John Franks, Patrice Le Calvez
In this paper we consider an analog of the regions of instability for twist maps in the context of area preserving diffeomorphisms which are not twist maps. Several properties analogous to those of classical regions of instability are proved.
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We construct a $C^2$ symplectic twist map f of the annulus that has an essential invariant curve C such that: - C is not differentiable; - the dynamics of f restricted to C is conjugated to the one of a Denjoy counter-example; - C is at the boundary of an instability zone for f.
For a twist map $f$ of the annulus preserving the Lebesgue measure, we give sufficient conditions to assure the existence of a set of positive measure of points with non-zero asymptotic torsion. In particular, we deduce that every bounded instability region for $f$ contains a set of positive measure of points with non-zero asymptotic torsion. Moreover, for an exact symplectic twist map $f$, we provide a simple, geometric proof of a result by Cheng and Sun (see [CS96]) which characterizes $\mathcal{C}^0$-integrability of $f$ by the absence of conjugate points.
We introduce a new measure of instability of area-preserving twist diffeomorphisms, which generalizes the notions of angle of splitting of separatrices, and flux through a gap of a Cantori. As an example of application, we establish a sharp >0 lower bound on the topological entropy in a neighbourhood of a hyperbolic, unique action-minimizing fixed point, assuming only no topological obstruction to diffusion, i.e. no homotopically non-trivial invariant circle consisting of orbits with the rotation number 0. The proof is based on a new method of precise construction of positive entropy invariant…
The twist interval of a twist map on the annulus $A=\mathbb{T}\times [0,1]$ has nonempty interior if $f$ preserves the area, but could be degenerate for general twist maps. In this note, we show that if a twist map $f$ is non-wandering, then the twist interval of $f$ is non-degenerate. Moreover, if there are two disjoint invariant curves of $f$, then their rotation numbers must be different (no matter if they are rational or irrational).
In this paper, we consider chaotic dynamics and variational structures of area-preserving maps. There is a lot of study on dynamics of their maps and the works of Poincare and Birkhoff are well-known. To consider variational structures of area-preserving maps, we define a special class of area-preserving maps called monotone twist maps. Variational structures determined from twist maps can be used for constructing characteristic trajectories of twist maps. Our goal is to define the variational structure such as giving infinite transition orbits through minimizing methods.
Variational Structures for Infinite Transition Orbits of Monotone Twist Maps
本文针对保面积映射中的单调扭转映射，利用变分结构，通过极小化方法证明了无限过渡轨道的存在性，该轨道在不动点间无限振荡。
We prove that symplectic twist maps defined on the cotangent bundle of the d-dimensional torus that have no conjugate points are $C^0$ integrable, i.e. the cotangent bundle is continuously foliated by a family of invariant Lagrangian graphs.
The Bernoulli property for counter-twisting linked twist maps
本研究证明了一类反向扭转连锁扭转映射的伯努利性质，聚焦弱剪切参数空间下临近双曲性丧失临界值的情形，所提方法有望推广至更多非一致双曲系统。
We study the dynamics of a symplectic twist map without conjugate points. We show that in a neighborhood of a totally periodic Lagrangian manifold, there exists a large family of invariant Lagrangian tori on which the twist map is conjugated to a translation of non-resonant vector.
We introduce the notion of abstract angle at a couple of points defined by two radial foliations of the closed annulus. We use this notion to give unified proofs of some classical results on area preserving positive twist maps of the annulus by using the Lifting Theorem and the Intermediate Value Theorem.
We consider the minimal average action (Mather's $β$ function) for area preserving twist maps of the annulus. The regularity properties of this function share interesting relations with the dynamics of the system. We prove that the $β$-function associated to a standard-like twist map admits a unique $C^1$-holomorphic complex extension, which coincides with this function on the set of real diophantine frequencies.
We investigate the break up of the last invariant curve for families of standard maps. Our main result is another evidence of how hard this problem is. We give an example of an analytic mapping with a "pathological" behavior, related to the break up of the last rotational invariant curve.
