Aubry sets vs Mather sets in two degrees of freedom

On Aubry sets and Mather's action functional

We study Lagrangian systems on a closed manifold. We link the differentiability of Mather's beta-function with the topological complexity of the complement of the Aubry set. As a consequence, when the dimension of the manifold is less than or equal to two, the differentiability of the beta-function at a given homology class is forced by the irrationality of the homology class. As an application we prove the two-dimensional case of a conjecture by Ricardo Mane.

接触哈密顿系统的 Aubry–Mather 理论 III

Aubry-Mather theory for contact Hamiltonian systems III

本文研究接触哈密顿系统的 Aubry–Mather 理论，刻画多类不变集关系，并揭示强静态集在弱 KAM 解表示与传递轨道存在性中的作用。

Ghost circles in lattice Aubry-Mather theory

Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice, arise in Hamiltonian lattice mechanics as models for fe?rromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multidimensional counterpart of monotone twist maps. They often admit a variational structure, so that the solutions are the stationary points of a formal action function. Classical Aubry-Mather theory establishes the existence of a large collection of solutions of any rotation vector. For irrational rotation vectors this is the well-known Aubry-Mather set. It consists of global minim…

Aubry-Mather theory on graphs

We formulate Aubry-Mather theory for Hamiltonians/Lagrangians defined on graphs and discuss its relationship with weak KAM theory developed in [24].

Aubry-Mather Theory for Lorentzian Manifolds

We introduce a version of Aubry-Mather theory for the length functional of causal curves in compact Lorentzian manifolds. Results include the existence of maximal invariant measures, calibrations and calibrated curves. We prove two versions of the Mather's graph theorem. A class of examples, the Lorentzian Hedlund examples, shows the optimality of the obtained results.

On the destruction of minimal foliations

Monotone variational recurrence relations such as the Frenkel-Kontorova lattice, arise in solid state physics, conservative lattice dynamics and as Hamiltonian twist maps. For such recurrence relations, Aubry-Mather theory guarantees the existence of solutions of every rotation number. They are the action minimizers that constitute the Aubry-Mather set. When the rotation number is irrational, the Aubry-Mather set is either connected or a Cantor set. A connected Aubry-Mather set is called a minimal foliation. In the case of twist maps, it describes an invariant circle, while in solid state phys…

Denjoy sub-systems and horseshoes

We introduce a notion of weak Denjoy subsystem (WDS) that generalizes the Aubry-Mather Cantor sets to diffeomorphisms of manifolds. We explain how a rotation number can be associated to such a WDS. Then we build in any horseshoe a continuous one parameter family of such WDS that is indexed by its rotation number. Looking at the inverse problem in the setting of Aubry-Mather theory, we also prove that for a generic conservative twist map of the annulus, the majority of the Aubry-Mather sets are contained in some horseshoe that is associated to a Aubry-Mather set with a rational rotation number.

The Aubry-Mather theorem for driven generalized elastic chains

We consider uniformly (DC) or periodically (AC) driven generalized infinite elastic chains (a generalized Frenkel-Kontorova model) with gradient dynamics. We first show that the union of supports of all the invariant measures, denoted by A, projects injectively to a dynamical system on a 2-dimensional cylinder. We also prove existence of ergodic invariant measures supported on a set of rotationaly ordered configurations with an arbitrary (rational or irrational) rotation number. This shows that the Aubry-Mather structure of ground states persists if an arbitrary AC or DC force is applied. The …

Aubry-Mather theory and Lipschitz continuity of the time separation

We consider Aubry-Mather theory for a subclass of class A spacetimes, i.e. compact vicious spacetimes with globally hyperbolic Abelian cover. In this subclass, called class A_1, we obtain improved results on timelike maximizers and Lipschitz continuity of the time separation of the Abelian cover on the i.g. optimal subsets.

