mu(\\nu)}$ is heteroclinically related to the continuation of $p$ when $\\nu$ is close to $-2$ and $n$ is sufficiently large. The point is that condition~(\\ref{eq:pvcond}) ensures both that $\\Phi$ is an isomorphism (which is obvious) and that the weakest contracting eigenvalue of $\\tilde{p}_{n, \\mu(\\nu)}$ is positive, at least when $\\nu$ is close to $-2$ and $n$ is large and even (see discussion after (6.10) at p. 244 of \\cite{PV}; in the notation of J.~Palis and M.~Viana condition~(\\ref{eq:pvcond}) has the form $\\det \\Delta_{\\mu = 0}(r_0) > 0$). For future use let us state as a proposition the exact statement that we need and that has been proved in~\\cite{PV} (but not stated as a proposition there).\n\n\\begin{prop}[\\cite{PV}]\n\\label{prop:pv}\nSuppose that the diffeomorphism $F\\in\\operatorname{Diff}^\\infty(M)$ has a sectionally dissipative saddle $p$ and a quadratic homoclinic tangency between $W^s(p)$ and $W^u(p)$. Suppose, moreover, that the following conditions hold:\n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n\\item[a)]{$p$ has a unique weakest contracting eigenvalue,}\n\\item[b)]{$F^{\\operatorname{per}(p)}$ is linearizable in the neighborhood of $p$,}\n\\item[c)]{there exists a transverse homoclinic point that belongs to the connected components of $W^u(p)\\setminus\\{p\\}$ and $W^s(p)\\setminus W^{ss}(p)$ that are involved in the tangency,}\n\\item[d)]{condition (\\ref{eq:pvcond}) holds for the orbit of tangency (with $N\\in\\mathbb N$ and points $r\\in W^s(p)\\cap W^u_{loc}(p)$ and $q \\in W^s_{loc}(p)\\cap W^u(p)$ as described above).}\n\\end{itemize}\nThen for $\\nu$ close to $-2$ and $n$ sufficiently large the saddle $\\tilde{p}_{n, \\mu(\\nu)}$ that appears when generically unfolding the tangency at $q$ in the one-parameter family $F_\\mu$ is heteroclinically related to the continuation of $p$.\n\\end{prop} \n\nWe will always assume $F_\\mu$ to be the special family defined in Section~\\ref{sect:gcfeds}, but in \\cite{PV} they consider an arbitrary one-parameter family that unfolds the tangency generically, i.e., with non-zero speed (see also Remark~\\ref{rem:genfamily}). The renormalization scheme works for such a family as well, and the saddles $\\tilde{p}_{n, \\mu(\\nu)}$ can be defined analogously, but since we use our special family only, we do not go into details.\n \n\nIf our diffeomorphism $F$ satisfies all conditions stated in the hypothesis of Proposition~\\ref{prop:pv}, then a small perturbation yields an extremely dissipative saddle $\\tilde{p}_{n, \\mu(\\nu)}$ heteroclinically related to the continuation of the original saddle. Due to Proposition~\\ref{prop:renormsaddletan}, we can also assume that our new saddle has a homoclinic tangency, so when we unfold this tangency, the unstable manifolds of both the new saddle and the continuation of the original one intersect a basin of a sink.\n \nHowever, for now we suppose that the assumptions of Proposition~\\ref{prop:pv} are not satisfied for the tangency that we consider. Let us show that after an arbitrarily small perturbation one can obtain a diffeomorphism with a homoclinic tangency associated with a saddle $\\hat{p}$ heteroclinically related to the continuation of the original saddle $p$ such that for this new tangency Proposition~\\ref{prop:pv} is applicable. For that we need to prove several auxiliary propositions first.\n\n\n\n\n\n\n\n\\subsubsection*{Upper and lower tangencies}\n\nLet $F$, as before, have a quadratic homoclinic tangency at $q\\in W^s(p)$. Take a connected component $\\Gamma$ of $W^u(p)\\setminus\\{p\\}$ that is involved\\footnote{\\label{footnote:involve}We say that the the connected component $\\Gamma$ of $W^u(p)\\setminus\\{p\\}$ is involved in the tangency at $q$ if $q\\in F^k(\\Gamma)$ for some $k$. Recall that we allow tangencies of invariant manifolds of different points of $O(p)$, so $q$ does not necessarily belong to $\\Gamma$, but it is still natural to talk about the component of $W^u(p)\\setminus\\{p\\}$ involved in the tangency. This remark applies to transverse homoclinic intersections as well.} in the tangency at $q$. If the expansive eigenvalue $\\sigma$ of $p$ is negative, then both unstable separatrices of $p$ are involved and one can choose any. \nThe stable manifold $W^s(p)$ is an injectively immersed open disk of dimension $m-1$. Let us call the ``side'' of $W^s(p)$ that faces $\\Gamma$ at $p$ \\emph{upper} and the opposite ``side'' \\emph{lower}.\n\nTo be precise, consider the local stable manifold of $p$ first. This manifold splits a small ball $B_0$ centered at $p$ in two parts; the part that corresponds to $\\Gamma$ will be called upper. Given any point $z\\in W^s(p)$, take an even number $2l$ such that $F^{2l\\cdot\\operatorname{per}(p)}(z)\\in W^s_{loc}(p)$. Now take a small ball $B\\subset M$ centered at $z$. The connected component of $z$ in $W^s(p)\\cap B$ splits $B$ in two parts. The upper part is the one that is mapped by $F^{2l\\cdot\\operatorname{per}(p)}$ inside the upper part of $B_0$ (we assume $B$ to be so small that its $F^{2l\\cdot\\operatorname{per}(p)}$-image lies inside $B_0$). Note that if $\\sigma $ is negative, $F^{\\operatorname{per}(p)}$ swaps the upper and the lower ``sides'' of $W^s(p)$.\n \n Thus, inside a small ball centered at the point $q$ of quadratic tangency the unstable manifold of $O(p)$ approaches $W^s(p)$ either from above or from below. Therefore, we can define the tangencies \\emph{from above} and \\emph{from below} (or \\emph{upper} and \\emph{lower} tangencies). This concept is well-defined for any quadratic tangency associated with a sectionally dissipative saddle. Note that if the expansive eigenvalue of the saddle is negative, then any quadratic tangency related to this saddle becomes a tangency from below when the proper unstable separatrix is chosen. Therefore, in this case we will regard any quadratic tangency as a tangency from below. \n\n\\begin{prop}\\label{prop:lowertan}\nSuppose that for a diffeomorphism $F\\in\\operatorname{Diff}^\\infty(M)$ a sectionally dissipative periodic saddle $p$ has an upper quadratic tangency at the point $q$ and also a transverse homoclinic orbit that involves the same connected component of $W^u(p)\\setminus\\{p\\}$ as the tangency. Then by an arbitrarily $C^\\infty$-small perturbation one can obtain a diffeomorphism $G$ with a lower homoclinic tangency associated with a sectionally dissipative saddle $\\hat p$ heteroclinically related to the continuation of $p$. If the original saddle had a unique weakest contracting eigenvalue, then one can take $\\hat p$ with the same property. \n\\end{prop}\n\\begin{proof}\nWe consider the case when the expansive eigenvalue $\\sigma$ of $p$ is positive, because otherwise any quadratic tangency is a tangency from below. However, if we did not make this agreement, the argument would stay the same.\n\nAs above, we can assume without loss of generality that $F^{\\operatorname{per}(p)}$ is linearizable in $R_0\\ni p$ and the linearizing coordinates and the points $r, q$ are as described in Section~\\ref{sect:gcfeds} ($q$ can be replaced with $F^{2l\\cdot\\operatorname{per}(p)}(q)$ for some $l > 0$ if necessary). \n\nLet the points $\\tilde{r} = (0, y_{\\tilde{r}}) \\in W^u_{loc}(p)\\cap R_0, y_{\\tilde r}\\in[0,1\/2],$ and $\\tilde{q} = (x_{\\tilde q}, 0)\\in W^s_{loc}(p)\\cap R_0$, ${\\|x_{\\tilde q}\\| < 1}, \\ {\\tilde{q} = F^{\\tilde{N}}(\\tilde{r})},$ belong to the transverse homoclinic orbit that involves the same connected component of $W^u(p)\\setminus \\{p\\}$ as the tangency at $q$. Without loss of generality we can assume that $\\tilde{N}$ is divisible by $\\operatorname{per}(p)$ (and so $\\tilde{q}\\in W^u(p)$): if this is not the case, then it is not difficult to show\\footnote{If the unstable separatrix $\\Gamma$ of $p$ intersects $W^s(F^k(p))$ transversely, then $\\Gamma$ accumulates on $W^u(F^k(p))$. Since $W^u(F^k(p))$ transversely intersects $W^s(F^{2k}(p))$, so does $\\Gamma$. Continuing this line of argument, one concludes that $\\Gamma$ has a point of transverse intersection with $W^s(F^{k\\cdot\\operatorname{per}(p)}(p)) = W^s(p)$.} with the help of the $\\lambda$-lemma that there is another transverse homoclinic orbit for which this is true. The existence of such an orbit implies (see, \\cite{Katok}, Thm 6.5.5, and also Chapter 3 of \\cite{PV}) that there is a basic set $\\Lambda\\ni p$. \n\nThis basic set $\\Lambda$ can be obtained in the following way. For $\\delta > 0$ small, take a cylinder \n$$V_\\delta = \\{(x, y)\\in R_0\\colon \\|x\\| \\le 1, |y| \\le \\delta\\}.$$\nSuppose that the value of $\\delta$ is adjusted so that there is $\\tilde{n}\\in \\mathbb N$ such that $F^{\\tilde{n}\\cdot\\operatorname{per}(p)}(V_\\delta)$ is a cylinder close to the segment $\\{0\\}\\times[-2y_{\\tilde{r}}, 2y_{\\tilde{r}}]\\subset W^u_{loc}(p)$ and, therefore, $F^{\\tilde{n}\\cdot\\operatorname{per}(p) + \\tilde{N}}(V_\\delta) \\cap V_\\delta$ has at least two connected components: one of these components is a cylinder $\\Delta_p\\ni p$ close to $\\{0\\}\\times[-\\delta, \\delta]$ and another component $\\Delta_{\\tilde{q}}$ contains $\\tilde{q}$. Denote $F^{\\tilde{n}\\cdot\\operatorname{per}(p) + \\tilde{N}}$ by $H$; then $H|_{\\Delta_p\\cup\\Delta_{\\tilde{q}}}$ is a horseshoe map (if $\\delta$ is small enough) and $\\hat\\Lambda = \\bigcap_{k\\in\\mathbb Z} H^k(\\Delta_p\\cup\\Delta_{\\tilde{q}})$ is its maximal invariant set. Then $\\Lambda = \\bigcup_{k \\in \\mathbb N} F^k(\\hat\\Lambda)$ is the required basic set of $F$. We describe this construction to note two important features. First, since $\\tilde r$ has a positive $y$-coordinate, this construction implies that in a sufficiently small neighborhood of $p$ every point $\\xi\\in\\Lambda$ lies either in $W^s_{loc}(p)$ or above it relative to the $y$-axis. Second, $F^{-\\tilde{N}}(\\Delta_{\\tilde{q}})$ is contained in a small neighborhood of $\\tilde{r}$ when $\\delta$ is small.\n \n Thus, it follows from the construction of $\\Lambda$ that there is a sequence $(p_j)_{j\\in\\mathbb N}$ of saddles $p_j\\in \\Lambda$ such that\n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n\\item[a)]{$\\operatorname{per}(p_j) = n_j\\cdot\\operatorname{per}(p) + \\tilde{N}$ (therefore $\\operatorname{per}(p_j)$ is divisible by $\\operatorname{per}(p)$), }\n\\item[b)]{$p_j\\to p$ and $n_j\\to\\infty$ as $j\\to\\infty$,}\n\\item[c)]{ every $p_j$ lies above $W^s_{loc}(p)$,}\n\\item[d)]{ there is $r_j\\in O(p_j)$ close to $\\tilde{r}$ such that $F^{\\tilde N}$ maps $r_j$ into a point $q_j$ close to $\\tilde{q}$, $F^{l\\cdot \\operatorname{per}(p)}(q_j)\\in R_0$ for $l = 1,\\dots, n_j$, and $F^{n_j\\cdot \\operatorname{per}(p)}(q_j) = r_j$.}\n\\item[e)]{ all orbits $O(p_j)$ are uniformly far from $F^{-1}(q)$. }\n\\end{itemize}\n\nFirst note that properties b) and d) imply that for large $j$ the saddles $p_j$ are sectionally dissipative. Indeed, let us show that in our coordinates they contract the 2-dimensional volume (and use Remark~\\ref{rem:volume}).\nSince $p$ is sectionally dissipative, we can assume that in the linearizing coordinates on $R_0$ we have $\\sigma\\|\\Lambda\\| < 1$, where the norm is generated by the euclidean vector norm. Then in these coordinates $dF^{\\operatorname{per}(p)}$ contracts the 2-dimensional volume.\\footnote{This can be checked in a straightforward way, e.g., by considering an arbitrary pair of vectors ${(v_i = u_i + s_i \\mid u_i \\in E^u_p, s_i \\in E^s_p)_{i = 1,2}}$ and comparing the Gram determinants of this pair and its image under $dF^{\\operatorname{per}(p)}$, expressed in terms of $u_i, s_i, \\sigma, \\Lambda$.} It follows from b), d) that for $r_j\\in O(p_j)$ we have\n$$dF^{\\operatorname{per}(p_j)}(r_j) = dF^{n_j\\cdot \\operatorname{per}(p)}(q_j) \\circ dF^{\\tilde N}(r_j).$$\nHere $\\tilde N$ is fixed, and so the norm of $dF^{\\tilde N}(r_j)$ is uniformly bounded, whereas $dF^{n_j\\cdot \\operatorname{per}(p)}(q_j)$ coincides with $\\left(dF^{\\operatorname{per}(p)}(p)\\right)^{n_j}$ in our coordinates and hence contracts 2-volume exponentially as $n_j\\to\\infty$. Thus, for sufficiently large values of $j$, the linear maps $dF^{\\operatorname{per}(p_j)}(r_j)$ contract 2-volumes, which proves that $p_j$ are sectionally dissipative.\n\n\\begin{comment}\nIndeed, by b) the expanding eigenvector $v_u(p_j)$ of the saddle $p_j$ (i.e., the expanding eigenvector of $dF^{\\operatorname{per}(p_j)}(p_j)$) is almost vertical. Then the expanding eigenvector $v_u(r_j)$ of the saddle $r_j\\in O(p_j)$ is even closer to the vertical direction. On the other hand, since we have a transverse homoclinic intersection at $\\tilde q$, the image of a vertical vector $e_y\\in T_{\\tilde r}M$ under $dF^{\\tilde N}(\\tilde r)$ is a vector transverse to the horizontal hyperplane. Since $r_j$ is close to $\\tilde r$, we may assume that $dF^{\\tilde N}(r_j)$ is close to $dF^{\\tilde N}(\\tilde r)$, therefore the angle between $(dF^{\\tilde N}(r_j))(v_u(r_j))$ and the horizontal hyperplane in $T_{q_j}M$ is bounded away from zero for $j$ larger than some $j_0$. Further iterations contract the horizontal component of the image of our eigenvector and multiply the vertical component by $\\sigma $. After a few iterations this image becomes almost vertical again. These considerations allow to conclude, that for $j > j_0$ we can write the following estimate for the expanding eigenvalue $\\sigma _j$ of $p_j$ (or $r_j$):\n\\begin{equation}\n\\label{eq:sigmaj}\nc_1^{-1}\\cdot (|\\sigma |-\\delta_1)^{n_j}\\le |\\sigma _j| \\le c_1\\cdot|\\sigma |^{n_j},\n\\end{equation}\nwhere $c_1 > 0$ does not depend on $j$ at all and $\\delta_1 > 0$ can be taken arbitrarily small provided that $j_0$ is sufficiently large. Likewise we can obtain (reversing the time) an estimate for the norm of $\\lambda_j := dF^{\\operatorname{per}(p_j)}(p_j)|_{E^s(p_j)}$ for $j > j_0$:\n\\begin{equation}\n\\label{eq:lambdaj}\nc_2^{-1}\\cdot \\|\\lambda\\|^{n_j} \\le \\|\\lambda_j\\| \\le c_2\\cdot (\\|\\lambda\\|+\\delta_2)^{n_j}.\n\\end{equation}\nOnce again, $c_2 > 0$ does not depend on $j$ and $\\delta_2 > 0$ can be chosen small when $j_0$ is large. Together second inequalities in (\\ref{eq:sigmaj}) and (\\ref{eq:lambdaj}) imply that $p_j$ is a sectionally dissipative saddle when $j$ is large.\n\\end{comment}\n\n\n\\bigskip\n\nFurthermore, if $j$ is large enough, then due to the local product structure on $\\Lambda$ the local stable manifold $W^s_{loc}(p_j)$ is a small nearly horizontal disk that lies above the $x$-hyperplane and transversally intersects $W^u_{loc}(p)$ at some point $a_j$. Likewise, $W^u_{loc}(p_j)$ is a nearly vertical segment that transversally intersects $W^s_{loc}(p)$ at $b_j$. Denote by $\\Gamma_j$ the unstable separatrix of $p_j$ that contains the latter intersection. It is the continuation of $\\Gamma_j$ that will be involved in the tangency we are going to obtain. As above, we can consider a ``side'' of $W^s(p_j)$ that faces $\\Gamma_j$ at $p_j$. Let us call this ``side'' \\emph{positive}, because in this case the word ``upper'' may cause ambiguity: note that the positive ``side'' of $W^s_{loc}(p_j)$ is directed down relative to the $y$-axis (see Figure \\ref{lt}).\n\n\\begin{figure}\n \\begin{center}\n \\begin{tikzpicture}[scale=3]\n\n \\node [below] at (1, -0.6) {$\\mu = 0$};\n\n \n \\coordinate (x) at (2.5, 0);\n \\coordinate (y) at (0, 1.5);\n \n \\draw[thick] (y) node[above] {$y$} -- (0,0) coordinate (p) node[below left] {$p$} -- (0,-0.5);\n \\filldraw [black] (p) circle (0.3pt);\n \\draw[thick] (-0.5, 0) -- (x) node[right] {$x$};\n \n \n \\coordinate (q) at (2,0);\n \\coordinate (r) at (0,1);\n\n\n \n \\draw[->] (-0.25, 0) -- (-0.24, 0);\n \\draw[<-] (0.24, 0) -- (0.25, 0);\n \n \\draw[->] (0, -0.24) -- (0, -0.25);\n \\draw[<-] (0, 0.25) -- (0, 0.24);\n \n \n \n \n\n \n \\begin{scope}[xshift = 0.75cm, yshift = 0.5cm]\n \\coordinate (pj) at (0, 0);\n \\draw[thick, color = red] (pj) -- +(-1.25, 0);\n \\draw[thick, color = red] (pj) -- +(0.25, 0) node[below left] {$+$} node[above left] {$-$};\n \\draw[color = red, >-<] (-0.15, 0) -- (0.15, 0);\n \\draw[color = blue, <-] (0, -0.15) -- (0, 0); \n \\draw[thick, color = blue] (0, -0.8) node[right, black] {\\small $\\Gamma_j$} -- (0, 0);\n \\draw[thick, color = magenta] (0,0) -- (0, 0.3);\n \\draw[color = magenta, ->] (0,0) -- (0, 0.15);\n \\node[below left] at (pj) {$p_j$} circle (0.3pt);\n \\filldraw [color = black] (pj) circle (0.3pt);\n\n \\end{scope}\n\n \n \\coordinate (aj) at (0, 0.5); \\coordinate (bj) at (0.75, 0);\n \\node[above left] at (aj) {$a_j$};\n \\filldraw [color = black] (aj) circle (0.3pt);\n \\node[below left] at (bj) {$b_j$};\n \\filldraw [color = black] (bj) circle (0.3pt);\n\n \n \\draw [thick, color = red] (-0.5, 0.04) node[above right, color = black] {$D_{k_2}$} -- (2.5, 0.04);\n \\begin{scope}[xshift = -0.03cm]\n \\draw [thick, color = blue] (0.08, -0.5) -- (0.08, 0.8);\n \\draw [thick, color = blue] (0.08, 1.2) -- (0.08, 1.5);\n \\draw [thick, color = green] (0.08, 0.8) -- (0.08, 1) node[right, black] {$I_{k_1}$} -- (0.08, 1.2);\n \\draw (0.06, 0.8) -- (0.1, 0.8); \\draw (0.06, 1.2) -- (0.1, 1.2);\n \\end{scope}\n\n \n \\draw[thick] (q) parabola +(0.3, 0.5);\n \\draw[thick] (q) parabola +(-0.3, 0.5);\n \\begin{scope}[yshift = -0.05cm]\n \\draw[thick, blue] (1.7, 0.5) parabola[bend pos = 0.5] bend +(0,-0.5) (2.3, 0.5);\n \\draw[very thick, green] (2, 0) parabola +(0.15, 0.125);\n \\draw[very thick, green] (2, 0) parabola +(-0.15, 0.125);\n \n \\draw (1.835, 0.12) -- (1.86, 0.135); \\draw (2.14, 0.135) -- (2.165, 0.12);\n \\end{scope}\n\n \n \\begin{scope}[xshift = 2cm, yshift = 0.2cm, scale = 0.4]\n \\coordinate (a) at (-1, 1);\n \\coordinate (b) at (1, 1);\n \\coordinate (c) at (1, -1);\n \\coordinate (d) at (-1, -1);\n \n \\path[draw, dashed] (a) -- (b) -- (c) -- (d) -- cycle; \n \\end{scope}\n\n \n \\begin{scope}[xshift = 3.8cm, yshift = 0.25cm, scale = 0.65]\n \\coordinate (a1) at (-1, 1);\n \\coordinate (b1) at (1, 1);\n \\coordinate (c1) at (1, -1);\n \\coordinate (d1) at (-1, -1);\n \n \\path[draw, dashed] (a1) -- (b1) -- (c1) -- (d1) -- cycle; \n \n \\draw[thick] (-1, -0.5) -- (1, -0.5);\n \\filldraw[black] (0, -0.5) node[below] {$q$} circle (0.5pt);\n \\draw[thick, red] (-1, -0.4) -- (1, -0.4);\n \n \\node[left, red] at (1, -0.6) {$+$};\n \\node[left, red] at (1, -0.2) {$-$};\n \\draw[thick, blue] (-0.75, 0.85) parabola bend (0,-0.4) (0.75, 0.85);\n \n \\draw[thick] (-0.75, 0.95) node[below right] {\\scriptsize $W^u(O(p))$} parabola bend (0,-0.3) (0.75, 0.95);\n \\draw[very thick, green] (-0.375, -0.088) parabola[bend pos = 0.5] bend (0,-0.4) (0.375, -0.088);\n \\draw (-0.405, -0.101) -- (-0.355, -0.07); \\draw (0.405, -0.101) -- (0.355, -0.07);\n \\node[below left] at (-0.37, 0.16) {\\scriptsize $F_{\\mu_i}^{\\tilde N}(\\Gamma_j)$};\n \\node at (0.55, -0.27) {\\scriptsize $I_{k_1, \\mu_i}$};\n \\filldraw[black] (0, -0.4) circle (0.5pt);\n \\end{scope}\n\n\n \\node[below] at (3.8, -0.6) {$\\mu = \\mu_i$};\n \\draw[dotted] (b) -- (a1); \\draw[dotted] (c) -- (d1);\n\n \\filldraw [black] (2,0) coordinate (q) node[below right, align = center] {$q$} circle (0.3pt);\n \\filldraw [black] (0,1) coordinate (r) node[left] {$r$} circle (0.3pt);\n \n\n\n\n\n \\end{tikzpicture}\n \\caption{Making a lower tangency.