On the existence of periodic invariant curves for analytic families of twist maps and billiards
本文证明环面扭转映射解析族中给定旋转数的周期不变曲线要么对所有参数值存在，要么至多存在于离散子集中，并将此结论推广并应用于台球的有理焦散线模型。
On Non-KAM Invariant Circles for Area-Preserving Twist Maps
本文研究保面积扭转映射中超出KAM理论适用范围的不变圆，结合构造与非构造技术证明其存在性与破裂，为Mather 1998年两个开放问题提供新证据。
Destruction of invariant circles for Gevrey area-preserving twist maps
该研究证明可积生成函数在 C^r 拓扑下的任意小 Gevrey-α 扰动可破坏精确保面积扭转映射中给定旋转数的不变圆。
In this article we investigate rigidity properties of integrable area-preserving twist maps of the cylinder. More specifically, we prove that if a deformation of the standard integrable map preserves rotational invariant circles (i.e., homotopically non-trivial invariant curves of the cylinder on which the dynamics is conjugate to a rotation) for any rational rotation number in an open interval (without any further assumption on its size), then the deformation must be trivial. Analogue phenomena for integrable Tonelli Hamiltonian flows in higher dimensions are discussed.
Poincare's last geometric theorem (Poincare-Birkhoff Theorem) states that any area-preserving twist map of annulus has at least two fixed points. We replace the area-preserving condition with a weaker intersection property, which states that any essential simple closed curve intersects its image under $f$ at least at one point. The conclusion is that any such map has at least one fixed point. Besides providing a new proof to Poincare's geometric theorem, our result also has some applications to reversible systems.
We study a class of homeomorphisms of surfaces collectively known as linked-twist maps. We introduce an abstract definition which enables us to give a precise characterisation of a property observed by other authors, namely that such maps fall into one of two classes termed co- and counter-twisting. We single out three specific linked-twist maps, one each on the two-torus, in the plane and on the two-sphere and for each prove a theorem concerning its ergodic properties with respect to the invariant Lebesgue measure. For the map on the torus we prove that there is an invariant, zero-measure Can…
We show that the dynamics of sufficiently dissipative semi-Siegel complex Hénon maps with golden-mean rotation number is not $J$-stable in a very strong sense. By the work of Dujardin and Lyubich, this implies that the Newhouse phenomenon occurs for a dense $G_δ$ set of parameters in this family. Another consequence is that the Julia sets of such maps are disconnected for a dense set of parameters.
Let $ R $ be a rational map with totally disconnected Julia set $ J(R). $ If the postcritical set on $ J(R) $ contains a non-persistently recurrent (or conical) point, then we show that the map $ R $ can not be a structurally stable map.
We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an application of our main theorems in the compact case we further improve on a result of A. Venkatesh, recently already improved by J. Tanis and P. Vishe, on a sparse equidistribution problem for classical horocycle flows proposed by N. Shah and G. Margulis, and in the general non-compact, finite area case we prove bounds on Fourier coefficients of cups forms w…
In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex closed curve G in the plane. It is shown that there do not exist invariant circles near G when there is a point on G where the radius of curvature vanishes or is discontinuous. In addition, when the radius of curvature is not $C^1$ there are examples with orbits that converge to a point of G. If the derivative of the radius of curvature is bounded, such orbits cannot exist. The final section of the paper concerns an impact oscillator whose dynamics are the same as a dua…
These are notes for a minicourse given at Regional Norte UdelaR in Salto, Uruguay for the conference CIMPA Research School-Hamiltonian and Lagrangian Dynamics. We will present Birkhoff and Aubry-Mather theory for the conservative twist maps of the 2-dimensional annulus and focus on what happens close to the Aubry-Mather sets: definition of the Green bundles, link between hyperbolicity and shape of the Aubry-Mather sets, behaviour close to the boundaries of the instability zones. We will also give some open questions. This course is the second part of a minicourse that was begun by R. Potrie. S…
We relate the properties of the postsingular set for the exponential family to the questions of stability. We calculate the action of the Ruelle operator for the exponential family. We prove that if the asymptotic value is a summable point and its orbit satisfies certain topological conditions, the map is unstable hence there are no Beltrami differentials in the Julia set. Also we show that if the postsingular set is a compact set, then the singular value is summable.