Spectral invariants towards a Non-convex Aubry-Mather theory

Aubry-Mather is traditionally concerned with Tonelli Hamiltonian (convex and super-linear). In \cite{Vi,MVZ}, Mather's $α$ function is recovered from the homogenization of symplectic capacities. This allows the authors to extend the Mather functional to non convex cases. This article shows that the relation between invariant measures and the subdifferential of Mather's functional (which is the foundational statement of Mather) is preserved in the non convex case. We give applications in the context of the classical KAM theory to the existence of invariant measures with large rotation vector af…

On the Aubry-Mather theory for symbolic dynamics

We propose a new model of ergodic optimization for expansive dynamical systems: the holonomic setting. In fact, we introduce an extension of the standard model used in this theory. The formulation we consider here is quite natural if one wants a meaning for possible variations of a real trajectory under the forward shift. In another contexts (for twist maps, for instance), this property appears in a crucial way. A version of the Aubry-Mather theory for symbolic dynamics is introduced. We are mainly interested here in problems related to the properties of maximizing probabilities for the two-si…

A Stochastic Analog of Aubry-Mather Theory

In this paper we discuss a stochastic analog of Aubry-Mather theory in which a deterministic control problem is replaced by a controlled diffusion. We prove the existence of a minimizing measure (Mather measure) and discuss its main properties using viscosity solutions of Hamilton-Jacobi equations. Then we prove regularity estimates on viscosity solutions of Hamilton-Jacobi equation using the Mather measure. Finally we apply these results to prove asymptotic estimates on the trajectories of controlled diffusions and study the convergence of Mather measures as the rate of diffusion vanishes.

Percival Lagrangian approach to Aubry-Mather theory

We present some streamlined proofs of some of the basic results in Aubry-Mather theory (existence of quasi-periodic minimizers, multiplicity results when there are gaps among minimizers) based on the study of hull functions. We present results in arbitrary number of dimensions We also compare the proofs and results with those obtained in other formalisms.

Aubry和Mather集的半离散近似

Semi-Discrete Approximation of Aubry and Mather sets

本文研究了Tonelli拉格朗日量在平直紧致流形上Aubry和Mather集的半离散近似，通过离散Lax-Oleinik方程引入自然离散类比，并分析其收敛性。结果表明半离散变分框架不仅捕捉到ergodic常数，还恢复了连续动力学的最小不变几何结构。

Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory

We consider Lagrangian submanifolds lying on a fiberwise strictly convex hypersurface in some cotangent bundle or, respectively, in the domain bounded by such a hypersurface. We establish a new boundary rigidity phenomenon, saying that certain Lagrangians on the hypersurface cannot be deformed (via Lagrangians having the same Liouville class) into the interior domain. Moreover, we study the "non-removable intersection set" between the Lagrangian and the hypersurface, and show that it contains a set with specific dynamical behavior, known as Aubry set in Aubry-Mather theory.

Aubry-Mather theory and idempotent eigenfunctions of the Bellman operator

We establish a connection between the Aubry-Mather theory of invariant sets of a 1D dynamical system described by a Lagrangian with potential periodic in space and time, on the one hand, and idempotent spectral theory of the Bellman operator of the corresponding optimization problem, on the other hand. This connection is applied to obtain a uniqueness result for an eigenfunction of the Bellman operator in the case of irrational rotation number.

On the Conley decomposition of Mather sets

In the context of Mather's theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian.

Aubry-Mather theory for contact Hamiltonian systems II

In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems $H(x,u,p)$ with certain dependence on the contact variable $u$. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set $\tilde{\mathcal{S}}_s$ consists of strongly static orbits, wh…

Diffusion along transition chains of invariant tori and Aubry-Mather sets

We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus, (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map, (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains, i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence, (iv) the tra…

Aubry-Mather and weak KAM theories for contact Hamiltonian systems. Part 1: Strictly increasing case

This paper is concerned with the study of Aubry-Mather and weak KAM theories for contact Hamiltonian systems with Hamiltonians $H(x,u,p)$ defined on $T^*M\times\mathbb{R}$, satisfying Tonelli conditions with respect to $p$ and $0<\frac{\partial H}{\partial u}\leqslant λ$ for some $λ>0$, where $M$ is a connected, closed and smooth manifold. First, we show the uniqueness of the backward weak KAM solutions of the corresponding Hamilton-Jacobi equation. Using the unique backward weak KAM solution $u_-$, we prove the existence of the maximal forward weak KAM solution $u_+$. Next, we analyse Aubry s…

Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry-Mather theory

We construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants developed by Oh and results by Abouzaid about the Fukaya category of a cotangent bundle. We also introduce the notion of Lipschitz-exact Lagrangians and prove that these admit an appropriate generalization of graph selector. We then, following Bernard-Oliveira dos Santos, use these results to give a new characterization of the Aubry and Mane sets of a Tonelli Hamiltonian and to generalize a result of Arnaud on Lagrangians…

Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics

We obtain a large deviation function for the stationary measures of twisted Brownian motions associated to the Lagrangians $L_λ(p,v)=\frac{1}{2}g_{p}(v,v)- λω_{p}(v)$, where $g$ is a $C^{\infty}$ Riemannian metric in a compact surface $(M,g)$ with nonpositive curvature, $ω$ is a closed 1-form such that the Aubry-Mather measure of the Lagrangian $L(p,v)=\frac{1}{2}g_{p}(v,v)-ω_{p}(v)$ has support in a unique closed geodesic $γ$; and the curvature is negative at every point of $M$ but at the points of $γ$ where it is zero. We also assume that the Aubry set is equal to the Mather set. The large d…

Hyperbolicity for conservative twist maps of the 2-dimensional annulus

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…

Locally Lipschitz graph property for lines

On a non-compact, smooth, connected, boundaryless, complete Riemannian manifold $(M,g)$, one can define its ideal boundary by rays (or equivalently, Busemann functions). From the viewpoint of Mather theory, boundary elements could be regarded as the static classes of Aubry sets, and thus lines should be think as the semi-statics curves connecting different static classes. In Mather theory, one core property is Lipschitz graph property for Aubry sets and for some kind of semi-static curves. In this article, we prove a such kind of result for a set of lines which connect the same pair of boundar…

On commuting Tonelli Hamiltonians: Time-periodic case

We show that the Aubry sets, the Mañé sets and Mather's barrier functions are the same for two commuting time-periodic Tonelli Hamiltonians.

离散Aubry-Mather模型在连续极限下的收敛性

Convergence of discrete Aubry-Mather model in the continuous limit

本文基于Aubry-Mather-Fathi理论，发展了两种求解单元方程和折扣单元方程的近似方案，研究了时间离散化后涉及离散Lax-Oleinik算子的加法特征值问题，并展示了如何通过时间步长趋近于零来逼近有效哈密顿量和弱KAM解。

Systèmes lagrangiens et fonction $β$ de Mather

We review the author's results on Mather's $β$ function : non-strict convexity of $β$ when the configuration space has dimension two, link between the size of the Aubry set and the differentiability of $β$, correlation between the rationality of the homology class and the differentiability of $β$, equality of the Mather set and the Aubry set for a large number of cohomology classes when the configuration space has dimension two, link beween the differentiability of $β$ and the integrability of the system. Mañé's conjectures are discussed in Chapters 6 and 7. A short list of open problems is gi…

On finite quotient Aubry set for generic geodesic flows

We study the structure of the Mather and Aubry sets for the family of lagrangians given by the kinetic energy associated to a riemannian metric $ g$ on a closed manifold $ M$. In this case the Euler-Lagrange flow is the geodesic flow of $(M,g)$. We prove that there exists a residual subset $ \mathcal G$ of the set of all conformal metrics to $g$, such that, if $ \overline g \in \mathcal G$ then the corresponding geodesic flow has a finitely many ergodic c-minimizing measures, for each non-trivial cohomology class $ c \in H^1(M,\mathbb{R})$. This implies that, for any $ c \in H^1(M,\mathbb{R})$…

Large Deviations for stationary probabilities of a family of continuous time Markov chains via Aubry-Mather theory