}\\label{lt}\n \\end{center} \n\\end{figure}\n\n\n\nSince $W^s_{loc}(p_j)$ transversally intersects $W^u_{loc}(p)$ at $a_j$, it follows from the $\\lambda$-lemma that there is a sequence of disks $D_k\\subset W^s(p_j), \\ k\\in \\mathbb N,$ such that \n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n\\item[1)]{the disk $D_k$ is a $F^{2k\\cdot\\operatorname{per}{p_j}}$-preimage of a small neighborhood $\\hat{D}_k$ of $a_j$ in $W^s_{loc}(p_j)$ (therefore, all images of these disks under the iterates of $F$ are uniformly far from $F^{-1}(q)$);}\n\\item[2)]{these disks tend to the disk $D_0 = \\{(x, y)\\colon y=0, \\|x\\| \\le 1 \\}\\subset W^s(p)$ as $C^\\infty$-immersed disks;}\n\\item[3)]{the positive ``sides'' of the disks $D_k$ are also directed downwards.}\n\\end{itemize}\n \nNote that 1) implies 3). Indeed, consider an oriented vertical segment $I = Oy \\cap R_0$ that goes upwards. It crosses $W^s_{loc}(p_j)$ at $a_j$ from the positive side to the negative side. On one hand, the preimage $F^{-2k\\cdot\\operatorname{per}(p_j)}(I)$ is also a vertical segment that goes upwards (because $2k\\cdot\\operatorname{per}(p_j) = 2k\\cdot l\\cdot\\operatorname{per}(p)$ for some $l\\in\\mathbb N$). On the other hand, a curve that approaches $W^s(p_j)$ from the positive side at $a_j$ is mapped by $F^{-2k\\cdot\\operatorname{per}(p_j)}$ to a curve that does so at $F^{-2k\\cdot\\operatorname{per}(p_j)}(a_j)$. Therefore, the positive side of any disk $D_k$ looks downwards.\n\nLikewise, there is a sequence of segments $I_k\\subset F^{k\\cdot\\operatorname{per}(p)}(\\Gamma_j), \\ k \\in \\mathbb N,$ that accumulate to some segment $I_0\\ni r$ (recall that $r = (0, 1)$) while their preimages are uniformly bounded away from~$F^{-1}(q)$.\n\nNow fix such a saddle $p_j$ with $j$ large and consider a special family $F_\\mu$ described in Section~\\ref{sect:gcfeds}. Recall that the maps of this family differ from $F_0 = F$ only inside a small neighborhood of $F^{-1}(q)$. If this neighborhood is small enough, $p_j$ is a periodic saddle for every $F_\\mu, \\ \\mu\\in[-\\varepsilon , \\varepsilon ],$ and the disks $D_k$ lie in the stable manifold of this saddle. Likewise, the segments $I_k$ lie in the unstable manifold of the orbit of this saddle. Recall\\footnote{See the properties of the special family in Section~\\ref{sect:gcfeds}.} that when we change the parameter $\\mu$, the intersection of $F^N_{\\mu}(I_{k})$ with a sufficiently small neighborhood of $q$ moves upwards or downwards, whereas the disk $D_k$ stands still. Here $k$ is arbitrary. Thus, there is a sequence of parameter values $\\mu_i\\to 0$ (as $i\\to\\infty$) such that for each $\\mu_i$ there exist $k_1$ and $k_2$ for which $F^N_{\\mu_i}(I_{k_1}) =: I_{k_1, \\mu_i}$ is tangent to $D_{k_2}$ at some point $\\hat{q}$. Moreover, this is a tangency from the negative side (see Figure \\ref{lt}). In other words, we have obtained a perturbation of $F$ with a lower tangency associated with $\\hat{p} = p_j$.\n\nNow we will prove the second statement of the proposition: if $p$ has a unique weakest contracting eigenvalue, then we can take $\\hat p$ with the same property. Let us assume that the image of $E^{ss}_{\\tilde q}$ under $dF^{-\\tilde N}(\\tilde q)$ is transverse to $E^{uw}_{\\tilde r}$. This condition can be satisfied by a small perturbation that preserves all relevant properties of $F$, so we suppose that $F$ satisfied it from the very beginning. \n\nFor any $\\alpha > 0$ and $z\\in R_0$ denote by $C^{ss}_\\alpha(z)$ the $\\alpha$-cone around $E^{ss}_z$:\n$$C^{ss}_\\alpha(z) = \\{v\\in T_{z}M \\mid v = v_{uw} + v_{ss}, \\ v_{uw}\\in E^{uw}_{z}, \\ v_{ss}\\in E^{ss}_{z}, \\ \\|v_{uw}\\| \\le \\alpha\\|v_{ss}\\| \\}.$$\nConsider again the point $q_j\\in O(p_j)$ (defined in {\\bf\\small{d)}} above) and a narrow $\\alpha$-cone $C^{ss}_\\alpha(q_j), \\ {\\alpha \\ll 1}$. Since all $q_j, \\ j\\in\\mathbb N,$ are close to $\\tilde q$, we suppose that the differentials $dF^{-\\tilde{N}}(q_j)$ are uniformly close to $dF^{-\\tilde{N}}(\\tilde q)$. Since $dF^{-\\tilde N}(\\tilde q)E^{ss}_{\\tilde q}$ is transverse to $E^{uw}_{\\tilde r}$, we can also assume that for any $j$ the cone $dF^{-\\tilde N}(q_j)(C^{ss}_\\alpha(q_j))$ is contained in $C^{ss}_{\\beta}(r_j)$, where $\\beta$ is some large number independent of $j$. \n\nThe map $F^{\\operatorname{per}(p)}$ is linearized in $R_0$, so we can assume that both this map and its differential at every point $z\\in R_0$ coincide with a linear map $L$ for which $E^{ss}$ is a strong stable bundle. Recall that (see {\\bf\\small{d)}} above) $q_j = F^{-n_j\\cdot \\operatorname{per}(p)}(r_j)$. The differential $dF^{-n_j\\cdot \\operatorname{per}(p)}(r_j)$ coincides with $L^{-n_j}$. If $j$ is large, $n_j$ is also large, and if $n_j$ is large enough, then $L^{-n_j}$ maps $C^{ss}_{\\beta}(r_j)$ inside $C^{ss}_\\alpha(q_j)$ and expands vectors in $C^{ss}_{\\beta}(r_j)$ considerably, so that $C^{ss}_\\alpha(q_j)$ becomes an expansive forward-invariant cone for $dF^{-(n_j\\operatorname{per}(p)+\\tilde{N})}(q_j)$. Then there is a unique invariant $(m-2)$-dimensional plane inside $C^{ss}_\\alpha(q_j)$ that can only be a span of strong stable eigenvectors of~$q_j$. So, $p_j$ has a unique weakest contracting eigenvalue, at least for $j$ large, and the same is true for its continuation that has a lower tangency related to it.\n\n\\end{proof}\n\n\\bigskip\n\n\\begin{rem}\\label{rem:samepoint}\nIf $q\\in W^s(p)\\cap W^u(p)$ for $F$, then one can assure that $\\hat{q}\\in W^u(\\hat{p}, G)\\cap W^s({\\hat p}, G)$.\n\nIndeed, one can argue as follows. The point ${\\hat q}$ always lies in $W^s({\\hat p}, G)$ by construction. \nIf $q\\in W^s(p, F)\\cap W^u(p, F)$, then $\\operatorname{per}(p)$ divides $N$. \nLet $N_1\\ge 0$ be the remainder of division of $N$ by $\\operatorname{per}(p_j)$. Since ${\\operatorname{per}(p)}$ divides $\\operatorname{per}(p_j)$ and $N$, it also divides $N_1$.\nConsider the segments $I_k\\subset F^{k\\cdot\\operatorname{per}(p)}(\\Gamma_j)$, as described in the proof, for $k = \\frac{l\\cdot\\operatorname{per}(p_j) - N_1}{\\operatorname{per}(p)}$, where $l\\in \\mathbb N$. For such $k$ we have that\n$$F_\\mu^{N}(I_k) \\subset F_\\mu^{N}\\left(W^u(F^{l\\cdot\\operatorname{per}(p_j) - N_1}(p_j), F_\\mu)\\right) = W^u(F^{l\\cdot\\operatorname{per}(p_j) - N_1 + N}(p_j), F_\\mu) = W^u(p_j, F_\\mu).$$\nThe last equality holds because ${N_1\\equiv N\\pmod{\\operatorname{per}(p_j)}}$.\nIf we obtain the new tangency using such $I_k$, this tangency will lie on both invariant manifolds of $\\hat{p} = p_j$.\n\\end{rem}\n\\bigskip\n\n\\begin{rem}\n\\label{rem:intersections}\nWe can also assume that the map $G$ in Proposition \\ref{prop:lowertan} has transverse homoclinic points that belong to the same connected components of $W^u(\\hat{p})\\setminus\\{\\hat p\\}$ and $W^s({\\hat p})\\setminus W^{ss}(\\hat p)$ as the newly obtained lower tangency. Indeed, after we fix the saddle $p_j$, whose continuation will play the role of $\\hat p$, we can ensure by a small perturbation that the point $a_j\\in W^s_{loc}(p_j)\\cap{W^u_{loc}(p)}$ does not belong to $W^{ss}(p_j)$. Then for a large $k_2$ the disk $D_{k_2}\\subset W^s(p_j)$ will not intersect $W^{ss}(p_j)$, because $D_{k_2}$ is a distant preimage of a small neighborhood of $a_j$. But this disk definitely will intersect $\\Gamma_j$ transversally at some point close to~$b_j$. This transverse intersection is preserved after the perturbation, it involves the same unstable separatrix as the new tangency, and both the new tangency and this intersection lie in the disk $D_{k_2}$ that does not intersect the strong stable manifold of $p_j$.\n\\end{rem}\n\n\n\n\n\n\\subsubsection*{From lower tangencies to condition (\\ref{eq:pvcond})}\n\n\\begin{prop}\\label{prop:goodtan}\nIf a diffeomorphism $F\\in\\operatorname{Diff}^\\infty(M)$ has a lower quadratic tangency associated with a sectionally dissipative periodic saddle $p$, then by an arbitrarily $C^\\infty$-small perturbation one can obtain a diffeomorphism $G$ with a tangency that is associated with the continuation of $p$ and satisfies condition (\\ref{eq:pvcond}).\n\\end{prop}\n\\begin{proof}\nAs above, we suppose that $F^{\\operatorname{per}(p)}$ is linearizable in $R_0\\ni p$ and the linearizing coordinates and the points $r, q$ are as described in subsection ``Linearizing coordinates'' of Section \\ref{sect:gcfeds}: if $q$ is not in $W^s_{loc}(p)$, replace it with $F^{2k\\cdot\\operatorname{per}(p)}(q)$ for an appropriate $k$. We also assume that $r$ belongs to the unstable separatrix of $p$ that defines the lower tangency: if the expansive eigenvalue $\\sigma $ is positive, this is always the case; otherwise replace $q$ by $F^{\\operatorname{per}(p)}(q)$ if necessary. Thus we can assume that in the neighborhood of $q$ the arc of $W^u(p)$ that is tangent to $W^s(p)$ at $q$ lies in the lower half-space $\\{(x, y)\\colon y\\le 0\\}$.\n\n\nIf the tangency at $q$ satisfies condition (\\ref{eq:pvcond}), one can take $G = F$, so in what follows we assume that this is not the case.\nIf $\\det(\\pi\\circ dF^{N}(r)|_{E^{uw}_{r}}) = 0$, this determinant can be made non-zero by a small perturbation that preserves the tangency at $q$, so we can assume that it is negative.\n\n\nWe will obtain a new tangency using the idea of the proof of Thm~1 in \\S~3.2 of~\\cite{PT}. Namely, consider a special\\footnote{In fact, this argument can be adapted for any one-parameter family that unfolds the tangency at $q$ generically.} one-parameter family $(F_\\mu)_{\\mu\\in[-\\varepsilon ,\\varepsilon ]}$ described in Section~\\ref{sect:gcfeds}. Let $D\\ni q$ be a small disk in $W^s_{loc}(p)$ such that $D_0 := F^{-N}(D)$ lies in $R_0$ and is $\\delta$-distant from $\\partial{R_0}$, while the boundary of $D_0$ is $\\delta$-distant from $W^u_{loc}(p)$, $\\delta$ being some small positive number. Let $I_0 = \\{0\\}\\times[1-\\varepsilon _1, 1+\\varepsilon _1]$ be a small neighborhood of the point $r$ in $W^u_{loc}(p)$. Then for small positive values of $\\mu$ the arc $F_{\\mu}^N(I_0)$ has two points $z_1(\\mu), z_2(\\mu) \\in D$ of transverse intersection with $W^s_{loc}(p)$. Denote by $\\Gamma_\\mu$ the segment of this arc that lies above $W^s_{loc}(p)$. Note that $D_{\\mu} := F^{-N}_{\\mu}(D)$ transversally intersects $W^u_{loc}(p)$ at points $w_i(\\mu) = F_{\\mu}^{-N}(z_i(\\mu)), \\ i = 1, 2,$ close to $r$. If $\\mu$ is sufficiently small, we can assume that $\\partial{D_\\mu}$ is $\\delta$-distant from $W^u_{loc}(p)$ and $D_\\mu$ itself is $\\delta$-distant from $\\partial{R_0}$. \n\nThe $\\lambda$-lemma implies that for sufficiently large even $n\\in\\mathbb N$ the arc $\\Gamma_{n, \\mu} := F_{\\mu}^{n\\cdot\\operatorname{per}(p)}(\\Gamma_\\mu)$ has points of transverse intersection with $D_{\\mu}$. Fix some $n$ such that this holds and, moreover, $\\|\\lambda^n(e)\\| < \\delta\/2$ (recall that $e$ is the $x$-coordinate of $q$). Let $\\hat{z}_i(\\mu) = F_\\mu^{n\\cdot\\operatorname{per}(p)}(z_i(\\mu)), \\: {i = 1,2},$ and let $\\gamma_i(\\mu)$ be a connected component of $\\hat{z}_i(\\mu)$ in $\\Gamma_{n, \\mu}\\cap R_0$. Note that $\\gamma_1(\\mu)$ coincides with $\\gamma_2(\\mu)$ when $\\Gamma_{n, \\mu}$ is contained in $R_0$.\n\nFor every point in $\\Gamma_\\mu$ its $x$-coordinate is close to that of the point $q = (e, 0)$, therefore the $x$-coordinate of any point that belongs to $\\gamma_1(\\mu)\\cup \\gamma_2(\\mu)$ is close to $\\lambda^n(e)$. In other words, $\\gamma_1(\\mu)\\cup \\gamma_2(\\mu)$ (viewed as a set) is very close to $W^u_{loc}(p)$. This property is preserved when $\\mu$ decreases to zero and the curve $\\Gamma_{n, \\mu}$ shrinks towards the point $(\\lambda^n(e), 0) = F^n(q)$. At the same time $D_{\\mu}$ stays in a small neighborhood of the point $r$ and far from $\\partial{R_0}\\cup W^s_{loc}(p)$, and $\\partial{D_\\mu}$ stays $\\delta$-far from $W^u_{loc}(p)$. Thus when we decrease $\\mu$, the curves $\\gamma_1(\\mu)$ and $\\gamma_2(\\mu)$ cannot intersect $\\partial{D_\\mu}$ because the latter is $\\delta$-far from $W^u_{loc}(p)$, and $D_\\mu$ cannot intersect $\\partial(\\gamma_1(\\mu)\\cup \\gamma_2(\\mu))\\subset \\partial{R_0}\\cup W^s_{loc}(p)$. However, for sufficiently small values of $\\mu$, $\\gamma_1(\\mu)$ coincides with $\\gamma_2(\\mu)$ and does not intersect $D_\\mu$. Therefore, for some $\\mu_n>0$ there is a point $r_0$ of tangency between one of the curves $\\gamma_i(\\mu_n)$ and $D_{\\mu_n}$. Note that this construction allows $\\mu_n$ to be taken arbitrarily close to zero and $r_0$ to be arbitrarily close to $r$. We will take $G = F_{\\mu_n}$. As always, we can assume without loss of generality, that the tangency at $r_0$ for $G$ is quadratic.\n\n\n\\begin{figure}\n \\begin{center}\n \\begin{tikzpicture}[scale=0.52]\n\n \\node [below] at (7, -6) {$\\mu = \\mu_n$};\n\n \n \\coordinate (x) at (17.5, 0);\n \\coordinate (y) at (0, 10);\n \n \\draw[thick, red] (-3, 0) -- (0,0) coordinate (p) node[below left, black] {$p$} -- (x); \n \n\n \n \\draw[->, red] (-2.3, 0) -- (-2.2, 0);\n \\draw[<-, red] (2.2, 0) -- (2.3, 0);\n\n \\draw[->, blue] (0, -2.4) -- (0, -2.5);\n \\draw[<-, blue] (-0.06, 2.5) -- (-0.055, 2.4); \n \n \n \\coordinate (q) at (11.2,0);\n \\coordinate (hatq) at (12.4,0);\n \\coordinate (hatr) at (-0.07,4);\n \\coordinate (r_0) at (1.2, 3);\n \\coordinate (q_0) at (11.2, 0.6);\n\n \\coordinate (right0) at (0.35, 0);\n \\coordinate (left1) at (1, 0);\n \\coordinate (right1) at (1.4, 0);\n \\coordinate (left2) at (3.5, 0);\n \\coordinate (right2) at (4.3, 0);\n \\coordinate (left3) at (10.3, 0);\n\n\n\n\n \n \\draw[thick, ->] plot[smooth, tension = 1] (10.5, 1.5) {[rounded corners] to [out = 175, in = -40] (2, 4)};\n \\node[above right] at (5.5, 2.) {$F_{\\mu}^n$};\n\n \n \\draw[thick, ->] plot[smooth, tension = 1] (2., 6.) {[rounded corners] to [out = 51, in = 87] (12, 2.5)};\n \\node at (8.3, 7.7) {$F_{\\mu}^N$};\n\n\n\n \n \\draw[thick, blue] plot [smooth, tension = 0.65] coordinates{(0,-5) (0,0) (0,5) (0.9,7.6) (3,9.7) (6,10.81) (10,10.35) (13, 8.2) (15, 4) (15, -1.5) (14.2, -3.7) (12.8, -3.5) (12, 0)};\n\n \n \\draw[thick, blue] plot [smooth, tension = 0.65] coordinates{(0.2,0) (0.2,4.9) (1.1,7.5) (3.2,9.6) (6,10.61) (10,10.15) (12.88, 8.08) (14.82, 3.92) (14.8, -1.3) (14.2, -3.4) (13, -3.4) (12.5, -1.5) (12.35, -0.5) (12.4, 0) (12.5, -0.22) };\n\n \n \\draw[thick, blue] plot [smooth, tension = 0.65] coordinates{(0.35,0) (0.36,4.83) (1.27,7.43) (3.4,9.5) (6,10.41) (10,9.95) (12.76, 7.9) (14.64, 3.87) (14.6, -1.2) (14.1, -3.1) (13.1, -3.1) (12.6, -0.7) (12.5, -0.22) (12.4, 0)};\n\n \n \\draw[thick, red] plot [smooth, tension = 1.7] coordinates{(2, 2.95) (-1,4) (2, 5.05)};\n\n \n \n \\draw[thick, blue] plot [smooth, tension = 1.5] coordinates{(left1) (r_0) (right1)};\n \n\n \n \\draw[thick, blue] plot [smooth, tension = 1.7] coordinates{(left2) (3.9,1.3) (right2)};\n\n \n \\draw[thick, blue] plot [smooth, tension = 1.5] coordinates{(10.3, 0) (11.24,0.62) (12, 0)};\n\n \n \n \n \n \n \\draw[dotted, blue] plot [smooth, tension = 1] coordinates{(right0) (0.53, -4) (0.58, 1.99) (0.85, -3) (left1)};\n \\draw[dotted, blue] plot [smooth, tension = 1.4] coordinates{(right1) (1.7, -2) (2.3, 1) (2.99, -1.5) (left2)};\n \\draw[dotted, blue] plot [smooth, tension = 1.1] coordinates{(right2) (5.2, -1.) (6.9, 0.4) (8.9, -0.6) (left3)};\n\n \n \\filldraw [black] (p) circle (2pt);\n \\node[above left] at (-1, 4) {$D_{\\mu}$};\n \\node[above right] at (1.3, 1.2) {$\\Gamma_{n,\\mu}$};\n \\filldraw [color = black] (q) circle (2pt) node[below] {$q$};\n \\filldraw [color = black] (hatq) circle (2pt) node[above right] {$\\hat q$};\n \\filldraw [color = black] (hatr) circle (2pt) node[left] {$\\hat r$};\n \\filldraw [color = black] (r_0) circle (2pt) node[above] {$r_0$};\n \\filldraw [color = black] (q_0) circle (2pt) node[above] {$q_0$};\n\n \n \\filldraw [color = black] (left1) circle (1.5pt);\n \\filldraw [color = black] (right1) circle (1.5pt);\n \\filldraw [color = black] (2, 2.95) circle (1.5pt);\n \\filldraw [color = black] (2, 5.05) circle (1.5pt);\n\n\n \\end{tikzpicture}\n \\caption{The new tangency.}\\label{pic3}\n \\end{center} \n\\end{figure}\n\n\n\n\n\n\nLet $q_0 = G^{-n\\cdot\\operatorname{per}(p)}(r_0)$, $\\hat{r} = G^{-N}(q_0)$ and $\\hat{q} = G^N(r_0) = G^{n\\cdot\\operatorname{per}(p) + 2N}(\\hat{r})$. The point $\\hat{r}$ lies in $W^u_{loc}(p)$ and is close to $r$. The points $q_0$ and $\\hat{q}$ are close to $q$, and $\\hat{q}\\in W^s_{loc}(p)$ (see Fig.~\\ref{pic3}). \nWe are going to prove that for the tangency at $\\hat{q}$ condition (\\ref{eq:pvcond}) holds. The point $\\hat{r}$ plays the same role for $\\hat{q}$ that $r$ played for $q$, so condition (\\ref{eq:pvcond}) takes the following form:\n\n\\begin{equation}\\label{eq:pvc2}\n\\Delta := \\det(\\pi\\circ dG^{n\\cdot\\operatorname{per}(p) + 2N}(\\hat{r})|_{E^{uw}_{\\hat r}}) > 0.\n\\end{equation} \n\nSince $dG^{n\\cdot\\operatorname{per}(p) + 2N}(\\hat{r}) = dG^{N}(r_0) \\circ dG^{n\\cdot\\operatorname{per}(p)}(q_0) \\circ dG^{N}(\\hat{r})$ we need to prove that\n$$ \\operatorname{sgn}\\det\\left(\\left.\\pi\\circ dG^{N}(r_0) \\circ dG^{n\\cdot\\operatorname{per}(p)}(q_0) \\circ dG^{N}(\\hat{r})\\right|_{E^{uw}_{\\hat r}}\\right) = 1.$$\n\n\\medskip\nRecall that $dG^{n\\cdot\\operatorname{per}(p)}(q_0) = dF_{\\mu_n}^{n\\cdot\\operatorname{per}(p)}(q_0) = L^{n}$. Let us introduce the following shorthand notation:\n$$\\Xi = dG^{N}(r_0), \\ \\ \\Theta = dG^{N}(\\hat{r}).$$\nNote that both $\\Xi$ and $\\Theta$ are close to $dF_{0}^{N}(r)$ since $r_0$ and $\\hat{r}$ are close to $r$ and $\\mu_n$ is close to zero.