We consider a sequences of symplectic twist maps corresponding to Frenkel Kontorova variational functional. For such a sequence without conjugate points we construct an invariant Lagrangian subbundle. Using the ideas by E.Hopf from Riemannian geometry we show that absence of conjugate points is a rare property for this class of maps.
Persistence of invariant graphs for twist maps under analytic perturbations
该研究针对与刚性旋转实解析共轭的扭转映射，在Gevrey-γ扰动下，采用参数化直接KAM方法，证明了满足更强条件时其不变图仍可持久存在，拓展了经典KAM理论的适用范围。
This paper concentrates on optical Hamiltonian systems of $T*\T^n$, i.e. those for which $\Hpp$ is a positive definite matrix, and their relationship with symplectic twist maps. We present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems. The novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system. We also present a theorem of suspension by Hamiltonian systems for the class of symplectic twist map that emerges in our study. Finally, we extend our results to manifolds of negative curvature.
In this note we consider $W$-shaped map $W_0=W_{s_1,s_2}$ with $\frac {1}{s_1}+\frac {1}{s_2}=1$ and show that eigenvalue 1 is not stable. We do this in a constructive way. For each perturbing map $W_a$ we show the existence of the "second" eigenvalue $λ_a$, such that $λ_a\to 1$, as $a\to 0$, which proves instability of isolated spectrum of $W_0$. At the same time, the existence of second eigenvalues close to 1 causes the maps $W_a$ behave in a metastable way. They have two almost invariant sets and the system spends long periods of consecutive iterations in each of them with infrequent jumps …
Linked-twist maps are area-preserving, piece-wise diffeomorphisms, defined on a subset of the torus. They are non-uniformly hyperbolic generalisations of the well-known Arnold Cat Map. We show that a class of canonical examples have polynomial decay of correlations for α-Hölder observables, of order 1/n.
We prove that for a large and important class of $C^1$ twist maps of the torus periodic and quasi-periodic orbits of a new type exist, provided that there are no rotational invariant circles (R.I.C's). These orbits have a non-zero ''vertical rotation number'' (V.R.N.), in contrast to what happens to Birkhoff periodic orbits and Aubry-Mather sets. The V.R.N. is rational for a periodic orbit and irrational for a quasi-periodic. We also prove that the existence of an orbit with a $V.R.N=a>0,$ implies the existence of orbits with $V.R.N=b,$ for all $0<b<a.$ In this way, related to a generalized de…
Foliations and minimality for twist maps
本文证明面积保持扭转映射在垂直旋转集为单点无理数时存在不变Lipschitz叶状结构，并在叶动力学与无理旋转双Lipschitz共轭时映射为极小。
Berkovich dynamics of twisted rational maps
该研究定义了非阿基米德域上的扭曲有理映射概念，并探究了这类映射在Berkovich射影直线上的动力学性质。
We investigate the connection between the instability of rational maps and summability methods applied to the spectrum of a critical point on the Julia set of a given rational map.
A class of piecewise $C^2$ Lozi-like maps in three-dimensional Euclidean spaces is introduced, and the existence of Sinai-Ruelle-Bowen measures is studied, where the dimension of the instability is equal to two. Further, an example with computer simulations is provided to illustrate the theoretical results.
We prove the existence of infinitely many periodic solutions and complicated dynamics, due to the presence of a topological horseshoe, for the classical Volterra predator--prey model with a periodic harvesting. The proof relies on some recent results about chaotic planar maps combined with the study of geometric features which are typical of linked twist maps.