We consider a family of continuous time symmetric random walks indexed by $k\in \mathbb{N}$, $\{X_k(t),\,t\geq 0\}$. For each $k\in \mathbb{N}$ the matching random walk take values in the finite set of states $Γ_k=\frac{1}{k}(\mathbb{Z}/k\mathbb{Z})$ which is a subset of the unitary circle. The stationary probability for such process converges to the uniform distribution on the circle, when $k\to \infty$. We disturb the system considering a fixed $C^2$ potential $V: \mathbb{S}^1 \to \mathbb{R}$ and we will denote by $V_k$ the restriction of $V$ to $Γ_k$. Then, we define a non-stochastic semigr…

Wigner measures and the semi-classical limit to the Aubry-Mather measure

In this paper we investigate the asymptotic behavior of the semi-classical limit of Wigner measures defined on the tangent bundle of the one-dimensional torus. In particular we show the convergence of Wigner measures to the Mather measure on the tangent bundle, for energy levels above the minimum of the effective Hamiltonian. The Wigner measures $μ_h$ we consider are associated to $ψ_h,$ a distinguished critical solution of the Evans' quantum action given by $ψ_h=a_h\,e^{i\frac{u_h}h}$, with $a_h(x)=e^{\frac{v^*_h(x)-v_h(x)}{2h}}$, $u_h(x)=P\cdot x+\frac{v^*_h(x)+v_h(x)}{2},$ and $v_h,v^*_h$ s…

On commuting Tonelli Hamiltonians: Autonomous case

We show that the Aubry sets, the Mañé sets, Mather's barrier functions are the same for two commuting autonomous Tonelli Hamiltonians. We also show the quasi-linearity of $α$-functions from the dynamical point of view and the existence of common $C^{1,1}$ critical subsolution for their associated Hamilton-Jacobi equations.

Symplectic and contact properties of the Mañé critical value of the universal cover

We discuss several symplectic aspects related to the Mañé critical value c_u of the universal cover of a Tonelli Hamiltonian. In particular we show that the critical energy level is never of virtual contact type for manifolds of dimension greater than or equal to three. We also show the symplectic invariance of the finiteness of the Peierls barrier and the Aubry set of the universal cover. We also provide an example where c_u coincides with the infimum of Mather's α-function but the Aubry set of the universal cover is empty and the Peierls barrier is finite. A second example exhibits all the e…

全移位系统中带有不可数字母表[0,1]的极小集的显式公式与拓扑描述

Explicit formulae and topological descriptions of action-minimizing sets of a full shift with an uncountable alphabet $[0,1]$

本文研究了全移位系统在不可数字母表[0,1]上的遍历优化问题，推导了关联的Mather集和Aubry集的显式公式，并探讨了Aubry集的完全不连通性，同时定义了Mañé集并证明其比Aubry集大得多。

Dynamics of the dominant Hamiltonian, with applications to Arnold diffusion

It is well known that instabilities of nearly integrable Hamiltonian systems occur around resonances. Dynamics near resonances of these systems is well approximated by the associated averaged system, called slow system. Each resonance is defined by a basis (a collection of integer vectors). We introduce a class of resonances whose basis can be divided into two well separated groups and call them dominant. We prove that the associated slow system can be well approximated by a subsystem given by one of the groups, both in the sense of the vector field and weak KAM theory. One of crucial ingredie…

KAM and geodesic dynamics of blackholes

In this paper we apply KAM theory and the Aubry-Mather theory for twist maps to the study of bound geodesic dynamics of a perturbed blackhole background. The general theories apply mainly to two observable phenomena: the photon shell (unstable bound spherical orbits) and the quasi-periodic oscillations. We discover there is a gap structure in the photon shell that can be used to reveal information of the perturbation.

A note on Tonelli Lagrangian systems on $\mathbb{T}^2$ with positive topological entropy on high energy level

In this work we study the dynamical behavior Tonelli Lagrangian systems defined on the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $ E_L^{-1}(c)$ (i.e $ c> c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale propriety in $ E_L^{-1}(c)$ (i.e, all closed orbit with energy $c$ are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_L^{-1}(c)$). The proof requires the use of well-known results in Aubry-Mather's Theory.