\n\n\\begin{rem*}\nLet us deal with the two-dimensional case first. If $\\dim M = 2$, (\\ref{eq:pvc2}) is reduced to $\\det(\\Xi \\circ L^n \\circ \\Theta) > 0$. Since $\\Xi \\approx dF^N_0(r) \\approx \\Theta$, we have $\\operatorname{sgn}\\det(\\Xi) = \\operatorname{sgn}\\det(\\Theta)$. Recall that $n$ is even and therefore $\\det(L^n) > 0$. Combining these two observations, we finish the proof. In the general case we implement the same idea.\n\\end{rem*}\n\nLet us assume that $dF^{N}(r)(E^{uw}_{r})$ is transversal to $E^{ss}_q$: this is a generic property compatible with tangency at $q$. Since $\\Theta$ is close to $dF^{N}(r)$, we suppose that $\\Theta(E^{uw}_{\\hat r})$ is transversal to $E^{ss}_{q_0}$. If $n$ is large, $L^n\\circ\\Theta(E^{uw}_{\\hat r})$ is a plane $E_0$ very close to $E^{uw}_{r_0}$. If it is close enough, we have\n$$\n\\operatorname{sgn}\\Delta = \\operatorname{sgn}\\det\\left(\\left.\\pi \\circ \\Xi \\circ L^n \\circ \\Theta\\right|_{E^{uw}_{\\hat r}}\\right) = \n\\operatorname{sgn}\\det\\left(\\pi \\circ \\Xi|_{E^{uw}_{r_0}}\\right) \\cdot \n\\operatorname{sgn}\\det\\left(\\pi \\circ L^n \\circ \\Theta|_{E^{uw}_{\\hat r}}\\right).\n$$\nSince $\\Xi$ is close to $dF^{N}(r)$, we have\n$$\n\\operatorname{sgn}\\det\\left(\\pi \\circ \\Xi|_{E^{uw}_{r_0}}\\right) = \\operatorname{sgn}\\det\\left(\\left.\\pi \\circ dF^{N}(r)\\right|_{E^{uw}_r}\\right) = -1.\n$$\nFurthermore, since the bundle $E^{uw}$ is invariant for $L^n$, we can write\n$$\n\\operatorname{sgn}\\det\\left(\\pi \\circ L^n \\circ \\Theta|_{E^{uw}_{\\hat r}}\\right) =\n\\operatorname{sgn}\\det\\left(\\pi \\circ L^n|_{E^{uw}_{q_0}} \\circ \\pi \\circ \\Theta|_{E^{uw}_{\\hat r}}\\right) =\n$$\n$$\n=\\operatorname{sgn}\\det\\left(\\pi \\circ L^n|_{E^{uw}_{q_0}}\\right) \\cdot \\operatorname{sgn}\\det\\left(\\pi \\circ \\Theta|_{E^{uw}_{\\hat r}}\\right).\n$$\nFinally, $\\operatorname{sgn}\\det\\left(\\pi \\circ L^n|_{E^{uw}_{q_0}}\\right) = 1$ since $n$ is even, and \n$$\n\\operatorname{sgn}\\det\\left(\\pi \\circ \\Theta|_{E^{uw}_{\\hat r}}\\right) = \\operatorname{sgn}\\det\\left(\\left.\\pi \\circ dF^{N}(r)\\right|_{E^{uw}_r}\\right) = -1.\n$$ \n\nTherefore, $\\operatorname{sgn}\\Delta = -1\\cdot 1\\cdot (-1) = 1$, which concludes the proof.\n\\end{proof}\n\n\\medskip\n\n\\begin{rem}\\label{rem:goodtan}\nNote that for the new point of tangency we have $$\\hat{q}\\in {W^u(G^{2N+n\\cdot\\operatorname{per}(p)}(p))\\cap W^s(p)} = W^u(G^{2N}(p))\\cap W^s(p)$$ (we denote the continuation of the original saddle $p$ of $F$ by the same symbol). If $q\\in W^u(p, F)\\cap W^s(p, F)$, then $N$ is divisible by $\\operatorname{per}(p)$ and, therefore, $\\hat{q}\\in W^u(p, G)\\cap W^s(p, G)$.\n\\end{rem}\n\n\n\\subsubsection*{From tangencies to transverse intersections}\n\n\\begin{prop}\\label{prop:trans}\nSuppose that a diffeomorphism $F$ has a homoclinic tangency associated with a sectionally dissipative saddle $p$. Then either there are transverse homoclinic intersections that involve\\footnote{See footnote~\\ref{footnote:involve}.} the same connected component of $W^u(p)\\setminus\\{p\\}$ as the tangency, or such intersections can be obtained by a small perturbation together with a new homoclinic tangency.\n\\end{prop}\n\\begin{proof}\nAs always, we can assume that $F$ is $C^\\infty$-smooth, $F^{\\operatorname{per}(p)}$ is linearizable in the neighborhood of $p$, and the tangency is quadratic. Denote the point of tangency by $q$. \n\nIf the tangency at $q$ is a tangency from below, we can argue as in the proof of Proposition~\\ref{prop:goodtan} and obtain a new tangency and transverse homoclinic intersections as required no matter if such transverse orbits existed prior to perturbation or not.\n\nSuppose now that we have a tangency from above at $q$. Denote by $\\Gamma$ the connected component of $W^u(p)\\setminus\\{p\\}$ involved in the tangency. Suppose that for the diffeomorphism $F$ there are no transverse intersections between $F^k(\\Gamma)$ and $W^s(p)$ for any $k\\in\\mathbb Z$.\nIf there is an orbit of tangency different from that of $q$, we can perturb one of these orbits into an orbit of transverse intersection while preserving the second orbit unchanged. \n\nThus, we need to consider only the case when there are no intersections between $\\bigcup_k F^k(\\Gamma)$ and $W^s(p)$ other than those that belong to $O(q)$. In this case the argument is analogous to the first part of the proof of Proposition~\\ref{prop:goodtan}, and even simpler.\n\nNamely, we can consider the linearizing neighborhood $R_0$ and assume that $q\\in W^s_{loc}(p)\\subset R_0$ and $r = F^{-N}(q)\\in W^u_{loc}(p)\\subset R_0$, just like in the proof of Proposition~\\ref{prop:goodtan}. Consider again a small disk $D\\colon q\\in D\\subset W^s_{loc}(p)$, and its $F^N$-preimage $D_0$ contained in a neighborhood of $r\\in\\Gamma$. We assume that $\\partial{D_0}$ is $\\delta$-far from $W^u_{loc}(p)$ and $D_0$ itself is $\\delta$-far from $\\partial{R_0}\\cup W^s_{loc}(p)$.\n\nConsider a special one-parameter family $(F_\\mu)$ that unfolds the tangency at $q$. Let $D_{\\mu} := F_{\\mu}^{-N}(D)$. Since we have a tangency from above when $\\mu = 0$, for small (in absolute value) $\\mu < 0$ there will be two transverse intersections between $F_{\\mu}^N(W^u_{loc}(p))$ and $W^s_{loc}(p)$ at the points $z_1(\\mu), z_2(\\mu)$ near $q$. Take some small $\\mu_0<0$ and denote by $\\Gamma_{\\mu_0}$ a small arc in $W^u(O(p))$ that starts at $z_1(\\mu_0)$ and goes up. For a sufficiently big even integer $n$ there is a transverse intersection between $F^{n\\cdot\\operatorname{per}(p)}_{\\mu_0}(\\Gamma_{\\mu_0})$ and $D_{\\mu_0}$ near~$r$.\n\nLet $R$ be the intersection of $R_0$ with the upper half-space and $\\gamma(\\mu), \\ \\mu < 0,$ be a connected component of the point $\\hat{z}_1 = F^{n\\cdot\\operatorname{per}(p)}_{\\mu}(z_1)$ in $W^u(O(p))\\cap R$. Then for large $n$ and small negative $\\mu$ the component $\\gamma(\\mu)$ continuously depends on $\\mu$. Define the curve $\\gamma(0)$ by continuity. If $n$ is sufficiently large, then for any $\\mu\\in[\\mu_0, 0]$ both endpoints of $\\gamma(\\mu)$ lie far from $D_\\mu$, while $\\gamma(\\mu)$ itself is at least $\\delta\/2$ close to $W^u_{loc}(p)$. At the same time we can assume that $\\partial{D_\\mu}$ is $\\delta$-far from $W^u_{loc}(p)$. Recall that for $\\mu = 0$ we have no intersections between $\\gamma(\\mu)$ and $D_\\mu$, but for $\\mu = \\mu_0$ there is a transverse intersection. Since for $\\mu\\in [\\mu_0, 0]$ the curve $\\gamma(\\mu)$ and the disk $D_\\mu$ can intersect by interior points only, there is some value of $\\mu$ when a point of tangency appears. Thus we have obtained both transversal intersections and a tangency as required. \n \n\\end{proof}\n\n\\begin{rem}\\label{rem:trans}\nIf for $F$ the point of tangency $q$ is not in $W^{ss}(p)$, then the same argument yields that we either have a transversal intersection that involves the same connected components of $W^u(p)\\setminus\\{p\\}$ and $W^s(p)\\setminus W^{ss}(p)$ or can obtain such an intersection together with a new tangency that involves the same connected components (to be precise, their continuations). It suffices to notice that the new transverse or tangential intersections are constructed near the original orbit of tangency with respect to the metric on $W^s(p)$.\n\nFurthermore, if $q\\in W^u(p)\\cap W^s(p)$, then this intersection also is in $W^u(p)\\cap W^s(p)$ (since $\\operatorname{per}(p)$ divides~$N$).\n\\end{rem}\n\n\n\\subsubsection*{Tangencies for invariant manifolds of the same saddle}\n\n\\begin{prop}\\label{prop:goodtan2}\nIf a diffeomorphism $F$ has a lower quadratic tangency between $W^s(p)$ and $W^u(F^N(p))$, where $p$ is a sectionally dissipative periodic saddle, then by an arbitrarily $C^\\infty$-small perturbation one can obtain a diffeomorphism $G$ with a tangency between the stable and the unstable manifolds of the continuation of $p$.\n\\end{prop}\n\\begin{proof}\nConsider a diffeomorphism $G$ and a point $\\hat{q}$ as in the proof of Proposition~\\ref{prop:goodtan}.\nNote that for $G$ we have transversal intersections between $W^u(p)$ and $W^s(G^{-N}(p))$ (at the points $w_i(\\mu_n)$ mentioned in the proof). The $\\lambda$-lemma implies that $W^u(p)$ accumulates on $W^u(G^{-N}(p))$ and, therefore, transversally intersects also $G^{-N}(W^s(G^{-N}(p))) = W^s(G^{-2N}(p))$. Arguing inductively, we obtain that for any $k\\in\\mathbb N$ the unstable manifold $W^u(p)$ transversally intersects $W^s(G^{-kN}(p))$ and accumulates on $W^u(G^{-kN}(p))$ . Take $k = 3\\operatorname{per}(p) - 2 > 0$. This yields that $W^u(p)$ transversally intersects $W^s(G^{2N}(p))$ and accumulates on $W^u(G^{2N}(p))$. But $W^u(G^{2N}(p))$ is tangent to $W^s(p)$ at $\\hat{q}$. Therefore, we can obtain a tangency between $W^u(p)$ and $W^s(p)$ by another small perturbation arguing as in the proof of Proposition~\\ref{prop:lowertan}. \n\\end{proof}\n\n\\begin{prop}\\label{prop:goodtan3}\nSuppose that a diffeomorphism $F$ has a sectionally dissipative periodic saddle $p$ and there is a quadratic tangency between $W^s(p)$ and $W^u(F^N(p))$ at the point $q$. Then by an arbitrarily $C^\\infty$-small perturbation one can obtain a diffeomorphism $G$ with a tangency between the stable and the unstable manifolds of the continuation of $p$.\n\\end{prop}\n\\begin{proof}\nSince the case of a lower tangency was already considered in the previous proposition, we assume that there is a tangency from above at $q$. \nWithout loss of generality we can also suppose that $q\\notin W^{ss}(p)$ and the saddle $p$ has transverse homoclinic orbits that involve the same connected components of $W^u(p)\\setminus\\{p\\}$ and $W^s(p)\\setminus W^{ss}(p)$ as the orbit of tangency (see Proposition~\\ref{prop:trans} and Remark~\\ref{rem:trans}). \n\nSince the tangency at $q$ is a tangency from above, Proposition \\ref{prop:lowertan} yields that we can switch to another saddle heteroclinically related to $p$ and obtain, by a small perturbation of $F$, a lower tangency associated with the continuation of this new saddle. Let us denote the perturbed map by $\\hat{F}$ and the new saddle by $\\hat{p}$ and preserve the notation $p$ for the continuation of the original saddle. Note that we can take $\\hat{p}$ such that $\\operatorname{per}(\\hat{p}) = l\\cdot\\operatorname{per}(p)$ (for some $l\\in\\mathbb N$), $W^u(p, \\hat{F})\\pitchfork W^s(\\hat{p}, \\hat{F})\\ne\\emptyset$, and $W^s(p, \\hat{F})\\pitchfork W^u(\\hat{p}, \\hat{F})\\ne\\emptyset$, as in the proof of Proposition~\\ref{prop:lowertan}. \n\nApplying Proposition~\\ref{prop:goodtan2} to $\\hat{F}$, we obtain a diffeomorphism $\\hat{G}$ such that there is a tangency between the stable and the unstable manifolds of the continuation of $\\hat{p}$. Again, let us preserve the notation $p, \\hat{p}$ for the continuations of our saddles.\n\nSince $W^u(p, \\hat{G})\\pitchfork W^s(\\hat{p}, \\hat{G})\\ne\\emptyset$ and $\\operatorname{per}(p) \\mid \\operatorname{per}(\\hat{p})$, we have that $W^u(p, \\hat{G})$ accumulates on $W^u(\\hat{p}, \\hat{G})$. Analogously, since $W^s(p, \\hat{G})\\pitchfork W^u(\\hat{p}, \\hat{G})\\ne\\emptyset$, we have that $W^s(p, \\hat{G})$ accumulates on $W^s(\\hat{p}, \\hat{G})$. Then, arguing as in the proof of Proposition~\\ref{prop:lowertan}, we can make a small perturbation and obtain a diffeomorphism $G$ with a tangency between the stable and the unstable manifolds of the continuation of $p$, which finishes the proof. \n\n\\end{proof}\n\n\n\n\n\n\n\\subsubsection*{End of the proof}\n\nNow when we proved the auxiliary propositions, we can get back to the proof of the capture lemma.\nRecall that we assumed $F\\in \\operatorname{Diff}^\\infty(M)$ to have a sectionally dissipative saddle $p$ with a quadratic homoclinic tangency between $W^s(p)$ and $W^u(F^{N}(p))$ at the point $q$. Our goal is to match all assumptions of Proposition~\\ref{prop:pv} and obtain an extremely dissipative saddle heteroclinically related to the continuation of $p$ and having an orbit of tangency. However, we assume that Proposition~\\ref{prop:pv} is not applicable to $F$ yet. \n\nWe are going to perform a series of perturbations, each of them arbitrarily $C^\\infty$-small, and obtain in the end a diffeomorphism, a saddle, and a tangency for which Proposition~\\ref{prop:pv} is applicable. In order to simplify the notation, we will repeatedly replace $F$ with the perturbed maps. At each step we also replace our saddle $p$ either by its hyperbolic continuation or by another saddle heteroclinically related to this continuation, and we obtain a new point of tangency, which we continue to denote by $q$ though it is a different point associated with the new saddle $p$.\n\n{\\bf 1.} First, by Proposition~\\ref{prop:goodtan3} we can obtain, by a small perturbation, a new tangency between the stable and the unstable manifolds of the continuation of $p$. Thus, we replace the diffeomorphism $F$ by the perturbed map and preserve the original notation for the continuation of $p$ and for the new point of tangency. If we already had a tangency between $W^s(p)$ and $W^u(p)$, this step is redundant.\n\n{\\bf 2.} Now we assume that $F$ has a quadratic tangency between $W^u(p)$ and $W^s(p)$ at $q$, the saddle $p$ being, as always, sectionally dissipative.\nIn Section 5 of \\cite{PV} J.~Palis and M.~Viana prove that in this case a small perturbation yields another sectionally dissipative saddle $\\tilde{p}$ that has a homoclinic tangency associated with it, has a unique weakest contracting eigenvalue and, moreover, is heteroclinically related to the original one. \n\n\\medskip\nThere might be a small gap in the argument of J.~Palis and M.~Viana, but it is easy to fix it.\nIn order to prove this result they first impose some genericity condition\\footnote{They suppose that the map $\\Phi$ (defined before condition~(\\ref{eq:pvcond})) is an isomorphism.} on $F$ and then consider a one-parameter family $(F_\\mu), \\ F_0 = F,$ that generically unfolds the tangency. Their argument suggests, however, that for $\\mu > 0$ the tangency is unfolded with creation of transverse intersections and that there is also a sequence $(\\mu_j)_{j\\in \\mathbb N}, \\ \\mu_j > 0, \\ \\mu_j\\to 0$ as $j\\to\\infty$, such that the maps $F_{\\mu_j}$ have new tangencies associated with the continuation of $p$. But it may happen that this is not the case: it is possible that the new tangencies appear only for negative $\\mu$.\nHowever, this is always the case for a lower tangency, as one can see from the first part of the proof of Proposition \\ref{prop:goodtan}.\nThus, one can use Proposition \\ref{prop:lowertan} (with Remark~\\ref{rem:samepoint}) to switch to a heteroclinically related saddle with a lower tangency first and then argue as J.~Palis and M.~Viana do.\n\\medskip\n\nAs before, we replace $F$ by the perturbed map and $p$ by $\\tilde{p}$ without changing the notation.\n\n{\\bf 3.} Now we assume that the saddle $p$ has a unique weakest contracting eigenvalue and there is\\footnote{We might need another perturbation to move the tangency off $W^{ss}(p)$ and make it non-degenerate.} a quadratic homoclinic tangency at $q\\in W^u(p)\\cap (W^s(p)\\setminus W^{ss}(p))$. Using Proposition~\\ref{prop:trans} with Remark~\\ref{rem:trans}, we can also assume that there is a transverse homoclinic intersection between the same connected components of $W^u(p)\\setminus\\{p\\}$ and $W^s(p)\\setminus W^{ss}(p)$. \n\nIf the tangency at $q$ is a tangency from above, we can apply Proposition \\ref{prop:lowertan} (with Remark~\\ref{rem:samepoint}) to obtain a new tangency from below between $W^u(p)$ and $W^s(p)\\setminus W^{ss}(p)$. In this case we replace the map, the saddle, and the tangency by the new ones again. By Remark~\\ref{rem:intersections}, we can assume that there still are transversal homoclinic orbits that involve the same connected components of $W^u(p)\\setminus\\{p\\}$ and $W^s(p)\\setminus W^{ss}(p)$ as the tangency. If we have accidentally lost linearizability of $F^{\\operatorname{per}(p)}$ in the neighborhood of $p$ (actually, we could not), we can restore it by a small perturbation that preserves all relevant properties of our map.\n\n{\\bf 4.} Thus, now we assume that the tangency at $q$ is a tangency from below. If this tangency does not satisfy condition~(\\ref{eq:pvcond}), then apply Proposition~\\ref{prop:goodtan} with Remark~\\ref{rem:goodtan} to obtain, after a small perturbation, a new tangency that satisfies condition~(\\ref{eq:pvcond}). We can assume that the linearizability is preserved. Replace the map and the tangency by the new ones. The new tangency belongs to the same connected components of $W^u(p)\\setminus\\{p\\}$ and $W^s(p)\\setminus W^{ss}(p)$ as before (recall that in the proof of Prop.~\\ref{prop:goodtan} $\\hat{q}$ is close to $q$), therefore there are still transversal homoclinic intersections as in condition c) of Proposition~\\ref{prop:pv}. Then Proposition~\\ref{prop:pv} can be applied to our new $F$.\n\n\n{\\bf 5.} Proposition~\\ref{prop:pv} yields that after another perturbation we finally obtain an extremely dissipative saddle heteroclinically related to the continuation of the saddle $p$. Using Proposition~\\ref{prop:renormsaddletan}, we can also suppose that for this new map there is a homoclinic tangency associated with this extremely dissipative saddle.\n\\medskip\n\nAt each step of our argument we could replace saddle $p$ either by its continuation or by a heteroclinically related saddle only, so this final extremely dissipative saddle is heteroclinically related to the continuation of the original saddle $p$ that existed prior to any perturbation.\nWe can apply the capture lemma to the tangency associated with the extremely dissipative saddle and conclude, as above, that after a proper perturbation the unstable manifolds of the continuations of both the extremely dissipative saddle and the original one intersect a basin of a sink. Now the proof of the capture lemma is complete. \n\n\n\n\n\n\n\n\n\\section{Dominated splitting or instability}\\label{dichotomy}\n\n\\subsection{Statement and plan of the proof}\n\nC. Bonatti, L. J. D{\\'i}az and E. R. Pujals prove in \\cite[Cor. 0.3]{BDP} the following dichotomy for a $C^1$-generic diffeomorphism of a closed manifold: for each periodic hyperbolic saddle its homoclinic class either admits a dominated splitting or is contained in the closure of an infinite set of sinks or sources. In this section by a relatively simple argument also based on \\cite{BDP} we will deduce Theorem~\\ref{thm:instability} from a technical statement that underlies the result we have just quoted. Let us recall the statement of Theorem~\\ref{thm:instability} for convenience.