On the Destruction of Invariant Lagrangian Graphs for Conformal Symplectic Twist Maps
本文研究耗散映射不变拉格朗日图的脆弱性，证明可积耗散扭转映射的C⁰不变拉格朗日图可被C¹⁻ᵉ拓扑中任意小扰动破坏，且结果具尖锐性，因C¹不变图在C¹扰动下可保持。
The key result of this article is key lemma: if a Jordan curve $γ$ is invariant by a given C 1+$α$ -diffeomorphism f of a surface and if $γ$ carries an ergodic hyperbolic probability $μ$, then $μ$ is supported on a periodic orbit. From this Lemma we deduce three new results for the C 1+$α$ symplectic twist maps f of the annulus: 1. if $γ$ is a loop at the boundary of an instability zone such that f |$γ$ has an irrational rotation number, then the convergence of any orbit to $γ$ is slower than exponential; 2. if $μ$ is an invariant probability that is supported in an invariant curve $γ$ with an…
For a symplectic twist map, we prove that there is a choice of weak K.A.M. solutions that depend in a continuous way on the cohomology class. We thus obtain a continuous function $u(θ, c)$ in two variables: the angle $θ$ and the cohomology class $c$. As a result, we prove that the Aubry-Mather sets are contained in pseudographs that are vertically ordered by their rotation numbers. Then we characterize the $C^0$ integrable twist maps in terms of regularity of $u$ that allows to see $u$ as a generating function. We also obtain some results for the Lipschitz integrable twist maps. With an exampl…
On automorphic measures, Lyapunov exponents and instability of rational maps
引入复特征值的自守测度，证明其存在导致有理映射的不稳定性，并建立遍历与组合条件确保此类测度存在。
Computer assisted proofs for hyperbolic quasi-periodic invariant tori in dissipative twist maps
本文提出并实现了一种基于后验KAM型定理的计算机辅助方法，用于证明耗散标准映射中双曲准周期不变环面的存在性，并为特定参数提供了不变圆的严格证明。
On the Dynamics of Invariant Graphs for Dissipative Twist Maps
本文研究了双参数族耗散扭转映射中不变图的动力学及其存在与崩溃阈值，证明了通过调参可实现预定旋转数的不变图，刻画了导致其完全破坏的尖锐扰动，并探讨了非$C^1$扰动下Lipschitz不变图的持续存在性。
We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For Gâteaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and, for Fréchet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fréchet differentiability is sufficient for linear exponential in…
On the fragility of periodic tori for families of symplectic twist maps
本文研究辛扭转映射族中Lagrange周期环面在势扰动下的脆弱性，证明非退化条件下指定旋转向量的周期环面仅能在有限参数值存在，并推出可积映射的刚性结果。
For exact symplectic twist maps of the annulus, we etablish a choice of weak K.A.M. solutions $u_c=u(\cdot, c)$ that depend in a Lipschitz-continuous way on the cohomology class $c$. This allows us to make a bridge between weak K.A.M. theory of Fathi, Aubry-Mather theory for semi-orbits as developped by Bangert and existence of backward invariant pseudo-foliations as seen by Katnelson \& Ornstein. We deduce a very precise description of the pseudographs of the weak K.A.M. solutions and many interesting results as --the Aubry-Mather sets are contained in pseudographs that are vertically ordered…
Since Moser's seminal work it is well known that the invariant curves of smooth nearly integrable twist maps of the cylinder with Diophantine rotation number are preserved under perturbation. In this paper we show that, in the analytic class, the result extends to Bryuno rotation numbers. First, we will show that the series expansion for the invariant curves in powers of the perturbation parameter can be formally defined, then we shall prove that the series converges absolutely in a neighbourhood of the origin. This will be achieved using multiscale analysis and renormalisation group technique…
Locally Maximizing orbits for multi-dimensional Twist maps and Birkhoff billiards
本文研究多维辛扭转映射（特别是Birkhoff台球映射）的局部极大化轨道，提出几何准则并证明该映射的两个生成函数具有相同的局部极大化轨道类。
Discrete and Continuous Weak KAM Theory: an introduction through examples and its applications to twist maps
本文以自洽方式介绍离散弱 KAM 理论，通过实例展示其作为经典弱 KAM 理论入门模型的价值，并给出离散与连续理论的新比较结果及保守扭转映射应用。
Regularity properties of critical invariant circles of twist maps, and their universality
通过数值计算圆柱面上面积保持扭转映射的黄金临界不变圆，分析相关函数的全局Hölder正则性，并基于傅里叶分析方法验证不同映射下正则性的普适性。
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