哈密顿系统中拉格朗日环面的定量破坏与持续性

Quantitative Destruction and Persistence of Lagrangian Torus in Hamiltonian Systems

本文在超出经典KAM方法适用范围的小扰动下，定量研究了$d$自由度可积哈密顿系统中给定频率的流不变拉格朗日环面的存在性与不存在性，并部分回答了Mather提出的两个关于不变圆周的公开问题。

双曲Aubry集的消失折扣问题的Sharp收敛速率

Sharp convergence rates for the vanishing discount problem with hyperbolic Aubry sets

针对闭连通流形上Tonelli哈密顿量的消失折扣问题，在提升Aubry集由双曲平衡点或双曲周期轨道构成的条件下，证明了解向临界解的收敛速率，给出最优收敛阶并建立了该问题首个系统定量理论。

奥布里集的$L^\u221e$变分近似

$L^\infty$ Variational Approximation of the Aubry Set

该研究针对周期托内利哈密顿量，通过Evans引入的变分泛函的极小值序列的极限，刻画了投影奥布里集的$L^\u221e$变分近似，为近似奥布里集提供了数值定位原理。

Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders

In the present paper we prove a form of Arnold diffusion. The main result says that for a "generic" perturbation of a nearly integrable system of arbitrary degrees of freedom $n\ge 2$ \[ H_0(p)+\eps H_1(þ,p,t),\quad þ\in \T^n,\ p\in B^n,\ t\in \T=\R/\T, \] with strictly convex $H_0$ there exists an orbit $(þ_{\e},p_{e})(t)$ exhibiting Arnold diffusion in the sens that [\sup_{t>0}\|p(t)-p(0) \| >l(H_1)>0] where $l(H_1)$ is a positive constant independant of $\e$. Our proof is a combination of geometric and variational methods. We first build 3-dimensional normally hyperbolic invariant cylinders…

Weak KAM for commuting Hamiltonians

For two commuting Tonelli Hamiltonians, we recover the commutation of the Lax-Oleinik semi-groups, a result of Barles and Tourin ([BT01]), using a direct geometrical method (Stoke's theorem). We also obtain a "generalization" of a theorem of Maderna ([Mad02]). More precisely, we prove that if the phase space is the cotangent of a compact manifold then the weak KAM solutions (or viscosity solutions of the critical stationary Hamilton-Jacobi equation) for G and for H are the same. As a corrolary we obtain the equality of the Aubry sets, of the Peierls barrier and of flat parts of Mather's $α$ fu…

Applications of Forcing Theory to Homeomorphisms of the Closed Annulus

This paper studies homeomorphisms of the closed annulus that are isotopic to the identity from the viewpoint of rotation theory, using a newly developed forcing theory for surface homeomorphisms. Our first result is a solution to the so called strong form of Boyland's Conjecture on the closed annulus: Assume $f$ is a homeomorphism of $\overline{\mathbb{A}}:=(\mathbb{R}/\mathbb{Z})\times [0,1]$ which is isotopic to the identity and preserves a Borel probability measure $μ$ with full support. We prove that if the rotation set of $f$ is a non-trivial segment, then the rotation number of the measu…

A strong form of Arnold diffusion for two and a half degrees of freedom

In the present paper we prove a strong form of Arnold diffusion. Let $\mathbb{T}^2$ be the two torus and $B^2$ be the unit ball around the origin in $\mathbb{R}^2$. Fix $ρ>0$. Our main result says that for a "generic" time-periodic perturbation of an integrable system of two degrees of freedom \[ H_0(p)+εH_1(θ,p,t),\quad \ θ\in \mathbb{T}^2,\ p\in B^2,\ t\in \mathbb{T}, \] with a strictly convex $H_0$, there exists a $ρ$-dense orbit $(θ_ε,p_ε,t)(t)$ in $\mathbb{T}^2 \times B^2 \times \mathbb{T}$, namely, a $ρ$-neighborhood of the orbit contains $\mathbb{T}^2 \times B^2 \times \mathbb{T}$. Our …