\n\n\\begin{repthm2}\nFor a Baire-generic diffeomorphism $F\\in\\operatorname{Diff}^1(M)$, $M$ being a closed manifold, either any homoclinic class of $F$ admits some dominated splitting, or the Milnor attractor is Lyapunov unstable for $F$ or $F^{-1}.$\n\\end{repthm2}\n\nNote that the two cases considered in Theorem~\\ref{thm:instability} are not mutually exclusive. Indeed, one can take locally generic diffeomorphisms with unstable attractors from the previous sections, multiply them by a strong contraction, and thus obtain a dominated splitting whereas the attractors will still be unstable. What Theorem \\ref{thm:instability} really says is that the absence of a dominated splitting over some homoclinic class generically implies instability of the Milnor attractor (perhaps, for the inverse map).\n\nWe will need the following fact that, though not stated explicitly, is proved in \\cite{BDP}.\n\n\\begin{thm}[\\cite{BDP}, Lemma 1.9 + Lemma 1.10 + Prop. 2.1]\\label{BDP_p1}\nSuppose that $p$ is a periodic hyperbolic saddle of the diffeomorphism $F\\in\\operatorname{Diff}^1(M)$ and the homoclinic class $H(p,F)$ does not admit a dominated splitting. Then, for any sufficiently small $\\varepsilon >0,$ in any neighbourhood of $p$ one can find a periodic saddle $q$ with the following properties: \n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n\\item[--]{ $q$ is heteroclinically related to $p$,}\n\\item[--]{ there are linear maps $A_j$ $\\varepsilon $-close to the differentials $dF(F^{j-1}(q)), \\ j = 1,\\dots, {\\rm per \\,}(q),$ such that the composition $A = A_{{\\rm per \\,}(q)}\\circ\\dots\\circ A_1$ is either a contraction or a dilation.\\footnote{In this context a contraction (dilation) is meant to be a linear map with moduli of eigenvalues less than~1 (greater than~1), and we do not assume that the norm of this map is necessarily less than~1 (resp., greater than~1). Then, since $\\varepsilon $ is arbitrary, we don't actually need to specify which norm we use to define the $\\varepsilon $-perturbation of $dF$. By default, we will assume the operator norm that corresponds to the vector norm provided by the Riemannian structure.}}\n\\end{itemize}\n\\end{thm}\n\n\n\n\\begin{rem} \\label{rem:sinks}\nActually, it follows from Remarks 5.5 and 5.6 in \\cite{BDP} that if $H(p,F)$ contains a dissipative saddle $p_1$ heteroclinically related to $p$ then one can take $q$ such that the perturbation of the differentials along $O(q)$ yields a contraction. Respectively, if there is an area-expanding $p_2$, one may assume that $A$ is a dilation. \n\\end{rem}\n\nRecall that according to Franks' lemma \\cite[Lemma 1.1]{Franks} an $\\varepsilon $-perturbation of the differential $dF$ over a finite subset $B\\subset M$ can be realized by a diffeomorphism $G$ that is $10\\varepsilon $-close to $F$ in $C^1$ and coincides with $F$ on $B$ and outside some neighborhood of $B$ that may be chosen to be arbitrary small. Therefore Theorem~\\ref{BDP_p1} combined with Franks' lemma may be viewed as a means of creating sinks near the (continuation of the) saddle $p$.\n\nIn order to prove Theorem \\ref{thm:instability} we will first deduce from Theorem \\ref{BDP_p1} an analogue of the capture lemma: if there is a saddle $q\\in H(p, F)$ that can be turned into a sink by a small perturbation, then there is another saddle $Q\\in H(p, F)$ that not only becomes a sink after an appropriate perturbation, but also catches a part of the unstable manifold of the continuation of $p$ into its basin of attraction. Since the proof of this lemma is a little technical, it is presented in a separate subsection.\n\nThen a localized version of Theorem~\\ref{thm:instability} can be proved similarly to Theorem~\\ref{thm:NI}, the only difference is that we should obtain new sinks with the help of Theorem~\\ref{BDP_p1} and Franks' lemma and use the new capture lemma instead of the old one.\n\nThe global version of Theorem~\\ref{thm:instability} is obtained from the local version by an argument of the Kupka-Smale type similar to the proof of Cor. 0.3 in~\\cite{BDP}.\n\n\\subsection{Another capture lemma}\n\n\\begin{lem}[another capture lemma]\\label{lem:capture2}\nFor any $\\varepsilon _0>0$ there exists $\\varepsilon <\\varepsilon _0$ such that the following holds.\nSuppose that $H(p,F)$ does not admit any dominated splitting and the point $q$ provided by Theorem~\\ref{BDP_p1} for a given $\\varepsilon $ yields a contraction. Then by an $\\varepsilon _0$-perturbation of $F$ in $\\operatorname{Diff}^1(M)$ we can obtain a diffeomorphism $G$ such that the point $p$ is a hyperbolic saddle\\footnote{and it is a hyperbolic continuation of the saddle $p$ of $F$.} for $G$ and $W^u(p,G)$ intersects the attraction basin of some sink. \n\\end{lem}\n\n\n\\subsubsection*{Idea of the proof}\nConsider the saddle $q$ given by Theorem~\\ref{BDP_p1}. Since it is heteroclinically related to $p$, there is a transverse heteroclinic orbit $O(z)$ that accumulates to $O(p)$ in the past and to $O(q)$ in the future.\nWhen we perturb the map in the neighborhood of $O(q)$ in order to turn $q$ into a sink, we cannot a priori be sure that $z$ will end up in the basin of this sink. It may happen, informally speaking, that the orbit of $z$ will leave the neighborhood where the map was perturbed just after a few iterations and long before $O(z)$ feels any attraction towards the new sink.\n\nIn order to circumvent this we will find another saddle $Q$ heteroclinically related to $p$ and such that the orbit of $Q$ makes $k$ winds close to the orbit of $q$, $k$ being arbitrarily large, and then closes after a few iterations. Denote by $x$ a heteroclinic point that goes from $O(p)$ to $O(Q)$. We will perform the perturbation in the neighborhood of $O(Q)$ in the following way. First, as in Franks lemma, the orbit of $Q$ itself will not be perturbed. Second, during the first $k_1$ winds there will be no perturbation, and we will let the images of $x$ approach the orbit of $Q$ as they do for the unperturbed map $F$. We need these images to come so close to $O(Q)$ that when we finally perturb the map in the neighborhood of the $(k_1+1)$-th wind, the corresponding points of the orbit of $x$ would stay in this neighborhood during the whole wind. Since the wind is close to $O(q)$, the perturbation can be so chosen that the image of $x$ is actually attracted to $O(Q)$ during the wind. Then we can repeat the same procedure during the rest of the winds. \n\nWe will make sure that the number of these winds $k_2 = k - k_1$ is very large, much larger then $k_1$. Then $Q$ will become a sink no matter that there was no perturbation done during the first $k_1$ winds and there would be no perturbation during those few iterations that the orbit of $Q$ may spend far from the orbit of $q$. Moreover, if $k_2$ is large enough, the future orbit of $x$ will be attracted to this sink, which will conclude the proof, because $x\\in W^u(O(p))$. \n\n\\subsubsection*{Saddle $Q$}\n\nSince $q\\in H(p,F)$, there are transverse homoclinic intersections associated with $q$. Then there is also a non-trivial basic set $\\Lambda\\subset H(p,F)=H(q,F)$ such that $q\\in\\Lambda$ (see, for example, \\cite[Thm 6.5.5]{Katok}). For a fixed $\\delta>0$ and any $k\\in\\mathbb N$, there is a periodic saddle $Q\\in\\Lambda$ with the following property: the orbit of $Q$ makes at least $k$ winds $\\delta$-close to the orbit of $q$, and then closes after a number of iterations limited by a fixed integer that does not depend on~$k$. \n\nIndeed, any basic set admits Markov partitions of arbitrarily small diameter (see \\cite[$\\S 18.7$]{Katok}). Take a Markov partition of diameter less than $\\delta$ for $\\Lambda$ and consider the corresponding\\footnote{See \\cite[Thm 18.7.4.]{Katok}.} transitive Markov chain $(\\Sigma_A, \\sigma_A)$ that is semi-conjugated to the dynamics on $\\Lambda$. Take a finite word $w$ defined by which rectangles of partition are consecutively visited by the orbit of $q$, then consider its $k$-th power (under concatenation) $[w]^k$ and take another finite word $w_0$ such that, first, $[w]^kw_0$ is not a power of any word and, second, our Markov chain admits a transition between the last letter of $w_0$ and the first letter of $w$. Note that $w_0$ can be taken independent of $k$, at least if we assume that there is a rectangle of the partition not visited by $O(q)$. Then the periodic sequence $\\omega\\in \\Sigma_A$ defined by the word $[w]^kw_0$ exists, and it is mapped to the required periodic point $Q$ by the semi-conjugacy map. \n\nLet us state explicitly that when we speak about the ``winds'' of the orbit $O(Q)$ around the orbit $O(q)$, we assume that $Q$ is $\\delta$-close to $q$ and the same is true for $F^j(Q)$ and $F^j(q)$ for $j=1,2,\\dots,kn$, where $n$ is the period of $q$. Each wind is a subset of $O(Q)$ that consists of $n$ consecutive points, namely the points $F^{n(j-1) + i}(Q), \\; i = 0,\\dots, n-1$, for the $j$-th wind. We will denote the period of $Q$ by $N$. Obviously, $N>kn$. \n\nObserve that by taking $\\delta$ small enough we can ensure that the differentials at the points of $O(Q)$ that belong to the winds are $\\varepsilon $-close to differentials at the corresponding points of $O(q)$. Therefore the composition of differentials along any wind can be turned into a contraction by $2\\varepsilon $-perturbations of these differentials. Since the number of iterations that $O(Q)$ spends far from $O(q)$ is limited whereas $k$ may be taken arbitrarily large, we conclude that the composition of differentials along the whole $O(Q)$ may be turned into a linear contraction as well.\n\nFrom now on we assume for simplicity, that for each point of the orbit $O(Q)$ we fix local coordinates with the origin at this point and whenever we consider $F$ restricted to a small neighborhood of the orbit $O(Q)$, we use these local coordinates. Then we may informally speak, for instance, about a linear mapping from the vicinity of $Q$ to the vicinity of $F(Q)$, or even about the mapping coinciding or $C^1$-close to $dF(Q)$. A formal way to say the same thing would be to consider an exponential mapping (given by the Riemannian metric) and then consider $\\exp\\circ\\, dF(Q)\\circ\\exp^{-1}$. Moreover, we will switch to the euclidean metric and vector norm given by our fixed coordinates. When we change the vector norm, $\\varepsilon $-perturbations of the differentials become $C\\varepsilon $-perturbations for some positive constant $C$, so let us redefine $\\varepsilon $ so that we do not need to write this constant every time.\n\nThe points $p$ and $Q$ are heteroclinically related (since this relation is transitive), therefore $W^u(O(p))$ transversally intersects $W^s(O(Q))$ at some point $x$. Replacing if necessary the points $p,Q,x$ by their images under some iteration of $F$ we can find a number $r<\\min(\\delta,\\varepsilon )$ such that for the $r$-neighborhood $W$ of $O(Q)$ the following holds:\n\n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n\\item{$W$ is a union of disjoint balls $B_j, \\; j =1,\\dots,N$ of radius $r$ centered at the points of $O(Q)$;}\n\\item{in each ball $B_j$ the map $F$ is $\\varepsilon \/10$-close in $C^1$-topology to the linear map that corresponds to the differential at the center of the ball, i.e., to $dF(F^{j-1}(Q))$;}\n\\item{$x\\in W$, but the past semi-orbit of $x$ does not intersect $W$;}\n\\item{$x\\in W^s_{loc, r}(Q)\\cap W^u(p)$.}\n\n\\end{itemize}\nIn what follows we do not perturb $F$ outside $W$, therefore the point $x$ stays at the unstable manifold of the saddle $p.$\nWe will perform the perturbation of $F$ in $W$ in two steps.\n\n\\subsubsection*{Step 1: no perturbation during the first $k_1$ winds}\n \nDuring the first step that consists of $k_1$ winds we do not perform any perturbations. The number $k_1$ should be chosen so large that the following inequality holds:\n\\begin{equation}\\label{ineq:ratio}\n\\frac{\\dist\\left(F^{nk_1}(x), F^{nk_1}(Q)\\right)}{\\dist(x, Q)} < \\frac{1}{10(L+1)^{n-1}},\n\\end{equation}\nwhere $L$ is the Lipschitz constant for $F$.\\footnote{Note that we can not argue like that in the case of an arbitrary saddle. Imagine a situation when there is a weak repulsion during nearly the whole periodic orbit and at the very end a few decisive iterations make it a saddle.} The distance between the images of $x$ and $Q$ decreases because $Q\\in\\Lambda$ and $x\\in W^s_{loc, r}(Q)$. Indeed, it follows from the definition of a hyperbolic set that there are constants $c>0$ and $\\lambda<1$ such that for any $z\\in\\Lambda$ and any $y\\in W^s_{loc, r}(z)$, if $r$ is small enough, the following holds: \n\n$$\\forall j\\in\\mathbb N,\\;\\;\\dist(F^j(y), F^j(z)) \\le c\\lambda^j\\cdot\\dist(y,z).$$\n\nThis inequality implies that (\\ref{ineq:ratio}) holds for large values of $k_1$.\n\nDenote $x_1 = F^{nk_1}(x)$ and $Q_1 = F^{nk_1}(Q)$. Since $\\dist(x, Q) \\le r$, inequality~(\\ref{ineq:ratio}) implies that $\\dist(x_1, Q_1) < r\/(10(L+1)^{n-1})$. \n\n\\subsubsection*{Step 2: actual perturbation}\n\nNow the second step begins. It consists of $k_2=k-k_1$ winds. \n\nTheorem \\ref{BDP_p1} provides $n$ linear maps $A_1,\\dots, A_n$ such that these are $\\varepsilon $-perturbations of the differential along the orbit of $q$ and the composition $A = A_n\\circ\\cdots\\circ A_1$ is a contraction. Let us assume for simplicity that $A$ contracts the euclidean norm. In the general case this is true for some power of $A$ and the argument should be modified accordingly. \n\nConsider the $(k_1+1)$-th wind of $O(Q)$ around $O(q)$ that starts at $Q_1$ and continues up to $Q_n := F^{n-1}(Q_1)$. \nRecall that, since $Q_1$ is $\\delta$-close to $F^{nk_1}(q)=q$, we assume that the map $A_1$ is a $2\\varepsilon $-perturbation of the differential at $Q_1$, and, analogously, each $A_j$ is a $2\\varepsilon $-perturbation of the differential at the corresponding point of the wind.\n\nTake a ball $B(Q_1, \\frac{r}{10})$ of radius $\\frac{r}{10}$ centered at $Q_1$ and a ten times larger ball $B(Q_1, r)$ with the same center (this larger ball actually coincides with the previously defined ball $B_j$ with $j = nk_1+1$). \nWe can modify $F$ inside $B(Q_1, r)$ in such a way that inside $B(Q_1, \\frac{r}{10})$ the resulting map would coincide with the map $A_1$. This can be made by a $(c_1\\cdot 2\\varepsilon )$-perturbation, where $c_1\\ge 1$ depends on the radii of the two balls, but not on $\\varepsilon $.\\footnote{Actually, making the radii small and their ratio large, we could take $c_1$ arbitrarily close to~1.} \n\nAnalogously, for any $j\\in\\{2,\\dots, n\\}$ we can take balls of radii $\\frac{r}{10}$ and $r$ centered at $Q_j := F^{j-1}(Q_1)$ and modify $F$ inside the bigger ball so that the restriction of the new map to the small ball coincide with $A_j$. Since for different $j$ the big balls do not intersect, it still takes merely a $(c_1\\cdot 2\\varepsilon )$-perturbation to perform the overall modification. \n\nWe will preserve the notation $F$ for the modified map. It is important that after this modification the point $x$, in general, does not belong to the stable manifold of $Q$. However, at the end of the wind the corresponding iterate of $x$ is still inside the ball $B(Q_n, r)$.\n\nIndeed, recall that we have denoted by $d$ the distance between $Q_1$ and $x_1\\in O(x)$. Obviously, the distance between $x_2 = F(x_1)$ and $Q_2=F(Q_1)$ is less than $d\\cdot (L+1)$, where $L$ is the Lipschitz constant for the original $F$: we assume that $\\varepsilon $ is small and add 1 to $L$ in order to take the perturbation into account. Likewise, we have for $x_j = F^{j-1}(x_1)$ that $\\dist(x_j, Q_j) < d\\cdot (L+1)^{j-1}$.\nSince, as we required during the first step, $d\\cdot 10\\cdot (L+1)^{n-1} < r$, we conclude that during this wind the points of $O(x)$ stay inside the union of smaller balls where the original map was replaced by the maps $A_1,\\dots, A_n$.\n \nRecall that we assume the composition $A = A_n\\circ\\cdots\\circ A_1$ to be a contraction map in our euclidean metric. Denoting by $\\lambda_1$ the minimal rate of this contraction we obtain \n$$\\dist(F^{n}(x_1),F^{n}(Q_1)) \\le \\lambda_1\\cdot d < d.$$\nThis means that we can repeat the same modification procedure for the next wind and further on up to the end of the $k$-th wind.\n\n\n\\subsubsection*{Attraction to the sink}\n\n If $k_2$ is large enough, after Step 2 the point $Q$ becomes a sink and $x$ is in its basin of attraction. Indeed, let us show that if $k_2$ is sufficiently large, then any point $y$ that is $d$-close to $Q_1$ (in particular, the point $x_1\\in O(x)$) is attracted to this sink. Recall that $N = {\\rm per \\,} (Q) = (k_1+k_2)n + r_0$, where $r_0 = \\mathrm{length}(w_0)$ is smaller than some constant whereas $k_1$ and $k_2$ may be chosen arbitrary large, and we have already chosen $k_1$. We have \n$$\\dist{(F^{N}(y),Q_1)} = \\dist{(F^{N}(y),F^N(Q_1))}\\le \\left(\\lambda_1^{k_2}\\cdot(L+1)^{k_1n+r_0}\\right)\\cdot\\dist{(y, Q_1)}.$$\nFor a large $k_2$ the inequality $\\lambda_1^{k_2}\\cdot(L+1)^{k_1n+r_0} < \\frac{1}{2}$ holds. Then we have \n$$\\dist{(F^{N}(y),Q_1)} \\le \\frac{1}{2}\\dist{(y, Q_1)}.$$ \nThis shows that $x_1$ and its preimage $x$ are both attracted to the sink. \n\nThus after a $2c_1\\varepsilon $-modification of the initial map inside the domain $W$ we see that:\n\\begin{itemize}\\setlength\\itemsep{-0.1em}\n\\item{the past semi-orbit of $x$ is unchanged, and consequently $x$ is still in $W^u(p)$;}\n\\item{the orbit of $Q$ is unchanged, but $Q$ becomes a sink, and $x$ is in its basin of attraction.}\n\\end{itemize}\nIf $\\varepsilon $ is small enough, we have $2c_1\\varepsilon < \\varepsilon _0$. Then we have obtained an $\\varepsilon _0$-modification of the initial $F$ with the required property. This completes the proof of the lemma. \n\n\n\\subsection{Local version of Theorem~\\ref{thm:instability}}\n\nThe following statement is a localized version of Theorem \\ref{thm:instability}.\n\n\n\\begin{thm} \\label{prop:localized}\nSuppose that $F\\in\\operatorname{Diff}^1(M)$ has a neighborhood $U$ where the hyperbolic continuation of a periodic saddle $p$ of $F$ is defined, and, moreover, for any $G\\in U$ this saddle is dissipative.\nThen for a Baire-generic $G\\in U$ either $H(p(G),G)$ admits a dominated splitting, or $A_M(G)$ is Lyapunov unstable. \n\\end{thm}\n\n\n\\begin{proof}\nWe will show, for a generic diffeomorphism $G\\in U$, that if $H(p(G),G)$ does not admit a dominated splitting, then $G$ satisfies both assumptions of Proposition \\ref{prop:suff}, i.e., there is a sequence of sinks accumulating to $p(G)$ and the unstable manifold of $p(G)$ intersects the basin of some sink. Then Proposition \\ref{prop:suff} will imply that the Milnor attractor $A_M(G)$ is unstable.\n \nDenote by $DS(U)$ the subset of $U$ that consists of diffeomorphisms $G$ for which $H(p(G),G)$ admits a dominated splitting.\nConsider the interior of $DS(U)$ and denote by $V$ the complement of the closure of this interior in $U$: $V = U\\setminus {\\rm Cl}({\\rm Int} DS(U))$. It follows from the definition that $V$ contains a dense subset of diffeomorphisms $G$ for which $H(p(G),G)$ does not admit any dominated splitting. If $V$ is empty, we are done: the homoclinic class $H(p(G),G)$ admits a dominated splitting for a topologically generic $G\\in U$.\n\nIf $V\\ne\\emptyset$, Theorem \\ref{BDP_p1} (combined with Franks' lemma) implies that any diffeomorphism $G\\in V$ can be approximated by a diffeomorphism with a sink or a source $s$ close to the continuation of $p$. Since we assume that $p(G)$ is dissipative, by Remark \\ref{rem:sinks} we can also assume that $s$ is a sink.\n\nThen we can use the Newhouse argument as in the proof of Theorem \\ref{thm:NI} (see Section \\ref{sec:thmNI}). The only difference is that new sinks are obtained not by unfolding homoclinic tangencies but with the help of Theorem \\ref{BDP_p1} as above. This argument yields that generically in $V$ sinks accumulate to the hyperbolic continuation of $p$. \n\nFurther note that Lemma \\ref{lem:capture2} can be applied to any diffeomorphism $G\\in V$ for which $H(p(G),G)$ does not admit a dominated splitting, and such diffeomorphisms are dense in $V$. Then there is an open and dense subset of $V$ where for any diffeomorphism $G$ the unstable manifold $W^u(p(G), G)$ intersects a basin of a sink. Thus, there is a residual subset $R$ of $V$ where both assumptions of Proposition \\ref{prop:suff} are satisfied and therefore $A_M$ is unstable.\n\nThe observation that the union $R\\cup DS(U)$ is a residual subset of $U$ completes the proof. \n\\end{proof}\n\n\n\\subsection{Global version of Theorem~\\ref{thm:instability}}\n\nTheorem \\ref{thm:instability} may be proved now by essentially the same argument as Cor. 0.3. in~\\cite{BDP}.\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:instability}]\n Diffeomorphisms for which all periodic points of period less than $n$ are hyperbolic form an open and dense subset $U_n$ of $\\operatorname{Diff}^1(M)$. Let us split $U_n$ into the union of open subsets $U_{n,\\alpha}$ such that for $F\\in U_{n,\\alpha}$ the number of saddles of period less than $n$ is constant and equal to $k(\\alpha)$, and these saddles vary continuously with the map. Take some $U_{n,\\alpha}$ and denote those saddles by $p_1,\\dots,p_k$. \n\nFor each $j$ consider a set $DS(p_j)\\subset U_{n,\\alpha}$, where $H(p_j(G), G)$ admits a dominated splitting. Then fix $j$ and consider an open set $V_j = U_{n,\\alpha}\\setminus {\\rm Cl}({\\rm Int\\,} DS(p_j))$. Denote by $V^+_j$ (resp. $V^-_j$) an open subset of $V_j$ that consists of diffeomorphisms for which $p_j$ is dissipative (resp. area-expanding). The union $V^+_j\\cup V^-_j$ is dense in $V_j$. Application of Theorem \\ref{prop:localized} to $V^+_j$ yields a residual subset $R^+_j\\subset V^+_j$ where diffeomorphisms have unstable Milnor attractors. An analogous argument for $V^-_j$ in the inversed time provides a residual $R^-_j\\subset V^-_j$ such that for each $F$ in this set the inverse map $F^{-1}$ has an unstable Milnor attractor. Then $R_j=R^-_j\\cup R^+_j$ is residual in $V_j$. The union of $R_j$ and $DS(p_j)$ is a residual subset of $U_{n,\\alpha}$. Intersecting these $R_j\\cup DS(p_j)$ we obtain a residual subset $R_{n,\\alpha}$ of $U_{n,\\alpha}$. For any $F\\in R_{n,\\alpha}$ either homoclinic classes of all saddles $p_j$ admit some dominated splittings or the Milnor attractor of $F$ or $F^{-1}$ is unstable. Finally, $R=\\bigcap\\limits_{n}\\bigcup\\limits_\\alpha R_{n,\\alpha}$ is a global residual subset such that for any $F\\in R$ either every homoclinic class admits a dominated splitting, or $A_M(F)$ is unstable for $F$, or $A_M(F^{-1})$ is unstable for~$F^{-1}.$ \n\\end{proof}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe game of plates and olives is a purely combinatorial process that has an interesting application to topology and Morse theory. Morse theory involves the study of topological manifolds by considering the smooth functions on the manifolds. An \\emph{excellent Morse function} on the $2$-sphere is a smooth function from $S^2\\to \\mathbb{R}$ such that all the critical points are non-degenerate (i.e. the matrix of second partial derivatives is non-singular) and take distinct values. \n\nIf $f$ is an excellent Morse function on the sphere, $S^2$, with critical points $x_1,\\dots,x_m$ with $f(x_1)<\\dots0$ and $\\frac{1}{342}\\leq c_1\\leq c_2\\leq \\frac23$ such that\n\\begin{equation}\\label{thm:eq1}\t\n\\mbox{{\\bf Pr}}(c_1t\\leq O_t \\leq c_2t) \\ge 1 - e^{-Ct}.\n\\end{equation}\n\\item Furthermore, there exists an absolute constants $A>0$ such that for every $\\delta\\ge 0$ we have\n\\begin{equation}\\label{thm:eq2}\t\n\\mbox{{\\bf Pr}}(|O_t-\\mbox{{\\bf E}}(O_t)|\\geq \\delta t )\\leq e^{-A\\delta^2t}\n\\end{equation}\n\\item Also, w.h.p. no plate, except for the first plate, has more than $B\\log t$ olives at any time, for some absolute constant $B>0$.\n\\end{enumerate}\n\\end{theorem}\n\nWe prove in Section~\\ref{section expected number of olives} that\n\\begin{equation}\\label{eq:eot}\n1\/342\\leq \\mbox{{\\bf E}}(O_t)\\leq 2\/3.\n\\end{equation}\nNext in Section~\\ref{section concentration} we derive the concentration result~\\eqref{thm:eq2}, which altogether will imply~\\eqref{thm:eq1}. In Section~\\ref{section markov chain} we consider an auxiliary Markov chain process. It follows from our proofs that constants $1\/342$ and $2\/3$ are not optimal. As a matter of fact a computer simulation suggests that the number of olives $O_t$ is concentrated around $c t$, where $c\\approx 0.096$.\n\n\n\\section{Bounds on the expected number of olives}\\label{section expected number of olives}\n\n\n\\subsection{Lower bound}\n\n\nWe would like to show that the number of olives at a given time grows linearly with time~$t$. Towards this, we will establish two facts:\n\\begin{itemize}\n\t\\item we expect to return to a single plate a linear number of times, and\n\t\\item each time we return to a single plate, we expect to gain a positive number of olives.\n\\end{itemize}\nThis will give us a linear expectation. \n\nNow let us show that we expect to return to a single plate a linear number of times. If we have $\\ell\\ge 1$ plates, then the probability we do a plate move is at least\n\\[\n\\frac{\\binom{\\ell}2+1}{2\\ell+\\binom{\\ell}2+1}\\geq 1\/3.\n\\]\nLet $t_{plate}$ be the random variable that counts the number of plate moves we have after $t$ moves overall. Then\n\\begin{equation}\\label{equation expected number of plate moves}\n\\mbox{{\\bf E}}(t_{plate})\\geq\\sum_{i=1}^t 1\/3 = t\/3.\n\\end{equation}\nNow let us consider only plate moves to get a lower bound on the random variable $X$, which counts the number of times we transition from two plates to one plate.\n\nWe consider a related Markov chain. In this process, we will consider a random walk on the positive integers. We will start this walk at $1$ (plate). If we are currently at $1$, then we will move to $2$ with probability $1$. If we are currently at $2$, we will move to $1$ with probability $1\/2$ and to $3$ with probability $1\/2$. If we are at $k\\geq 3$, we will move to $k-1$ with probability $3\/4$ and to $k+1$ with probability $1\/4$. This Markov chain will be indexed by time $t_{plate}$ as it only models moves made when there is at least one plate.\n\nObserve that in our model, for $\\ell\\geq 3$,\n\\begin{equation}\\label{eq:rm_plate}\n\\mbox{{\\bf Pr}}(P^-|\\text{there are }\\ell\\text{ plates currently and we perform a plate move})=\\frac{\\binom{\\ell}{2}}{\\binom{\\ell}2+1}\\geq 3\/4,\n\\end{equation}\nand\n\\[\n\\mbox{{\\bf Pr}}(P^+|\\text{there are }\\ell\\text{ plates currently and we perform a plate move})=\\frac{1}{\\binom{\\ell}2+1}\\leq 1\/4.\n\\]\nThus the Markov chain gives an underestimate for how often we transition from two plates to one plate. Let $N_{1,1}(t_{plate})$ be the random variable that tracks the total number of times this Markov chain returns to a state with a single plate, given that we start at a state with a single plate and a total of $t_{plate}$ plate moves have been made.\n\nBy Theorem \\ref{theorem Markov chain}, we have that $\\mbox{{\\bf E}}(N_{1,1}(t_{plate}))\\geq \\mbox{{\\bf E}}(t_{plate})\/19$. Note that $\\mbox{{\\bf E}}(N_{1,1}(t_{plate}))\\leq \\mbox{{\\bf E}}(X)$, so\n\\[\n\\mbox{{\\bf E}}(X)\\geq \\mbox{{\\bf E}}(t_{plate})\/19\\geq t\/57.\n\\]\n\n\nNow we explore what happens each time we transition from two plates to one plate. Consider a state in the process that currently has two plates. If the second plate currently has olives on it, then the probability that next time we make a plate move, plate $2$ still has olives is at least $1\/2$. (We can immediately make the plate move.) If the second plate currently has no olives on it, then the probability there is at least one olive on it when we make the next plate move is at least $1\/6$. ($1\/3$ probability to add an olive, $1\/2$ probability to perform a plate move once we have added the olive.) Thus we have at least a $1\/6$ chance of adding an olive to the first plate each time we reduce the number of plates to $1$. Let $Y$ be a random variable that counts the number of times we add at least one olive to the first plate from a plate move, given that we transition from $2$ plates to $1$ plate $X$ times. Then\n\\[\n\\mbox{{\\bf E}}(Y)\\geq \\mbox{{\\bf E}}(X)\/6.\n\\]\n\nNow we put everything together. We will only consider olives on the first plate. Let $O_t^{(1)}$ denote the number of olives on plate $1$ at time $t$. Let $O^{(1)+}_t$ and $O^{(1)-}_t$ denote the total number of olives that were added to (respectively subtracted from) plate $1$ from $O^+$ (respectively $O^-$) moves. Note that $\\mbox{{\\bf E}}(O^{(1)+}_t-O^{(1)-}_t)\\geq 0$ since the probability of performing an $O^+$ move is always at least the probability of performing an $O^-$ move. Finally, let $O^{plate}_t$ denote the total number of olives added to the first plate from plate moves. Then $O^{plate}_t\\geq Y$. This gives us that\n\\beq{Ot}\n\\mbox{{\\bf E}}(O_t)\\geq \\mbox{{\\bf E}}(O_t^{(1)})=\\mbox{{\\bf E}}(O^{(1)+}_t-O^{(1)-}_t+O^{plate}_t)\\geq \\mbox{{\\bf E}}(Y)\\geq \\mbox{{\\bf E}}(X)\/6\\geq t\/342.\n\\end{equation}\n\n\n\\subsection{Upper bound}\nNow we bound the expected value of~$O_t$ from above. Let $O_t^+$ and $O_t^-$ are the random variables that count the number of $O^+$ moves and the number of $O^-$ moves after $t$ total moves, respectively. Clearly,\n\\[\nO_t=O_t^+-O_t^-=t-t_{plate}-2O_t^- \\le t-t_{plate}.\n\\]\nThus, by \\eqref{equation expected number of plate moves} we conclude that\n\\[\n\\mbox{{\\bf E}}(O_t)\\leq t-\\mbox{{\\bf E}}(t_{plate})\\leq 2t\/3.\n\\]\nAnd this proves~\\eqref{eq:eot}.\n\n\n\n\n\\section{Concentration}\\label{section concentration}\n\nSuppose that we transition from a state with two plates to a state with a unique plate at times $t_1,t_2,\\ldots,t_m$ and recall that $O_t$ denotes the number of olives at time $t$. Define $t_0:=1$. Let $X_i=O_{t_{i+1}}-O_{t_i}$. Then the $X_i$ are independent random variables and based on the previous section we have $\\mbox{{\\bf E}}(X_i)\\geq 1\/342$. Then $S_m:=O_{t_m}=\\sum_{i=0}^mX_i$. We can argue for concentration of $S_m$ as follows.\n\nNote that from \\eqref{equation expected number of plate moves} $\\mbox{{\\bf E}}(t_{plate})\\geq t\/3$, and by the Chernoff bound we have for any $0\\le \\delta\\le 1$,\n\\[\n\\mbox{{\\bf Pr}}(t_{plate}<(1-\\delta)t\/3)\\leq e^{\\frac{-\\delta^2t}6}.\n\\]\nAs we are not trying to optimize the constant $A$, we can be imprecise here and choose $\\delta=1\/4$, giving\n\\begin{equation}\\label{equation plate probability}\n\\mbox{{\\bf Pr}}(t_{plate}0$ are constants. Let $\\m_i=\\mbox{{\\bf E}}(X_i)$ and $\\m=\\m_1+\\cdots+\\m_m$. Note that we have\n\\begin{equation}\\label{eq:mui}\n\\m_i\\leq \\sum_{k=1}^\\infty kC\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma^k = \\frac{C\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma}{(\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma-1)^2}<\\infty.\n\\end{equation}\nNow we can easily prove a concentration result for this situation. We modify an argument from~\\cite{FP}. We prove\n\\beq{conc1}\n\\mbox{{\\bf Pr}}(|S_m-\\m|\\geq \\delta m)\\leq e^{-A\\delta^2m}\n\\end{equation}\nfor some constant $A>0$. That means we have replaced \\eqref{Ot} by a concentration inequality.\n\nWe write, for $\\lambda>0$ such that $e^\\lambda<1\/\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma$,\n\\[\n\\mbox{{\\bf E}}(X_i^2e^{\\lambda X_i})=\\sum_{k=0}^\\infty k^2e^{\\lambda k}\\mbox{{\\bf Pr}}(X_i=k)\\leq C\\sum_{k=0}^\\infty k^2(\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma e^\\lambda)^k\\leq \\frac{3C}{(1-\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma e^\\lambda)^3}.\n\\]\nNow $e^x\\leq 1+x+x^2e^x$ for $x\\geq 0$, and so, using the above, we have\n\\[\n\\mbox{{\\bf E}}(e^{\\lambda X_i})\\leq 1+\\lambda\\m_i+\\lambda^2\\brac{\\frac{3C}{(1-\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma e^{\\lambda})^3}}< 1+\\lambda\\m_i+\\lambda^2\\brac{1+\\frac{3C}{(1-\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma e^{\\lambda})^3}}.\n\\]\nSince $\\mbox{{\\bf Pr}}(S_m\\geq \\m_i m+\\delta m) = \\mbox{{\\bf Pr}}(e^{\\lambda S_m} \\geq e^{\\lambda (\\m_i m+\\delta m)})$ and $X_i$s are independent, the Markov bound implies that\n\\begin{align*}\n\\mbox{{\\bf Pr}}(S_m\\geq \\m_i m+\\delta m)&\\leq e^{-\\lambda(\\m_i m+\\delta m)}\\prod_{i=1}^m\\mbox{{\\bf E}}(e^{\\lambda X_i})\\\\ \n&\\leq \\exp\\set{-\\lambda(\\m_i m+\\delta m)}\\cdot \\brac{1+\\lambda\\m_i+\\lambda^2\\brac{1+\\frac{3C}{(1-\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma e^{\\lambda})^3}}}^m\\\\\n&\\leq \\exp\\set{-\\lambda(\\m_i m+\\delta m)}\\cdot \\exp\\set{\\brac{\\lambda\\m_i+\\lambda^2\\brac{1+\\frac{3C}{(1-\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma e^{\\lambda})^3}}}m}\\\\\n&= \\exp\\set{-\\lambda\\delta m + \\lambda^2\\brac{1+\\frac{3C}{(1-\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma e^{\\lambda})^3}}m}\\\\\n&\\leq \\exp\\set{-\\lambda\\delta m + \\lambda^2(1+3C\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma^{-3})m},\n\\end{align*}\nwhere $\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma=\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma(\\delta)>0$ is a constant such that\n\\beq{star}\ne^\\lambda\\leq (1-\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma)\/\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma.\n\\end{equation}\nNow choose $\\lambda=\\delta\/(2(1+3C\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma^{-3}))$ and $\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma$ such that \\eqref{star} holds. Such a choice of $\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma$ is always possible since as $\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma\\to 0$, $\\exp\\set{\\delta\/(2(1+3C\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma^{-3}))}\\to 1$ and $(1-\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma)\/\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma\\to 1\/\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma>1$. Then\n\\[\n\\mbox{{\\bf Pr}}\\brac{S_m\\geq \\m+\\delta m} \\leq\n\\max_{1\\le i\\le m}\\mbox{{\\bf Pr}}\\brac{S_m\\geq \\m_i m+\\delta m)}\\leq \\exp\\set{-\\frac{\\delta^2m}{4(1+3\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma^{-3})}}.\n\\]\nTo bound $\\mbox{{\\bf Pr}}(S_m\\leq \\m-\\delta m)$, we proceed similarly. We have $e^{-x}\\leq 1-x+x^2e^x$, so\n\\[\n\\mbox{{\\bf E}}(e^{-\\lambda X_i})\\leq 1-\\lambda\\m_i+\\lambda^2\\brac{\\frac{3C}{(1-\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma e^{\\lambda})^3}}< 1-\\lambda\\m_i+\\lambda^2\\brac{1+\\frac{3C}{(1-\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma e^{\\lambda})^3}}\n\\]\nand\n\\begin{align*}\n\\mbox{{\\bf Pr}}(S_m\\leq \\m_i m-\\delta m)&\\leq e^{\\lambda(\\m_i m-\\delta m)} \\prod_{i=1}^m\\mbox{{\\bf E}}(e^{-\\lambda X_i})\\\\ \n&\\leq \\exp\\set{\\lambda(\\m_i m-\\delta m)}\\cdot \\brac{1-\\lambda\\m_i+\\lambda^2\\brac{1+\\frac{3C}{(1-\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma e^{\\lambda})^3}}}^m\\\\\n&\\leq \\exp\\set{\\lambda(\\m_i m-\\delta m)}\\cdot \\exp\\set{\\left(-\\lambda\\m_i+\\lambda^2\\brac{1+\\frac{3C}{(1-\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma e^{\\lambda})^3}}\\right)m}\\\\ \n&\\leq \\exp\\set{-\\lambda\\delta m + \\lambda^2(1+3C\\varepsilon} \\def\\f{\\phi} \\def\\F{{\\Phi}} \\def\\vp{\\varphi} \\def\\g{\\gamma^{-3})m},\n\\end{align*}\nand we can proceed as before. This completes the proof of~\\eqref{conc1}.\n\nFor \\eqref{conc1} to be useful, we need to show that w.h.p. $m$ is linear in $t$. We condition on performing a plate move. Let the random variables $\\tau_i$ for $1\\leq i<\\infty$ count how many times we have exactly $i$ plates after $t$ steps, where we only count when the number of plates change. So, if we are at e.g. two plates and we make three olive moves before the next plate move, we only count this as having two plates once. Then $t_{plate}=\\sum_{i=1}^\\infty \\tau_i$ and $\\tau_1=m$. Note that we can express $\\tau_1$ as a sum of $\\tau_2$ indicator random variables that denotes if on the $j$th time we are at two plates, we then transition to one plate. Note that the probability of such a transition is 1\/2 (when we have exactly two plates, there is one way to remove a plate and one way to add a plate, giving equal probability of moving to $1$ plate vs. $3$ plates) and so $\\mbox{{\\bf E}}(\\tau_1) = \\tau_2\/2$.\n\n\nWe will consider two cases based on the value of $\\tau_2$. If $\\tau_2\\geq 3t_{plate}\/19$, we have from equation \\eqref{equation plate probability} and the Chernoff bound,\n\\begin{align*}\n\\mbox{{\\bf Pr}}(\\tau_1\\leq t\/76=(1\/3)&(1\/4)(3\/19)t)\\leq \\mbox{{\\bf Pr}}(\\tau_1<\\tau_2\/3)+\\mbox{{\\bf Pr}}(t_{plate}\\tau_2$, a contradiction. Thus, there exists an absolute constant $D>0$ such that \n\\begin{equation}\\label{no2}\nm=\\tau_1\\geq \\frac{t}{76}\\text{ with probability at least }1-e^{-Dt}.\n\\end{equation}\n\nNow observe that~\\eqref{tail1} also implies that for $k = \\log_\\rho (1\/Ct^2) = B\\log t$ (for some constant $B>0$) we have\n\\begin{equation}\\label{no1}\n\\mbox{{\\bf Pr}}(t_{i+1}-t_i\\geq k) \\le C\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma^k = 1\/t^2\n\\end{equation}\nand so\n\\begin{equation}\\label{eq:log}\n\\mbox{{\\bf Pr}}\\left(\\bigcup_{1\\le i\\le m} (t_{i+1}-t_i\\geq k)\\right) \\le t \\cdot \\frac{1}{t^2} = o(1).\n\\end{equation}\nNote that between time $t_{i}$ and $t_{i+1}$ the number of olives at any plate different from the first one is at most $t_{i+1}-t_i$ and so~\\eqref{eq:log} implies that w.h.p. no plate, except for the first plate has more than $B\\log t$ olives at any time. Part (c) of Theorem \\ref{theorem main thorem} follows directly from \\eqref{eq:log}.\n\nNow let $T = O_t - O_{t_m} = O_t - S_m$. Then, by~\\eqref{tail1} we have $\\mbox{{\\bf Pr}}(T\\geq k)\\leq C\\rho^k$ and the triangle inequality implies\n\\[\n|O_t - \\mbox{{\\bf E}}(O_t)| = |T+S_m - \\mbox{{\\bf E}}(O_t)| \\le |S_m - \\mu| + |T + \\mu - \\mbox{{\\bf E}}(O_t)| = |S_m - \\mu| + |T - \\mbox{{\\bf E}}(T)|.\n\\]\nThus,\n\\[\n\\mbox{{\\bf Pr}}(|O_t - \\mbox{{\\bf E}}(O_t)| \\ge \\delta t) \\le \\mbox{{\\bf Pr}}(|S_m - \\mu| \\ge \\delta t\/2)+\\mbox{{\\bf Pr}}(|T-\\mbox{{\\bf E}}(T)|\\geq \\delta t\/2).\n\\]\nFurthermore, since $T\\ge 0$ and $\\mbox{{\\bf E}}(T)=O(1)$ (cf.~\\eqref{eq:mui}) we get that $\\mbox{{\\bf Pr}}(|T-\\mbox{{\\bf E}}(T)|\\geq \\delta t\/2) \\le \\mbox{{\\bf Pr}}(T\\geq \\delta t\/2)$. Hence, \n\\begin{align*}\n\\mbox{{\\bf Pr}}(|O_t - \\mbox{{\\bf E}}(O_t)| \\ge \\delta t) &\\le \\mbox{{\\bf Pr}}(|S_m - \\mu| \\ge \\delta t\/2)+\\mbox{{\\bf Pr}}(T\\geq \\delta t\/2)\\\\\n&\\le \\mbox{{\\bf Pr}}(|S_m - \\mu| \\ge \\delta m\/2)+\\mbox{{\\bf Pr}}(T\\geq \\delta m\/2)\\\\\n& \\le e^{-A\\delta^2m\/4} +C\\rho} \\def\\R{\\Rho} \\def\\s{\\sigma} \\def\\S{\\Sigma^{\\delta m\/2}.\n\\end{align*}\nThis together with \\eqref{no2} proves Theorem \\ref{theorem main thorem}(b) and this completes the proof of Theorem~\\ref{theorem main thorem}.\n\n\\section{A related Markov chain}\\label{section markov chain}\n\n\\begin{theorem}\\label{theorem Markov chain}\n\tConsider a random walk on the positive integers: If we are currently at $1$, then we will move to $2$ with probability $1$. If we are currently at $2$, we will move to $1$ with probability $1\/2$ and to $3$ with probability $1\/2$. If we are at $k\\geq 3$, we will move to $k-1$ with probability $3\/4$ and to $k+1$ with probability $1\/4$.\n\t\n\tLet $N_{1,1}(t)$ be the random variable that counts the number of times we return to state $1$ when we start the walk at state $1$. Then w.h.p. we have that $N_{1,1}(t)\\geq t\/19$.\n\\end{theorem}\n\n\\begin{proof}\n\tNotice that we cannot return to $1$ in an odd number of steps. Let $f_{1,1}(2t)$ be the probability that the first time we return to state~1 after $2t$ steps, given that we start at state~1 i.e. the probability that the \\emph{first return time} is $2t$. Let $X_j$ be the location at time~$j$. So $X_0 = X_{2t} = 1$, $X_1 = X_{2t-1}=2$ and $X_j\\neq 1$ for each $20$.\n \n For $z$ belonging to a small disc $D_\\epsilon = \\{z\\in \\mathbb{C}|\\ |z| < \\epsilon\\}$, the space $V_z$ is a complex manifold with boundary, homotopically \n equivalent to a bouquet $\\vee S^{N-1}$ \n of $\\mu$ spheres, \\cite{M}. \n \n The family of free abelian groups \n \n \\begin{equation}\n Q(f;z) := \\tilde H_{N-1}(V_z;\\mathbb{Z})\\overset{\\sim}{=} \\mathbb{Z}^\\mu,\\ z\\in \\overset{\\bullet}D_\\epsilon := \n D_\\epsilon \\setminus \\{0\\}, \n \\end{equation}\n\n\n ($\\tilde H$ means that we take the reduced homology for $N = 1$), \n carries a flat Gauss - Manin conection. \n \n Take $t\\in\\mathbb{R}_{>0}\\cap \\overset{\\bullet}D_\\epsilon$; the lattice \n $Q(f;t)$ does not depend, up to a canonical isomorphism, \n on the choice of $t$. Let us call this lattice $Q(f)$. The linear operator \n \n \\begin{equation}\n T(f): Q(f) \\overset{\\sim}{\\longrightarrow} Q(f)\n \\end{equation}\n\n\n induced by the path $p(\\theta) = e^{i\\theta}t,\\ 0\\leq \\theta\\leq 2\\pi$, is called the classical monodromy of the germ $(f,0)$. \n \n In all the examples below $T(f)$ has finite order $h$. The \n eigenvalues of $T(f)$ have the form $e^{2\\pi i k\/h},\\ k\\in \\mathbb{Z}$. The set of suitably chosen $k$'s for each eigenvalue are called \n the {\\it spectrum} of our singularity. \n\n\\subsection{Morse deformations}\n\\label{sub22}\n\n The $\\mathbb{C}$-vector space $\\operatorname{Miln}(f,0)$ may be identified with the \n tangent space to the base $B$ of the miniversal defomation \n of $f$. For \n $$\n \\lambda \\in B^0 = B\\setminus \\Delta\n $$ \n where $\\Delta\\subset B$ is an analytic subset of codimension $1$, \n the corresponding function $f_\\lambda: \\mathbb{C}^N\\to \\mathbb{C}$ has $\\mu$ nondegenerate Morse critical points with distinct critical values, \n and the algebra $\\operatorname{Miln}(f_\\lambda)$ is semisimple, \n isomorphic to $\\mathbb{C}^\\mu$. \n \n Let $0\\in B$ denote the point corresponding to $f$ itself, so that \n $f = f_0$, and pick $t\\in \\mathbb{R}_{>0}\\cap \\overset{\\bullet}D_\\epsilon$ as in \\S \\ref{sub21}. \n \n Afterwards \n pick $\\lambda\\in B^0$ close to $0$ in such a way that the critical values $z_1, \\ldots z_\\mu$ of $f_\\lambda$ have absolute values $\\ll t$. \n \n As in \\S \\ref{sub21}, \n for each \n $$\n z\\in \\tilde D_\\epsilon := D_\\epsilon\\setminus\\{z_1, \\ldots z_\\mu\\}\n $$\n the Milnor fiber $ V_z$ has the homotopy type of \n a bouquet $\\vee S^{N-1}$ \n of $\\mu$ spheres, and we will be interested in the middle \n homology \n $$\n Q(f_\\lambda;z) = \\tilde H_{N-1}(V_z;\\mathbb{Z})\\overset{\\sim}{=} \\mathbb{Z}^\\mu\n $$\n\n \n The lattices $Q(f_\\lambda;z)$ carry a natural bilinear product induced by the cup product in the homology which is symmetric (resp. skew-symmetric) when $N$ is \n odd (resp. even). \n \n The collection of these lattices, \n when $z\\in \\tilde D_\\epsilon$ varies, carries a flat Gauss - Manin connection. \n \n Consider an ''octopus'' \n $$\n Oct(t)\\subset\\mathbb{C}\n $$ \n with the head at $t$: a collection of non-intersecting \n paths $p_i$ (''tentacles'') connecting $t$ with $z_i$ and not meeting the critical values $z_j$ otherwise. It gives rise \n to a base \n $$\n \\{b_1,\\ldots, b_\\mu\\}\\subset Q(f_\\lambda) := Q(f_\\lambda;t)\n $$ \n (called ''distinguished'') where $b_i$ is the cycle vanishing \n when being transferred from $t$ to $z_i$ along the tentacle $p_i$, \n cf. \\cite{Gab}, \\cite{AGV}. \n \n The Picard - Lefschetz formula describes the action of the fundamental group $\\pi_1(\\tilde D_\\epsilon;t)$ on \n $Q(f_\\lambda)$ with respect to this basis. Namely, \n consider a loop $\\gamma_i$ which turns around $z_i$ along the tentacle $p_i$, \n then the corresponding transformation of $Q(f_\\lambda)$ is the reflection \n (or transvection) $s_i := s_{b_i}$, cf. \\cite{Lef}, Th\\'eor\\`eme fondamental, Ch. II, p. 23. \n \n The loops $\\gamma_i$ generate the fundamental group \n $\\pi_1(\\tilde D_\\epsilon)$. Let \n $$\n \\rho:\\ \\pi_1(\\tilde D_\\epsilon;t)\\to GL(Q(f_\\lambda))\n $$\n denote the monodromy representation. The image of $\\rho$, denoted by $G(f_\\lambda)$ and called the {\\it monodromy group of $f_\\lambda$}, \n lies inside the subgroup \n \\newline $O(Q(f_\\lambda))\\subset GL(Q(f_\\lambda))$ \n of linear transformations respecting the above mentioned \n bilinear form on $Q(f_\\lambda)$. \n \n The subgroup $G(f_\\lambda)$ is generated by $s_i, 1\\leq i \\leq \\mu$. \n \n As in \\S \\ref{sub21}, we have the monodromy operator \n $$\n T(f_\\lambda)\\in G(f_\\lambda),\n $$\n the image by $\\rho$ of the path $p\\subset \\tilde D_\\epsilon$ starting at $t$ and going around all points $z_1, \\ldots, z_\\mu$.\n \n This operator \n $T(f_\\lambda)$ is now a product of $\\mu$ simple reflections\n $$\n T(f_\\lambda) = s_1s_2\\ldots s_\\mu,\n $$\n - this is because the only critical value $0$ of $f$ became \n $\\mu$ critical values $z_1, \\ldots, z_\\mu$ of $f_\\lambda$. \n \n One can identify the relative (reduced) homology\n $\\tilde H_{N-1}(V_t, \\partial V_t;\\mathbb{Z})$ with the dual group \n $\\tilde H_{N-1}(V_t;\\mathbb{Z})^*$, and one defines a map \n $$\n \\text{var}: \\tilde H_{N-1}(V_t, \\partial V_t;\\mathbb{Z}) \\to \\tilde H_{N-1}(V_t;\\mathbb{Z}),\n $$\n called a {\\it variation operator}, which translates to a map\n $$\n L: Q(f_\\lambda)^* \\overset{\\sim}{\\longrightarrow} Q(f_\\lambda)\n $$\n (''Seifert form'') \n such that the matrix $A(f_\\lambda)$ of the bilinear form \n in the distinguished basis is \n $$\n A(f_\\lambda) = L + (-1)^{N-1}L^t,\n $$\n and \n $$\n T(f_\\lambda) = (-1)^{N-1}LL^{-t}.\n $$\n \n\n \n \n A choice of a path $q$ in $B$ connecting $0$ with $\\lambda$, \n enables one to identify $Q(f)$ with $Q(f_\\lambda)$, and \n $T(f)$ will be identified with $T(f_\\lambda)$. \n \n The image $G(f)$ of the monodromy group $G(f_\\lambda)$ in \n $GL(Q(f)) \\overset{\\sim}{=} GL(Q(f_\\lambda))$ is called the monodromy group \n of $f$; it does not depend on a choice of a path $q$.\n \n \\subsection{Sebastiani - Thom factorization}\n\\label{sub23}\n\nIf $g\\in \\mathbb{C}[y_1,\\ldots, y_M]$ is another function, the sum, or \n{\\bf join} of two singularities \n$f\\oplus g:\\ \\mathbb{C}^{N+M}\\to \\mathbb{C}$ is defined by \n$$\n(f\\oplus g)(x,y) = f(x) + g(y)\n$$\nObviously we can identify \n$$\n\\operatorname{Miln}(f\\oplus g)\\overset{\\sim}{=} \\operatorname{Miln}(f)\\otimes \\operatorname{Miln}(g)\n$$\nNote that the function $g(y) = y^2$ is a unit for this operation.\n\nIt follows that the singularities \n$f(x_1,\\ldots, x_N)$ and \n$$\nf(x_1,\\ldots, x_N) + x_{M+1}^2 + \\ldots + x^2_{N+M}\n$$\nare ''almost the same''. In order to have good signs (and for other purposes) it is \nconvenient to add some squares to a given $f$ to get \n$N\\equiv 3\\mod(4)$. \n\nThe fundamental Sebastiani - Thom theorem, \\cite{ST}, says that there exists a natural isomorphism of lattices \n$$\nQ(f\\oplus g) \\overset{\\sim}{=} Q(f)\\otimes_\\mathbb{Z} Q(g),\n$$\nand under this identification \nthe full monodromy decomposes as \n$$\nT_{f\\oplus g} = T_f\\otimes T_g\n$$\nThus, if \n$$\n\\operatorname{Spec}(T_f) = \\{e^{\\mu_p\\cdot 2\\pi i\/h_1}\\},\\ \n\\operatorname{Spec}(T_f) = \\{e^{\\nu_q\\cdot 2\\pi i\/h_2}\\}\n$$ \nthen \n$$\n\\operatorname{Spec}(T_{f\\oplus g}) = \\{e^{(\\mu_p h_2 + \\nu_qh_1)\\cdot 2\\pi i\/h_1h_2}\\}\n$$\n\n\\subsection{Simple singularities}\n\\label{sub24}\n\n Cf. \\cite{AGV} (a), 15.1. They are:\n $$\n x^{n+1},\\ n\\geq 1,\n \\eqno{(A_n)}\n $$\n $$\n x^2y + y^{n-1},\\ n\\geq 4\n \\eqno{(D_n)}\n $$\n $$\n x^4 + y^3\n \\eqno{(E_6)}\n $$\n $$\n xy^3 + x^3\n \\eqno{(E_7)}\n $$\n $$\n x^5 + y^3\n \\eqno{(E_8)}\n $$\n \n Their names come from the following facts: \n \n --- their lattices of vanishing cycles may be identified with the corresponding root lattices;\n \n --- the monodromy group is identified with the corresponding Weyl group; \n \n --- the classical monodromy $T_f$ is a Coxeter element, therefore its order\n $h$ is equal to the Coxeter number, and \n $$\n \\operatorname{Spec}(T_f) = \\{e^{2\\pi i k_1\/h},\\ldots, e^{2\\pi i k_r\/h}\\}\n $$ \n where the integers\n $$\n 1 = k_1 < k_2 < \\ldots < k_r = h - 1,\n $$ \n are the exponents of our root system.\n \n We will discuss the case of $E_8$ in some details below. \n \n\\section{Cartan - Coxeter correspondence}\n\\label{sec3}\n\n\\subsection{Lattices, polarization, Coxeter elements}\n\\label{sub31}\n\n\nLet us call {\\it a lattice} a pair $(Q, A)$ where $Q$ is a free abelian group, and \n$$\nA: Q\\times Q\\to \\mathbb{Z}\n$$\na symmetric bilinear map (''Cartan matrix''). We shall identify \n$A$ with a map\n$$\nA: Q \\to Q^\\vee := Hom(Q,\\mathbb{Z}).\n$$\n{\\it A polarized lattice} is a triple $(Q, A, L)$ where \n$(Q, A)$ is a lattice, and \n$$\nL:\\ Q\\overset{\\sim}{\\longrightarrow} Q^\\vee\n$$\n(''variation'', or ''Seifert matrix'') is an isomorphism such that\n\n\\begin{equation}\nA = A(L) := L + L^\\vee\n\\end{equation}\n\nwhere\n$$\nL^\\vee: Q = Q^{\\vee\\vee}\\overset{\\sim}{\\longrightarrow} Q^\\vee\n$$\nis the conjugate to $L$. \n\n\nThe {\\it Coxeter automorphism} of a polarized lattice is defined by \n\n\\begin{equation}\nC = C(L) = - L^{-1}L^\\vee \\in GL(Q).\n\\end{equation}\n\n\nWe shall say that the operators $A$ and $C$ are in a {\\it Cartan - Coxeter correspondence}.\n\n\\vspace{2mm}\n\n\n\\textbf{Example} Let $(Q, A)$ be a lattice, and $\\{e_1, \\ldots, e_n\\}$ \nan ordered $\\mathbb{Z}$-base of $Q$. With respect to this base $A$ is expressed as a symmetric matrix $A = (a_{ij}) = A(e_i, e_j)\\in \\mathfrak{gl}_n(\\mathbb{Z})$. Let us suppose that all $a_{ii}$ are even. We define the matrix of $L$ to be the unique upper \ntriangular matrix $(\\ell_{ij})$ such that $A = L + L^t$ (in particular $\\ell_{ii} = a_{ii}\/2$; in our examples we will have \n$a_{ii} = 2$.) We will call $L$ the {\\it standard polarization} \nassociated to an ordered base. $\\square$\n\n\\vspace{2mm}\n\nPolarized lattices form a groupoid: \n\nan isomorphosm of polarized lattices \n$f:\\ (Q_1, A_1, L_1) \\overset{\\sim}{\\longrightarrow} (Q_2, A_2, L_2)$ is by definition an isomorphism of abelian groups $f: Q_1\\overset{\\sim}{\\longrightarrow} Q_2$ such that \n$$\nL_1(x, y) = L_2(f(x), f(y))\n$$ \n(and whence $A_1(x, y) = A_2(f(x), f(y))$). \n\n\\subsection{Orthogonality}\n\\label{sub32}\n\n\\begin{lemma} (i) (orthogonality) {\\it \n\t$$\n\tA(x,y) = A(Cx, Cy).\n\t$$}\n\n(ii) (gauge transformations) {\\it For any $P\\in GL(Q)$ \n\t$$\n\tA(P^\\vee L P) = P^\\vee A(L)P,\\ C(P^\\vee L P) = P^{-1}C(L)P.\n\t$$}\n\\end{lemma}\n\n$\\square$\n\n\n\n\n\\subsection{Black\/white decomposition and a Steinberg's theorem}\n\\label{sub34}\n\nCf. \\cite{Stein}, \\cite{C}. Let $\\alpha_1, \\ldots, \\alpha_r$ \nbe a base of simple roots of a finite reduced irreducible root \nsystem $R$ (not necessarily simply laced). \n\nLet \n$$\nA = (a_{ij}) = (\\langle \\alpha_i, \\alpha_j^\\vee \\rangle)\n$$\nbe the Cartan matrix. \n\nChoose a black\/white coloring of the set of vertices of the corresponding Dynkin graph $\\Gamma(R)$ in such a way that \nany two neighbouring vertices have different colours; this is possible since $\\Gamma(R)$ is a tree (cf. \\ref{sub-tree}). \n\nLet us choose an ordering of simple roots in such a way that the first $p$ roots are black, and the last $r - p$ roots are white. \nIn this base $A$ has a block form\n$$\nA = \\left(\\begin{matrix} 2I_p & X\\\\ Y & 2I_{r-p}\n\\end{matrix}\\right)\n$$\nConsider a Coxeter element\n\n\\begin{equation}\nC = s_1s_2\\ldots s_r = C_B C_W,\n\\end{equation}\n\nwhere \n$$\nC_B = \\prod_{i=1}^p s_i,\\ C_W = \\prod_{i=p+1}^r s_i.\n$$\nHere $s_i$ denotes the simple reflection corresponding to the root \n$\\alpha_i$.\n\nThe matrices of $ C_B, C_W$ with respect to the base $\\{\\alpha_i\\}$ \nare\n$$ \nC_B = \\left(\\begin{matrix} -I & -X\\\\ 0 & I\\end{matrix}\\right),\nC_W = \\left(\\begin{matrix} I & 0\\\\ -Y & -I\\end{matrix}\\right), \n$$\nso that \n\n\\begin{equation}\nC_B + C_W = 2I - A.\n\\end{equation}\n\nThis is an observation due to R.Steinberg, cf. \\cite{Stein}, p. 591. \n\n\\bigskip\n\nWe can also rewrite this as follows. \nSet \n$$\nL = \\left(\\begin{matrix} I & 0\\\\ Y & I\\end{matrix}\\right),\\\nU = \\left(\\begin{matrix} I & X\\\\ 0 & I\\end{matrix}\\right). \n$$\nThen $A = L + U$, and one checks easily that \n\n\\begin{equation}\nC = - U^{-1}L,\n\\end{equation}\n\nso we are in the situation \\ref{sub31}. This explains the name ''Cartan - Coxeter coresspondence''.\n\n\n\\subsection{Eigenvectors' correspondence}\n\\label{sub35}\n\n\\begin{theorem}\\label{white-bl} Let \n$$\nL = \\left(\\begin{matrix} I_p & 0\\\\ Y & I_{r-p}\\end{matrix}\\right),\\ \nU = \\left(\\begin{matrix} I_p & X\\\\ 0 & I_{r-p}\\end{matrix}\\right)\\ \n$$\nbe block matrices. Set \n$$\nA = L + U,\\ C = - U^{-1}L.\n$$ \nLet $\\mu\\neq 0$ be a complex number, $\\sqrt{\\mu}$ be any of its square roots, and \n\n\\begin{equation}\\label{lambda}\n\\lambda = 2 - \\sqrt{\\mu} - 1\/\\sqrt{\\mu}.\n\\end{equation}\n\n\nThen a vector $v_C = \\left(\\begin{matrix} v_1\\\\ v_2\\end{matrix}\\right)$ \nis an eigenvector of $C$ with eigenvalue $\\mu$ if and only if \n$$\nv_A = \\left(\\begin{matrix} v_1\\\\ \\sqrt{\\mu}v_2\\end{matrix}\\right)\n$$\nis an eigenvector of $A$ with the eigenvalue $\\lambda$\\footnote{this formulation has been suggested by A.Givental.}.\n\n\\end{theorem}\n\n{\\bf Proof}: a direct check. $\\square$\n\n\\bigskip\n\n\\subsubsection{Remark}\nNote that the formula (\\ref{lambda}) gives two possible values of $\\lambda$ corresponding \nto $\\pm \\sqrt{\\mu}$. On the other hand, $\\lambda$ does not change \nif we replace $ \\mu$ by $\\mu^{-1}$. \n\nIn the simplest case of $2\\times 2$ matrices the eigenvalues of $A$ are \n$2 \\pm (\\sqrt{\\mu} + \\sqrt{\\mu^{-1}})$, whereas the eigenvalues of \n$C$ are $\\mu^{\\pm 1}$. \n\n\\bigskip\n\n\n\n\\begin{corollary}\\label{vp}\n\t{\\it In the notations of} \\ref{sub31}, {\\it a vector\n\t\t$$\n\t\tx = \\sum x_j\\alpha_j\n\t\t$$\n\t\tis an eigenvector of $A$ with the eigenvalue $2(1 - \\cos\\theta)$ iff \n\t\tthe vector\n\t\t$$\n\t\tx_c := \\sum e^{\\pm i\\theta\/2}x_j \\alpha_j\n\t\t$$\n\t\twhere the sign in $e^{\\pm i\\theta\/2}$ is plus if $i$ is a white vertex, and minus otherwise, is an eigenvector of $C$ with eigenvalue $e^{2i\\theta}$.}\n\\end{corollary}\n\nCf. \\cite{F}.\n\n\\begin{proof}\nWithout loss of generality, we can suppose that $A$ is expressed in a basis of simple roots such that the first $r-p$ ones are white, and the last $p$ roots are black. \n\nThen $A$ has a block form \n\n\\[A = \\begin{pmatrix}\n2 I_{r-p} & X \\\\\nY & 2 I_{p}\n\\end{pmatrix} = \\begin{pmatrix}\nI_{r-p} & 0 \\\\\nY & I_{p}\n\\end{pmatrix} + \\begin{pmatrix}\nI_{r-p} & X \\\\\n0 & I_{p} \n\\end{pmatrix} = L + U \\]\n\nApplying Theorem 1 with \n\n\\[ v_1 = \\begin{pmatrix}\ne^{i\\theta\/2} x_1 \\\\\n.. \\\\\ne^{i\\theta\/2} x_{r-p} \\end{pmatrix} \\text{ and } v_2 = \\begin{pmatrix} e^{-i\\theta\/2} x_{r-p+1} \\\\\n.. \\\\\ne^{-i\\theta\/2} x_r \\end{pmatrix} \\]\t\n\nand the well-known eigenvalues of the Cartan matrix $A$, \n\\[\\lambda = 2 - 2 \\cos \\theta_k , \\text{ with } \\theta_k = 2\\pi k\/h , k\\in \\operatorname{Exp}(R) \\]\nwe obtain : $x_c := \\sum e^{\\pm i\\theta\/2}x_j \\alpha_j $ is an eigenvector of $C$ with the eigenvalue $e^{2 i \\theta_k}$ iff $e^{i\\theta_k}x = e^{i\\theta_k} \\sum x_j \\alpha_j$ is an eigenvector of $A$ with the eigenvalue $2 - 2 \\cos \\theta_k $. $\\square$ \n\\end{proof}\n\n\\subsection{Example: the root systems $A_n$.}\n\\label{sub41}\n\nWe consider the Dynkin graph of $A_n$ \nwith the obvious numbering of the vertices. \n\nThe Coxeter number $h = n + 1$, the set of exponents:\n$$\n\\operatorname{Exp}(A_n) = \\{1, 2, \\ldots, n\\}\n$$ \n\nThe eigenvalues of any Coxeter element are $e^{i\\theta_k}$, and \nthe eigenvalues of the Cartan \nmatrix $A(A_n)$ are $2 - 2\\cos\\theta_k$, $\\theta_k = 2\\pi k\/h$, \n$k\\in \\operatorname{Exp}(A_n)$.\n\nAn eigenvector of $A(A_n)$ with the eigenvalue $2 - 2\\cos\\theta$ \nhas the form \n\n\\begin{equation}\nx(\\theta) = (\\sum_{k=0}^{n-1} e^{i(n-1 - 2k)\\theta},\n\\sum_{k=0}^{n-2} e^{i(n-2 - 2k)\\theta} , \\ldots, \n1)\n\\end{equation}\n\nDenote by $C(A_n)$ the Coxeter element \n$$\nC(A_n) = s_1s_2\\ldots s_n\n$$\nIts eigenvector with the eigenvalue $e^{2i\\theta}$ is:\n$$\nX_{C(A_n)} = (\\sum_{k=0}^{n-j} e^{2ik\\theta})_{1\\leq j\\leq n}\n$$\nFor example, for $n = 4$:\n\n\\[ C_{A_4} = \\begin{pmatrix}\n0 & 0 & 0 & -1 \\\\\n1 & 0 & 0 &-1 \\\\\n0 & 1 & 0 & -1 \\\\\n0 & 0 & 1 & -1 \\end{pmatrix} \\text{ and } X_{C(A_4 )} = \\begin{pmatrix} 1 + e^{2i\\theta} + e^{4i\\theta} + e^{6i\\theta} \\\\ 1 + e^{2i\\theta} + e^{4i\\theta} \\\\ 1 + e^{2i\\theta} \\\\ 1 \\end{pmatrix}\\]\n\nis an eigenvector with eigenvalue $ e^{2i\\theta}$. \n\nSimilarly, for $n = 2$:\n\n\\[ C_{A_2} = \\begin{pmatrix}\n0 & -1 \\\\\n1 & -1 \\end{pmatrix},\\ X_{C(A_2 )} = \\begin{pmatrix} 1+ e^{2i\\gamma} \\\\ 1 \\end{pmatrix} \\]\n\n\n$\\square$\n\n\\section{Sebastiani - Thom product; factorization of $E_8$ and $E_6$}\n\\label{sec4}\n\n\\subsection{Join product}\n\\label{sub33}\nSuppose we are given two polarized lattices \n$(Q_i, A_i, L_i)$, $i = 1, 2$. \n\nSet $Q = Q_1\\otimes Q_2$, whence \n$$\nL:= L_1\\otimes L_2: Q\\overset{\\sim}{\\longrightarrow} Q^\\vee,\n$$\nand define\n$$\nA: = A_1*A_2 := L + L^\\vee: Q\\overset{\\sim}{\\longrightarrow} Q^\\vee \n$$\nThe triple $(Q, A, L)$ will be called the {\\bf join}, or {\\bf Sebastiani - Thom}, product of the polarized lattices \n$Q_1$ and $Q_2$, and denoted by $Q_1*Q_2$. \n\nObviously\n$$\nC(L) = - C(L_1)\\otimes C(L_2)\\in GL(Q_1\\otimes Q_2).\n$$\n\nIt follows that if $\\operatorname{Spec}(C(L_i)) = \\{e^{2\\pi i k_{i}\/h_i},\\ k_i\\in K_i\\}$ then\n\n\\begin{equation}\n\\operatorname{Spec}(C(L)) = \\{ - e^{2\\pi i(k_{1}\/h_1 + k_{2}\/h_2)},\\ (k_1,k_2)\\in K_1\\times K_2\\}\n\\end{equation}\n\n\n\n\n\n\\subsection{$E_8$ versus $A_4*A_2*A_1$: elementary analysis}\n\\label{sub42}\n\nThe ranks:\n$$\nr(E_8) = 8 = r(A_4)r(A_2)r(A_1);\n$$\nthe Coxeter numbers: \n$$\nh(E_8) = h(A_4)h(A_2)h(A_1) = 5\\cdot 3\\cdot 2 = 30.\n$$\nIt follows that\n$$\n|R(E_8)| = 240 = |R(A_4)||R(A_2)||R(A_1)|.\n$$ \n\nThe exponents of $E_8$ are:\n$$\n1, 7, 13, 19, 11, 17, 23, 29.\n$$\nAll these numbers, except $1$, are primes, and these are all primes $\\leq 30$, not dividing $30$. \n\nThey may be determined from the formula \n$$\n\\frac{i}{5} + \\frac{j}{3} + \\frac{1}{2} = \\frac{30 + k(i,j)}{30},\\ \n1\\leq i\\leq 4,\\ 1\\leq j\\leq 2,\n$$\nso\n$$\nk(i, 1)= 1 + 6(i-1) = 1, 7, 13, 19;\\ \n$$\n$$\nk(i,2) = 1 + 10 + 6(i-1) = 11, 17, 23, 29. \n$$\n\n\nThis shows that the exponents of $E_8$ are the same as the exponents \nof \n\\newline $A_4*A_2*A_1$.\n\n\nThe following theorem is more delicate. \n\n\\subsection{Decomposition of $Q(E_8)$}\n\\label{sub43}\n\n\\begin{theorem}\\label{gab} (Gabrielov, cf. \\cite{Gab}, Section 6, Example 3). There exists a polarization \n\tof the root lattice $Q(E_8)$ and an isomorphism \n\tof polarized lattices\n\t\n\t\\begin{equation}\\label{gam}\n\t\t\\Gamma: Q(A_4)*Q(A_2)*Q(A_1) \\overset{\\sim}{\\longrightarrow} Q(E_8).\n\t\\end{equation}\n\n\\end{theorem}\n\nIn the left hand side $Q(A_n)$ means the root lattice of $A_n$ with \nthe standard Cartan matrix and the standard polarization\n$$\nA(A_n) = L(A_n) + L(A_n)^t\n$$\nwhere the Seifert matrix $L(A_n)$ is upper triangular.\n\nIn the process of the proof, given in \\S \\ref{sub44} - \\ref{sub46} below, the isomorphism $\\Gamma$ will be written down explicitly. \n\n\\subsection{Beginning of the proof}\n\\label{sub44}\n\nFor $n = 4, 2, 1$, \nwe consider the bases of simple roots $e_1,\\ldots, e_n$ in $Q(A_n)$, with scalar products \ngiven by the Cartan matrices $A(A_n)$. \n\nThe tensor product of three lattices \n$$\nQ_* = Q(A_4)\\otimes Q(A_2)\\otimes Q(A_1) \n$$\nwill be equipped with the ''factorizable'' basis in the lexicographic order: \n$$\n(f_1,\\ldots, f_8) := (e_1\\otimes e_1\\otimes e_1, e_1\\otimes e_2\\otimes e_1, \ne_2\\otimes e_1\\otimes e_1, e_2\\otimes e_2\\otimes e_1,\n$$\n$$\ne_3\\otimes e_1\\otimes e_1, e_3\\otimes e_2\\otimes e_1, \ne_4\\otimes e_1\\otimes e_1, e_4\\otimes e_2\\otimes e_1).\n$$\nIntroduce a scalar product $(x, y)$ on $Q_*$ given, in the basis $\\{f_i\\}$, by the matrix\n$$\nA_* = A_4*A_2*A_1.\n$$\n\n\n\n\\subsection{Gabrielov - Picard - Lefschetz transformations $\\alpha_m, \n\t\\beta_m$}\n\\label{sub45}\n\n\nLet $(Q, ( , ))$ be a lattice of rank $r$. We introduce the following \ntwo sets of transformations $\\{\\alpha_m\\}, \\{\\beta_m\\}$ on the set $Bases-cycl(Q)$ of cyclically ordered bases of $Q$.\n\nIf $x = (x_i)_{i\\in \\mathbb{Z}\/r\\mathbb{Z}}$ is a base, and $m\\in \\mathbb{Z}\/r\\mathbb{Z}$, \nwe set\n\n$$\n(\\alpha_m(x))_i = \\left\\lbrace \\begin{matrix}\nx_{m+1} + (x_{m+1}, x_{m})x_m\\ & \\text{if}\\ i = m\\\\\nx_m & \\text{if}\\ i = m + 1 \\\\\nx_i & \\text{otherwise}\n\\end{matrix}\\right.\n$$\nand \n$$\n(\\beta_m(x))_i = \\left\\lbrace \\begin{matrix}\nx_m & \\text{if}\\ i = m - 1 \\\\\nx_{m-1} + (x_{m-1}, x_{m})x_m\\ & \\text{if}\\ i = m\\\\\nx_i & \\text{otherwise}\n\\end{matrix}\\right.\n$$\nWe define also a transformation $\\gamma_m$ by\n$$\n(\\gamma_m(x))_i = \\left\\lbrace \\begin{matrix}\n- x_m & \\text{if}\\ i = m \\\\\nx_i & \\text{otherwise}\n\\end{matrix}\\right.\n$$ \n\n\\subsection{Passage from $A_4*A_2*A_1$ to $E_8$}\n\\label{sub46}\n\nConsider the base \n$ f = \\{f_1, \\ldots f_8\\}$ of the lattice $Q_* := Q(A_4)\\otimes Q(A_2) \\otimes Q(A_1)$ described in \\S \\ref{sub44}, and apply to it the following transformation\n\n\\begin{equation}\\label{g'}\nG' = \\gamma_2\\gamma_1\\beta_4\\beta_3\\alpha_3\\alpha_4\\beta_4\\alpha_5\\alpha_6\n\\alpha_7\\alpha_1\\alpha_2\\alpha_3\\alpha_4\\beta_6\\beta_3\\alpha_1,\n\\end{equation}\ncf. \\cite{Gab}, Example 3. Note that\n\\begin{equation}\\label{eqn:gamma}\n\\gamma_2\\gamma_1 = \\alpha_1^6,\n\\end{equation}\ncf. \\cite{Br}. \n\n\nThen the base $G'(f)$ has the intersection matrix given by the Dynkin \ngraph of $E_8$, with the ordering indicated in Figure \\ref{fig1} below.\n\n\\begin{figure}\n\t$$\n\t1 \\ -\\ 2\\ - \\ 3 \\ - \\ 5 \\ - \\ 6 \\ - \\ 7\\ - 8\n\t$$\n\t$$\n\t|\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\t$$\n\t$$\n\t4\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\t$$\n\t\n\t\\caption{Gabrielov's ordering of $E_8$.}\n\t\\label{fig1} \n\\end{figure}\n\n\nThis concludes the proof of Theorem \\ref{gab} $\\square$\n\n\\subsection{The induced map of root sets}\n\\label{sub47}\n\n \nBy definition, the isomorphism of lattices $\\Gamma$, (\\ref{gam}), induces \na bijection between the bases\n$$\ng:\\ \\{f_1,\\ldots, f_8\\} \\overset{\\sim}{\\longrightarrow} \\{\\alpha_1,\\ldots, \\alpha_8\\}\\subset R(E_8).\n$$\nwhere in the right hand side we have the base of simple roots, \nand a map\n$$\nG:\\ R(A_4)\\times R(A_2)\\times R(A_1)\\to R(E_8),\\ \nG(x,y,z) = \\Gamma(x\\otimes y \\otimes z) \n$$\nof sets of the same cardinality $240$ which is not a bijection however: its image consists of $60$ elements. \n\nNote that the set of vectors $\\alpha\\in Q(E_8)$ with \n$(\\alpha, \\alpha) = 2$ coincides with the root system $R(E_8)$, \ncf. \\cite{Serre}, Premi\\`ere Partie, Ch. 5, 1.4.3. \n\n\\subsection{Passage to Bourbaki ordering}\n\\label{sub48}\n\n\nThe isomorphism $G'$ (\\ref{g'}) is given by a matrix \n$G'\\in GL_8(\\mathbb{Z})$ such that \n$$\nA_G(E_8) = G^{\\prime t}A_*G'\n$$\nwhere we denoted\n$$\nA_* = A(A_4)*A(A_2)*A(A_1),\n$$\nthe factorized Cartan matrix, \nand $A_G$ denotes the Cartan matrix of $E_8$ with respect to the \nnumbering of roots indicated on Figure \\ref{fig1}. \n\nNow let us pass to the numbering of vertices of the Dynkin graph \nof type $E_8$ indicated in \\cite{B} (the difference with Gabrielov's \nnumeration is in three vertices $2, 3$, and $4$).\n\n\\begin{figure}[h]\n$$\n1 \\ -\\ 3\\ - \\ 4 \\ - \\ 5 \\ - \\ 6 \\ - \\ 7\\ - 8\n$$\n$$\n|\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n$$\n$$\n2\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n$$\t\n\t\n\t\\caption{Bourbaki ordering of $E_8$.}\n\t\\label{fig2} \n\\end{figure} \n\n\nThe Gabrielov's Coxeter element (the full monodromy) in the Bourbaki numbering looks as follows: \n$$\nC_{G}(E_8 ) = s_1 \\circ s_3 \\circ s_4 \\circ s_2 \\circ s_5 \\circ s_6 \\circ s_7 \\circ s_8\n$$\n\n\n\\begin{lemma}{\\it Let $A(E_8)$ be the standard Cartan matrix of \n\t$E_8$ from} [B]:\n\\[ A(E_8) = \\begin{pmatrix}\n2 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 2 & 0 & -1 & 0 & 0 & 0 & 0 \\\\\n-1 & 0 & 2 & -1 & 0 & 0 & 0 & 0 \\\\\n0 & -1 & -1 & 2 & -1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 \\end{pmatrix}. \\]\n{\\it Then\n\t$$\n\tA(E_8) = G^{ t}A_*G\n\t$$\n\tand\n\t$$\n\tC_G(E_8) = G^{-1}C_* G\n\t$$\n\twhere\n\t$$\n\tC_* = C(Q(A_4)*Q(A_2)*Q(A_1)) = C(A_4)\\otimes C(A_2)\\otimes C(A_1),\n\t$$\n\tis the factorized Coxeter element, \n\tand \n\t\\[ G = \\begin{pmatrix}\n\t0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 \\\\\n\t-1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 \\\\\n\t-1 & 1 & -1 & 0 & 0 & 1 & 0 & 0 \\\\\n\t0 & 1 & -1 & 0 & 0 & 0 & 1 & 0 \\\\\n\t-1 & 1 & -1 & 0 & 0 & 0 & 1 & 0 \\\\\n\t0 & 1 & -1 & 0 & 0 & 0 & 0 & 1 \\\\\n\t0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\end{pmatrix} \n\t\\eqno{(3.8.1)} \\]\n\tHere\n\t$$\n\tG = G'P\n\t$$\n\twhere $P$ is the permutation matrix of passage from the Gabrielov's ordering in Figure \\ref{fig1} to the Bourbaki ordering in Figure \\ref{fig2}}\n\\end{lemma}\n\n\\subsection{Cartan eigenvectors of $E_8$}\n\\label{sub491}\n\nTo obtain the Cartan eigenvectors of $E_8$, one should pass from $C_G(E_8)$ to \nthe ''black\/white'' Coxeter element (as in \\S \\ref{sub34}) \n\n\\[C_{BW}(E_8) = s_1 \\circ s_4 \\circ s_6 \\circ s_8 \\circ s_2 \\circ s_3 \\circ s_5 \\circ s_7 \\]\n\nAny two Coxeter elements are conjugate in the Weyl group $W(E_8)$.\n\nThe elements $C_G(E_8)$ and $C_{BW}(E_8)$ \nare conjugate by the following element of $W(E_8)$:\n\n\\[ C_G(E_8) = w^{-1}C_{BW}(E_8)w \\]\n\nwhere\n\n\\[ w = s_7 \\circ s_5 \\circ s_3 \\circ s_2 \\circ s_6 \\circ s_4 \\circ s_5 \\circ s_1 \\circ s_3 \\circ s_2 \\circ s_4 \\circ s_1 \\circ s_3 \\circ s_2 \\circ s_1 \\circ s_2 \\]\n\nThis expression for $w$ can be obtained using an algorithm described in \\cite{C}, \ncf. also \\cite{Br}. \n\nThus, if $x_*$ is an eigenvector of $C_*(E_8)$ then \n\n\\[ x_{BW} = wG^{-1}x_* \\]\n\nis an eigenvector of $C_{BW}(E_8)$. But we know the eigenvectors of \n$C_*(E_8)$, they are all factorizable. \n\nThis provides the eigenvectors of $C_{BW}(E_8)$, which in turn \nhave very simple relation to the eigenvectors of $A(E_8)$, \ndue to Theorem \\ref{vp}.\n\n\\bigskip \n\n\n{\\bf Conclusion: an expression for the eigenvectors of $A(E_8)$.}\n\n\\bigskip \n\nLet $\\theta = \\frac{a\\pi}{5},\\ 1\\leq a\\leq 4$, $\\gamma = \\frac{b\\pi}{3},\\ 1\\leq b\\leq 2$, $\\delta = \\frac{\\pi}{2}$,\n$$\n\\alpha = \\theta + \\gamma + \\delta = \\pi + \\frac{k\\pi}{30},\n$$\n$$\nk\\in \\{1, 7, 11, 13, 17, 19, 23, 29\\}.\n$$\nThe $8$ eigenvalues of $A(E_8)$ have the form\n$$\n\\lambda(\\alpha) = \\lambda(\\theta,\\gamma) = 2 - 2\\cos \\alpha\n$$ \nAn eigenvector of $A(E_8)$ with the eigenvalue \n$\\lambda(\\theta,\\gamma)$ is \n\n\\[ X_{E_8}(\\theta,\\gamma) = \\begin{pmatrix} \n\\cos(\\gamma + \\theta - \\delta) + \\cos ( \\gamma - 3 \\theta - \\delta) + \\cos (\\gamma - \\theta - \\delta) \\\\\n\\cos( 2 \\gamma + 2 \\theta)\\\\\n\\cos(2\\gamma) + \\cos (2\\gamma + 2 \\theta) + \\cos ( 2\\gamma - 2 \\theta) + \\cos (4\\theta) + \\cos(2 \\theta) \\\\\n\\cos(\\gamma + 3 \\theta - \\delta) + \\cos ( \\gamma + \\theta - \\delta ) + \\cos ( -\\gamma + 3 \\theta - \\delta) \\\\\n2 \\cos(2\\gamma) + 2 \\cos(2 \\gamma + 2 \\theta) + \\cos(2\\gamma - 2 \\theta) + \\cos(2 \\gamma + 4 \\theta) + \\cos(4\\theta) + 2 \\cos (2 \\theta) + 1 \\\\\n\\cos(\\gamma + 3 \\theta - \\delta) + \\cos( \\gamma + \\theta - \\delta) \\\\\n\\cos (2 \\gamma) + \\cos ( 2 \\theta - 2 \\delta) \\\\\n\\cos (\\gamma - \\theta - \\delta ) \\end{pmatrix} \\]\n\nOne can simplify it as follows:\n\n\\begin{equation}\n X_{E_8}(\\theta,\\gamma) = - \\begin{pmatrix} \n 2 \\cos (4\\theta) \\cos(\\gamma - \\theta - \\delta) \\\\\n - \\cos( 2 \\gamma + 2 \\theta ) \\\\\n 2 \\cos^{2} (\\theta) \\\\\n -2 \\cos(\\gamma) \\cos (3\\theta - \\delta) - \\cos ( \\gamma + \\theta - \\delta ) \\\\\n -2 \\cos(2\\gamma + 3 \\theta) \\cos (\\theta) + \\cos (2 \\gamma ) \\\\\n -2 \\cos \\theta \\cos (\\gamma + 2\\theta - \\delta ) \\\\\n -2\\cos(\\gamma + \\theta - \\delta) \\cos(\\gamma -\\theta + \\delta) \\\\\n - \\cos (\\gamma - \\theta - \\delta) \\end{pmatrix} \n\\end{equation}\n\n\n\\subsection{Perron - Frobenius and all that}\n\\label{sub492}\n\nThe Perron - Frobenius eigenvector corresponds to the eigenvalue\n\n\\[ 2 - 2 \\cos \\frac{\\pi}{30}, \\]\n\nand may be chosen as\n\n\\[ v_{PF} = \\begin{pmatrix} \n2 \\cos \\frac{\\pi}{5} \\cos \\frac{11\\pi}{30} \\\\\n\\cos \\frac{\\pi}{15} \\\\\n2\\cos^{2}\\frac{\\pi}{5} \\\\ \n2\\cos \\frac{2\\pi}{30} \\cos \\frac{\\pi}{30} \\\\ \n2 \\cos \\frac{4\\pi}{15} \\cos \\frac{\\pi}{5} + \\frac{1}{2} \\\\ \n2 \\cos \\frac{\\pi}{5} \\cos \\frac{7\\pi}{30} \\\\\n2 \\cos \\frac{\\pi}{30} \\cos \\frac{11\\pi}{30}\\\\ \n\\cos \\frac{11\\pi}{30} \\end{pmatrix} \\] \n\nOrdering its coordinates in the increasing order, we obtain\n\n\\[ v_{PF <} = \\begin{pmatrix} \n\\cos \\frac{11\\pi}{30} \\\\\n2 \\cos \\frac{\\pi}{5} \\cos \\frac{11\\pi}{30} \\\\\n2 \\cos \\frac{\\pi}{30} \\cos \\frac{11\\pi}{30} \\\\\n\\cos \\frac{\\pi}{15} \\\\\n2 \\cos \\frac{\\pi}{5} \\cos \\frac{7\\pi}{30} \\\\\n2\\cos^{2}\\frac{\\pi}{5} \\\\\n2 \\cos \\frac{4\\pi}{15} \\cos \\frac{\\pi}{5} + \\frac{1}{2} \\\\\n2\\cos \\frac{2\\pi}{30} \\cos \\frac{\\pi}{30} \\end{pmatrix} \\]\n\nIn the Ref. \\cite{Z}, A. B. Zamolodchikov obtains \nthe following expression for the PF vector: \n\n\\[ v_{Zam}(m) = \\begin{pmatrix}\nm \\\\\n2 m \\cos \\frac{\\pi}{5} \\\\\n2 m \\cos \\frac{\\pi}{30} \\\\\n4 m \\cos \\frac{\\pi}{5} \\cos \\frac{7\\pi}{30} \\\\\n4 m \\cos \\frac{\\pi}{5} \\cos \\frac{2\\pi}{15} \\\\\n4 m \\cos \\frac{\\pi}{5} \\cos \\frac{\\pi}{30} \\\\\n8 m \\cos^{2} \\frac{\\pi}{5} \\cos \\frac{7\\pi}{30} \\\\\n8 m \\cos^{2} \\frac{\\pi}{5} \\cos \\frac{2\\pi}{15} \\end{pmatrix} \\] \n\nSetting $m = \\cos \\frac{11\\pi}{30}$, we find indeed : \n\n\\[ v_{PF <} = v_{Zam} (\\cos \\frac{11\\pi}{30} ) \\]\n\n\n\n\n\\subsection{Factorization of $E_6$}\n\\label{sub-E6}\n\n\\begin{theorem}\n\t(Gabrielov, cf. \\cite{Gab}, Section 6, Example 2). There exists a polarization \n\tof the root lattice $Q(E_6)$ and an isomorphism \n\tof polarized lattices\n\t\n\t\\begin{equation}\\label{gam}\n\t\\Gamma_{E_6} : Q(A_3)*Q(A_2)*Q(A_1) \\overset{\\sim}{\\longrightarrow} Q(E_6).\n\t\\end{equation}\n\\end{theorem}\n\nThe proof is exactly the same as for $Q(E_8)$. The passage from $A_3 * A_2 * A_1$ to $E_6$ is obtained by the following transformation\n\n\\[ G'_{E_6} = \\gamma_4 \\gamma_1 \\alpha_1 \\alpha_2 \\alpha_3 \\alpha_4 \\beta_6 \\beta_3 \\alpha_1 \\]\n\ncf. \\cite{Gab}, Example 2.\n\nAfter a passage from Gabrielov's ordering to Bourbaki's, we obtain a transformation \n\n\\[ G_{E_6} = \\begin{pmatrix}\n0&-1&1&0&0&0\\\\\n-1&0&1&0&0&0\\\\\n0&-1&0&1&0&0\\\\\n-1&0&0&0&1&0\\\\\n0&0&0&0&0&1\\\\\n-1&0&0&0&0&1\n\\end{pmatrix} \\in GL_{6} ( \\mathbb{Z} )\\]\n\nsuch that \n\\[ A(E_6) = G_{E_6}^{t} A_{*} G_{E_6} \\text{ and }C_{G} (E_6) = G_{E_6}^{-1} C_{*} G_{E_6} \\] \nwhere $ A_{*} = A(A_3) * A(A_2) * A(A_1)$ and $ C_{*} = C(A_3) \\otimes C(A_2) \\otimes C(A_1)$ and \n\\[ C_{G}(E_6) = s_1 \\circ s_3 \\circ s_4 \\circ s_2 \\circ s_5 \\circ s_6 \\]\n $C_{G}(E_6)$ is the Gabrielov's Coxeter element in the Bourbaki numbering, cf. \\cite{B}. \n \n Let $C_{BW} (E_6) = s_1 \\circ s_4 \\circ s_6 \\circ s_2 \\circ s_3 \\circ s_5$ be the \"black\/white\" Coxeter element. $C_{G}(E_6)$ and $C_{BW}(E_6)$ are conjugated by the following element of the Weyl group $W(E_6)$ :\n \n \\[ v = s_5 \\circ s_3 \\circ s_2 \\circ s_4 \\circ s_1 \\circ s_3 \\circ s_3 \\circ s_1 \\circ s_2 \\]\n\nThus, if $x_{*}$ is an eigenvector of $C_{*} ( E_6)$ then $x_{BW} = v G^{-1}_{E_6} x_{*}$ is an eigenvector of $C_{BW}(E_6)$. \n\nFinally, let $\\theta = \\frac{a\\pi}{4} , 1 \\leq a \\leq 3$, $\\gamma = \\frac{b\\pi}{3}, 1 \\leq b \\leq 2$, $\\delta = \\frac{\\pi}{2}$ and\n\n\\[ \\alpha = \\theta + \\gamma + \\delta \\]\n\nThe 6 eigenvalues of $A(E_6)$ have the form $\\lambda (\\alpha) = \\lambda(\\theta, \\gamma) = 2 - 2\\cos \\alpha$. An eigenvector of $A(E_6)$ with the eigenvalue $\\lambda (\\alpha)$ is \n\n\n\\[ X_{E_6} ( \\theta, \\lambda) = \\begin{pmatrix}\n\\cos \\left(3\\gamma + 3 \\theta - \\delta \\right) \\\\\n2 \\cos^{2} \\theta \\\\\n-2 \\cos \\left( 3 \\gamma + 3 \\theta - \\delta\\right) \\cos \\left(\\gamma + \\theta - \\delta\\right) \\\\\n-4 \\cos^{2} \\theta \\cos \\left( \\gamma + \\theta - \\delta\\right) \\\\\n1 - 2 \\cos \\left( 2 \\gamma + 3 \\theta\\right) \\cos \\theta \\\\\n-2 \\cos(\\gamma) \\cos \\left(\\theta - \\delta\\right)\n\\end{pmatrix} \\]\n\n\n\n\\section{Givental's $q$-deformations}\n\\label{sec5}\n\n\n\n\\subsection{$q$-deformations of Cartan matrices}\n\\label{sub-cartan}\n\nLet $A = (a_{ij})$ be a $n\\times n$ complex matrix. We will say that $A$ is \na {\\it generalized Cartan matrix} if\n\n\\bigskip\n\n(i) for all $i\\neq j$, $a_{ij}\\neq 0$ implies $a_{ji}\\neq 0$;\n\n(ii) all $a_{ii} = 2$.\n\n\\bigskip\n\nIf only (i) is fulfilled, we will say that $A$ is \na {\\it pseudo-Cartan matrix}. \n\n\\bigskip\n\nWe associate to a pseudo-Cartan matrix $A$ an unoriented graph $\\Gamma(A)$ with vertices \n$1, \\ldots, n$, two vertices $i$ and $j$ being connected by an edge $e = (ij)$ iff $a_{ij}\\neq 0$. \n\nLet $A$ be a generalized Cartan matrix. \nThere is a unique decomposition \n$$\nA = L + U\n$$\nwhere $L = (\\ell_{ij})$ (resp. $U = (u_{ij})$) is lower (resp. upper) triangular, with $1$'s \non the diagonal. \n\nWe define a $q$-deformed Cartan matrix by \n\t$$\n\tA(q) = qL + U\n\t$$\n\t\nThis definition is inspired by the $q$-deformed Picard - Lefschetz \ntheory developed by Givental, \\cite{Giv}.\n\n\n\n\\begin{theorem}\\label{q-thm} Let $A$ be a generalized Cartan matrix \nsuch that $\\Gamma(A)$ is a tree. \n\n(i) The eigenvalues of $A(q)$ have the form\n\\begin{equation}\\label{q-lambda}\n\\lambda(q) = 1 + (\\lambda - 2)\\sqrt{q} + q\n\\end{equation}\nwhere $\\lambda$ is an eigenvalue of $A$. \n\n(ii) There exist integers $k_1, \\ldots, k_n$ such that if $x = (x_1, \\ldots, x_n)$ \nis an eigenvector of $A$ for the eigenvalue $\\lambda$ then\n\\begin{equation}\\label{q-vect}\nx(q) = (q^{k_1\/2}x_1,\\ldots, q^{k_n\/2}x_n) \n\\end{equation}\nis an eigenvector of $A(q)$ for the eigenvalue $\\lambda(q)$. \n\\end{theorem}\n\nThe theorem will be proved after some preparations. \n\n\\subsection{}\\label{sub-tree} Let $\\Gamma$ be an unoriented tree \nwith a finite set of vertices $I = V(\\Gamma)$.\n\nLet us pick a root of $\\Gamma$, and partially order its vertices \nby taking the minimal vertex $i_0$ to be the bottom of the root, and then \ngoing ''upstairs''. This defines an orientation on $\\Gamma$.\n\n\\begin{lemma}\\label{tree-lemma} Suppose we are given a nonzero complex number $b_{ij}$ for each \nedge $e = (ij), i < j$ of $\\Gamma$. \nThere exists a collection of nonzero complex numbers $\\{c_i\\}_{i\\in I}$ such that \n$$\nb_{ij} = c_j\/c_i,\\ i < j.\n$$\nfor all edges $(ij)$. \n\nWe can choose the numbers $c_i$ in such a way that they are products \nof some numbers $b_{pq}$.\n\n\\end{lemma}\n\n\n{\\bf Proof.} Set $c_{i_0} = 1$ for the unique minimal vertex $i_0$, \nand then define the other $c_i$ one by one, by going upstairs, \nand using as a definition\n$$\nc_j := b_{ij}c_i,\\ i < j.\n$$\nObviously, the numbers $c_i$ defined in such a way, are products \nof $b_{pq}$. $\\square$ \n\n\\begin{lemma}\\label{two-matrices} Let $A = (a_{ij})$ and $A' = (a'_{ij})$ be two \npseudo-Cartan matrices with $\\Gamma(A) = \\Gamma(A')$. Set \n$b_{ij}:= a'_{ij}\/a_{ij}$. Suppose that\n\\begin{equation}\\label{eq-two}\nb_{ij} = b_{ji}^{-1}. \n\\end{equation} \n\nfor all $i\\neq j$, and $a_{ii} = a'_{ii}$ for all $i$. Then there exists a diagonal matrix \n$$\nD = \\operatorname{Diag}(c_1,\\ldots, c_r)\n$$\nsuch that $A' = D^{-1}AD$.\n\nMoreover, the numbers $c_i$ may be chosen to be products of some \n$b_{pq}$.\n\\end{lemma}\n\n{\\bf Proof.} Let us choose a partial order $<_p$ on the set of vertices $V(\\Gamma)$ as in \\ref{sub-tree}. \n\n{\\it Warning.} This partial order {\\it differs} in general from \nthe standard total order on $\\{1,\\ldots, n\\}$. \n\nLet us apply Lemma\\ \\ref{tree-lemma} to the collection of numbers \n$\\{b_{ij},\\ i<_p j\\}$. We get a sequence of numbers $c_{ij}$ such \nthat \n$$\nb_{ij} = c_j\/c_i\n$$\nfor all $i <_p j$. The condition (\\ref{eq-two}) implies that this \nholds true for all $i\\neq j$.\n\nBy definition, this is equivalent to \n$$\na'_{ij} = c_i^{-1}a_{ij}c_j,\n$$\ni.e. to $A' = D^{-1}AD$. $\\square$\n\n\\subsection{\\bf{Proof of Theorem \\ref{q-thm}.}}\n\\label{proof-q-thm} \n\nLet us consider two matrices: $A(q) = (a(q)_{ij})$ with $a(q)_{ii} = \n1 + q$\n$$\na(q)_{ij} = \\left\\{\\begin{matrix} a_{ij} & \\text{if\\ }i < j\\\\\n qa_{ij} & \\text{if\\ }i > j\n\\end{matrix}\\right.\n$$\nand\n$$\nA'(q) = \\sqrt{q}A + (1 - \\sqrt{q})^2 I = (a(q)'_{ij})\n$$\nwith $a(q)'_{ii} = 1 + q$ and $a(q)'_{ij} = \\sqrt{q}a(q)_{ij}$, \n$i\\neq j$. \n\nThus, we can apply Lemma\\ \\ref{two-matrices} to $A(q)$ and \n$A'(q)$. \nSo, there exists a diagonal matrix $D$ as above such that \n$$\nA(q) = D^{-1}A'(q)D.\n$$\nBut the eigenvalues of $A'(q)$ are obviously \n$$\n\\lambda(q) = \\sqrt{q}\\lambda + (1 - \\sqrt{q})^2 = \n1 + (\\lambda - 2)\\sqrt{q} + q.\n$$\nIf $v$ is an eigenvector of $A$ for $\\lambda$ then $v$ is an eigenvector of $A'(q)$ for $\\lambda(q)$, and $Dv$ will be an eigenvector of $A(q)$ for $\\lambda(q)$. $\\square$\n\n \t\n\\subsection{{\\bf Remark} (M.Finkelberg)}\n\\label{sub53}\n\nThe expression (\\ref{q-lambda}) resembles \nthe number of points of an elliptic curve $X$ over a finite field \n$\\mathbb{F}_q$. To appreciate better this resemblance, note that in all \nour examples $\\lambda$ has the form \n$$\n\\lambda = 2-2 \\cos \\theta,\n$$\nso if we set \n$$\n\\alpha = \\sqrt{q}e^{i\\theta}\n$$\n(''a Frobenius root'') then $|\\alpha| = \\sqrt{q}$, and \n$$\n\\lambda(q) = 1 - \\alpha - \\bar\\alpha + q,\n$$ \ncf. \\cite{IR}, Chapter 11, \\S 1, \\cite{Kn}, Chapter 10, Theorem 10.5. \n\nSo, the Coxeter eigenvalues $e^{2i\\theta}$ may be seen as analogs of ''Frobenius roots of an elliptic curve over $\\mathbb{F}_1$''. \n\n\n\\subsection{\\bf Examples. }\n\\label{sub54}\n\n\\subsubsection{Standard deformation for $A_n$}\n\n\n\nLet us consider the \nfollowing $q$-deformation of $A = A(A_n)$: \n$$\nA(q) = \\left(\\begin{matrix} \n1 + q & - 1 & 0 & \\ldots & 0\\\\\n- q & 1 + q & - 1 & \\ldots & 0\\\\\n\\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\\n0 & \\ldots & 0 & - q & 1 + q\n\\end{matrix}\\right)\n$$\nThen \n$$\n\\operatorname{Spec}(A(q)) = \\{\\lambda(q) := 1 + (\\lambda - 2)\\sqrt{q} + q|\\ \\lambda\\in \\operatorname{Spec}(A(1))\\}.\n$$\nIf $x = (x_1,\\ldots, x_n)$ is an eigenvector of $A = A(1)$ \nwith eigenvalue $\\lambda$ then\n$$\nx(q) = (x_1, q^{1\/2}x_2,\\ldots, q^{(n-1)\/2}x_n)\n$$\nis an eigenvector of $A(q)$ with eigenvalue $\\lambda(q)$. \n\n\\subsubsection{Standard deformation for $E_8$}\n\\label{sub55}\n\nA $q$-deformation: \n\\[ A_{E_8}(q) = \\begin{pmatrix}\n1+q & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1+q & 0 & -1 & 0 & 0 & 0 & 0 \\\\\n-q &0& 1+q & -1 & 0 & 0 & 0 & 0 \\\\\n0&-q&-q& 1+q & -1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -q & 1+q & -1 & 0 & 0\\\\\n0& 0 & 0 & 0 & -q & 1+q & -1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & -q & 1+q & -1 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & -q & 1+q \\end{pmatrix} \\]\n\nIts eigenvalues are \n$$\n\\lambda(q) = \n1+q+(\\lambda-2)\\sqrt{q} = 1+q-2 \\sqrt{q} \\cos \\theta\n$$ \nwhere $\\lambda = 2-2 \\cos \\theta$ is an eigenvalue of $A(E_8)$. \n\nIf $X = (x_1 , x_2 , x_3 , x_4 , x_5 , x_6 , x_7 , x_8)$ is an eigenvector of $A(E_8)$ for the eigenvalue $\\lambda$, then \n\n\\begin{equation}\\label{eq1}\nX = ( x_1 , \\sqrt{q} x_2 , \\sqrt{q} x_3 , q x_4 , q \\sqrt{q} x_5 , q^2 x_6 , q^2 \\sqrt{q} x_7 , q^3 x_8 )\n\\end{equation}\nis an eigenvector of $A_{E_8}(q)$ for the eigenvalue $\\lambda(q)$.\n\n\\bigskip\n\n\n\n\\section{A physicist's appendix: \ncobalt niobate producing an $E_8$ chord}\n\nIn this Section, we briefly describe the relation of Perron-Frobenius components, \nin the case of $R = E_8$, to the physics of certain magnetic systems as anticipated \nin a pioneering theoretical work \\cite{Z} and possibly observed \nin a beautiful neutron scattering experiment \\cite{Coldea}. \n\n\\subsection{One-dimensional Ising model in a magnetic field}\n\n(a) {\\it The Ising Hamiltonian}\n\nLet $W = \\mathbb{C}^2$. Recall three Hermitian Pauli matrices: \n$$\n\\sigma^x = \\left(\\begin{matrix}0 & 1\\\\1 & 0\\end{matrix}\\right), \\,\\,\n\\sigma^y = \\left(\\begin{matrix}0 & -i\\\\i & 0\\end{matrix}\\right), \\,\\, \n\\sigma^z = \\left(\\begin{matrix}1 & 0\\\\0 & -1\\end{matrix}\\right). \n$$\nThe $\\mathbb{C}$-span of $\\sigma^x, \\sigma^y, \\sigma^z$ inside $End(W)$ \nis a complex Lie algebra $\\mathfrak g = \\mathfrak{sl}(2,\\mathbb{C})$; the $\\mathbb{R}$-span of the \nanti-Hermitian matrices $i\\sigma^x, i\\sigma^y, i\\sigma^z$ is a real Lie subalgebra $\\mathfrak k = \\mathfrak{su}(2)\\subset \\mathfrak g$. The resulting representation \nof $\\mathfrak g$ (or $\\mathfrak k$) on $W$ is what \nphysicists refer to as the ``spin-$\\frac{1}{2}$ \nrepresentation''. \n\nFor a natural $N$, consider a $2^N$-dimensional tensor product\n$$\nV = \\otimes_{n=1}^N W_n\n$$\nwith all $W_n = W$. We are interested in the spectrum of the following \nlinear operator $H$ acting on $V$: \n\\begin{equation}\\label{eqn:ising-ham}\nH = H(J,h_z, h_x) = - J\\sum_{n=1}^{N} \\sigma^z_n\\sigma^z_{n+1} - \nh_z\\sum_{n=1}^{N} \\sigma^z_n\n- h_x\\sum_{n=1}^{N} \\sigma^x_n ,\n\\end{equation} \nwhere $J, h_x, h_z$ are positive real numbers. Here for $A:\\ W \\longrightarrow W$, $A_n: V\\longrightarrow V$ denotes an operator acting as $A$ \non the $n$-th tensor factor and as the identity on all the other factors. \nBy definition, $A_{N+1} := A_1$. \n\nIn keeping with the conditions of the experiment \\cite{Coldea}, everywhere \nbelow we assume that $N$ is very large ($N >> 1$), and that $0 < h_z << J$.\n\n\nThe space $V$ arises as the space of states of a quantum-mechanical \nmodel describing a chain of $N$ atoms on the plane $\\mathbb{R}^2$ with \ncoordinates $(x,z)$. The chain is parallel to the $z$ axis, and is subject \nto a magnetic field with a component $h_z$ along the chain, and \na component $h_x$ along the $x$-axis. The $W_n$ is the space of \nstates of the $n$-th atom. Only the nearest-neighbor atoms interact, \nand the $J$ parameterizes the strength of this interaction.\n\nThe operator $H$ in the Eq. (\\ref{eqn:ising-ham}) is \ncalled the Hamiltonian, and its eigenvalues $\\epsilon$ correspond to the \nenergy of the system. It is an Hermitian operator (with respect to an \nobvious Hermitian scalar product on $V$), thus all its eigenvalues are real. \n\nConsider also the {\\it translation} operator $T$, acting as follows: \n\\begin{equation}\nT(v_1\\otimes v_2\\otimes\\ldots \\otimes v_N) = \nv_2\\otimes v_3\\otimes\\ldots \\otimes v_N\\otimes v_1,\n\\end{equation} \nThe operator $T$ is unitary, and commutes with $H$.\n\nAn eigenvector $v_0\\in V$ of $H$ with the lowest \nenergy eigenvalue $\\epsilon_0$ is called the ground state. \n\nWhat happens as $h_x$ varies, at fixed $J$ and $h_z$?\nWhen $h_x << J$, the ground state $v_0$ is close to the ground \nstate $v_J$ of the operator $H_J = H(J,0,0)$: \n$$\nv_J = \\otimes_{n=1}^N v_n^z ,\n$$\nwhere $v^z_n$ is an eigenvector of $\\sigma^z$ in $W_i$ \nwith eigenvalue $1$. Thus, the state $v_J$ is \ninterpreted as ``all the spins pointing along the $z$-axis''.\n\nIn the opposite limit, when $h_x >> J$, \nthe ground state $v_0$ is close to the ground state $v_x$ \nof the operator $H_x = H(0,0,h_x)$: \n$$\nv_x = \\otimes_{n=1}^N v_n^x ,\n$$\nwhere $v^x_n$ is an eigenvector of $\\sigma^x$ in \n$W_n$ with eigenvalue $1$. Thus, the state $v_x$ is \ninterpreted as ``all the spins pointing along the $x$-axis''.\n\nAs a function of $h_x$ at fixed $J$ and $h_z$, \nthe system has two phases. There is a critical value $h_x = h_c$, \nof the order of $J\/2$ : for $h_x < h_c$, the ground state $v_0$ is \nclose to $v_J$, and one says that the chain is in the {\\it ferromagnetic} \nphase. By contrast, for $h_x > h_c$, the ground state $v_0$ is close \nto $v_x$, and one says that the chain is in the {\\it paramagnetic} phase. \n(The transition between the two phases is far less trivial \nthan the spins simply turning to follow the field upon increasing $h_x$: to find \nout more, curious reader is encouraged to consult the Ref. \\cite{Chakrabarti}.)\n\n(b) {\\it Elementary excitations at $h_x = h_c$}\n\nZamolodchikov's theory, \\cite{Z}, says something \nspectacularly precise about the next few, \nafter $\\epsilon_0$, eigenvalues (``energy levels'') of a nearly-critical \nHamiltonian $H_c := H(J,h_z << J, h_c)$. \nTo see this, notice that the possible eigenvalues of the translation operator $T$ \nhave the form $e^{2\\pi i k\/N}$, with $-N\/2\\leq k \\leq N\/2$; let us call the number \n$$\np =2\\pi k \/N\n$$\nthe \n{\\it momentum} of an eigenstate. \nSince $H$ commutes with $T$, each eigenspace \n\\newline \n$V_\\epsilon := \\{v\\in V|\\ H_c v = \\epsilon v\\}$ decomposes further as per\n$$\nV_\\epsilon = \\oplus_p\\ V_{p,\\epsilon}, \n$$ \nwhere \n$$\nV_{p,\\epsilon} := \\{v\\in V|\\ H_cv = \\epsilon v, \nTv = e^{i p}v\\}\n$$ \nLet us add a constant to $H_c$ in such a way that the ground state energy\n$\\epsilon_0$ becomes $0$ and, on the plane $P$ with coordinates $(p, \\epsilon)$, \nlet us mark all the points, for which $V_{p,\\epsilon} \\neq 0$.\n\nZamolodchikov predicted \\cite{Z}, that there exist $8$ numbers \n$0 < m_1 < \\ldots < m_8$ with the following property. Let us draw on $P$ eight \nhyperbolae\n\\begin{equation} \n\\label{eq:dispersion}\n\\operatorname{Hyp}_i: \\ \\epsilon = \\sqrt{m_i^2 + p^2},\\ 1\\leq i \\leq 8.\n\\end{equation}\n\nAll the marked points will be located: \n\n---- either in a vicinity of one of the hyperbolae \n$\\operatorname{Hyp}_i$ (in the limit $N\\longrightarrow \\infty$ they will all lie on these hyperbolae). \n\n---- or in a shaded region separated from these hyperbolae \nas shown in the Fig. \\ref{fig:hyperbol}. \n\n\\begin{figure}\n\\includegraphics[scale=0.3]{ising-hyperb3.pdf}\n\\caption{The expected joint spectrum of the operators $T, H$.} \n\\label{fig:hyperbol}\n\\end{figure}\n\nThe states $v\\in V_{p,\\epsilon}$ \nwith $(p, \\epsilon)\\in \\operatorname{Hyp}_i$ are called {\\it elementary \nexcitations}. The numbers $m_i$ are called their {\\it masses}. \n\n\nThe vector\n\\begin{equation} \n\\label{eq:masses}\n\\vec{m} = (m_1, \\ldots, m_8)\n\\end{equation} \nis proportional to the Perron - Frobenius $v_{PF<}$ for $E_8$ from \\ref{sub492}, \nwhose normalized approximate value is\n\\begin{equation}\\label{eq:e8-masses}\nv_{PF<} = (1, 1.62, 1.99, 2.40, 2.96, 3.22, 3.89, 4.78)\n\\end{equation} \n\nThese low-lying excitations (hyperbolae) are \nobservable: one may be able to see them \n\n(a) in a computer simulation, or \n\n(b) in a neutron scattering experiment. \n\n\n\n\n\n\\subsection{Neutron scattering experiment}\n\nThe paper \\cite{Coldea} reports the results of a magnetic neutron scattering \nexperiment on cobalt niobate CoNb$_2$O$_6$, a material that can be pictured \nas a collection of parallel non-interacting\none-dimensional chains of atoms. We depict such a chain as a straight line, \nparallel to the $z$-axis in our physical space $\\mathbb{R}^3$ with coordinates $x, y, z$. \n\nThe sample, at low temperature $T < 2.95$K (Kelvin), was subject to an \nexternal magnetic field with components $(h_z,h_z)$, with the $h_x$ at \nthe critical value $h_x = h_c$, \nand with $h_z << h_c$. \nThe system may be described as the Ising chain \nwith a nearly-critical Hamiltonian $H = H(J, h_z << h_c, h_c)$ of the Eq. \n(\\ref{eqn:ising-ham}). The experiment \\cite{Coldea} may be interpreted \nwith the help of the following (oversimplified) theoretical picture.\n\nConsider a neutron scattering off the sample. If the incident neutron has \nenergy $\\epsilon$ and momentum $p$, and scatters off with energy $\\epsilon'$ \nand momentum $p'$, the energy and momentum conservation laws imply that \nthe differences, called energy and momentum transfers \n$\\omega = \\epsilon - \\epsilon', q = p - p'$, are absorbed by the sample.\n\nThe energy transfer cannot be arbirtary. Suppose that, prior to scattering the neutron, \nthe sample was in the ground state $v_0$; upon scattering the neutron, it undergoes \na transition to a state that is a linear combination of the eight elementary \nexcitations $v\\in V_{p,\\epsilon}$.\n\nWe will be interested in neutrons that scatter off with zero \nmomentum transfer. The Zamolodchikov theory \\cite{Z} predicted, that the neutron scattering intensity $\\mathcal{S}(0,\\omega)$ should have \npeaks at $\\omega = m_a$, ($a = 1, ..., 8$) of the Eq. (\\ref{eq:e8-masses}). \nAt zero momentum transfer, a neutron scattering experiment would \nmeasure the proportion of neutrons that scattered off with the energies \n$m_1, \\ldots, m_8$: the resulting $\\mathcal{S}(0,\\omega)$ would look \nas in the schematic Fig. \\ref{fig:mass.peaks}.\nMetaphorically speaking, the crystal would thus ``sound'' \nas a ``chord'' of eight ``notes'': the eigenfrequencies $m_i$.\n\n\\begin{figure}\n\\includegraphics[scale=0.3]{fig2ac.pdf}\n\\caption{A sketch of the scattering \nintensity $\\mathcal{S}(0,\\omega)$ at zero momentum peaks \nrelative to $\\mathcal{S}(0, m_1)$, against the $\\omega \/ m_1$ \nratio. The two leftmost peaks shown by thick lines correspond \nto the excitations with the masses $m_1$ and $m_2$, \nthat were resolved in the experiment \\cite{Coldea}. \nThe experimentally found mass ratio $m_2\/m_1$ is consistent \nwith $\\frac{m_2}{m_1} = \\frac{1 + \\sqrt{5}}{2}$, as per the \nexpression for the $v_{Zam}(m)$ in the Subsection \\ref{sub492}. \n}\n\\label{fig:mass.peaks}\n\\end{figure}\n\n\nAt the lowest temperatures, and in the immediate vicinity of $h_x = h_c$, \nthe experiment \\cite{Coldea} succeeded to resolve the first two excitations, \nand to extract their masses $m_1$ and $m_2$. \nThe mass ratio $m_2\/m_1$ was found to be $\\frac{m_2}{m_1} = 1.6 \\pm 0.025$, \nconsistent with $\\frac{m_2}{m_1} = \\frac{1 + \\sqrt{5}}{2} \\approx 1.618$ of the \nexpression for the $v_{Zam}(m)$ in the Subsection \\ref{sub492}. \nIn other words, the experimentalists were able to hear two of the \neight notes of the Zamolodchikov $E_8$ chord. \n\nA reader wishing to find out more about various facets of the story is \ninvited to turn to the references \\cite{Rajaraman,Delfino,Goss,Borthwick}.\n\n\\begin{acknowledgements}\nWe are grateful to Misha Finkelberg, Andrei Gabrielov, and Sabir Gusein-Zade \nfor the inspiring correspondence, and to Patrick Dorey for sending us his thesis. \nOur special gratitude goes to Sasha Givental whose remarks enabled us to \ngeneralize some statements and to simplify the exposition. A.V. \nthanks MPI in Bonn for hospitality; he was supported in part by NSF grant \nDMS-1362924 and the Simons Foundation grant no. 336826.\n\n\n\n\\end{acknowledgements}\n \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Acknowledgements}\n\n{\\it {\\bf Acknowledgements -}} This work is supported in part by DOE Grant\nNo. DE-FG02-13ER42020 and DE-FG02-12ER41848 and NSF Award PHY-1206044. T.K.\nwas also supported in part by Qatar National Research Fund under project NPRP\n5 - 464 - 1 - 080. K.S. is supported by NASA Astrophysics Theory Grant\nNNH12ZDA001N. Z.W. is supported by National Science Foundation under Grant No.\nPHY-1306951. Z.W. would like to thank Richard Cavanaugh for useful\ndiscussions.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}