By exploiting the contact Hamiltonian dynamics $(T^*M\times\mathbb R,Φ_t)$ around the Aubry set of contact Hamiltonian systems, we provide a relation among the Mather set, the $Φ_t$-recurrent set, the strongly static set, the Aubry set, the Mañé set and the $Φ_t$-non-wandering set. Moreover, we consider the strongly static set, as a new flow-invariant set between the Mather set and the Aubry set, in the strictly increasing case. We show that this set plays an essential role in the representation of certain minimal forward weak KAM solution and the existence of transitive orbits around the Aubry set.
We prove that the so-called Aubry and Mane sets introduced by John Mather in Lagrangian dynamics are symplectic invariants. In order to do so, we introduce a barrier in phase space, and propose definitions of Aubry and Mane sets for non-convex Hamiltonian systems. On montre que les ensembles dits d'Aubry et de Mane introduits par John Mather en dynamique Lagrangienne sont des invariants symplectiques. Pour ceci on introduit une barriere sur l'espace des phases, et on definit des ensembles d'Aubry et de Mather pour des systemes Hamiltoniens non convexes.
We consider the irrational Aubry-Mather sets of an exact symplectic monotone twist map and explain what is the link between the Lyapunov exponents and the shape of such a set. The main tools that we use in the proofs are the so-called Green bundles.
We study autonomous Tonelli Lagrangians on closed surfaces. We aim to clarify the relationship between the Aubry set and the Mather set, when the latter consists of periodic orbits which are not fixed points. Our main result says that in that case the Aubry set and the Mather set almost always coincide.
In this article we develop an analogue of Aubry-Mather theory for a class of dissipative systems, namely conformally symplectic systems, and prove the existence of interesting invariant sets, which, in analogy to the conservative case, will be called the Aubry and the Mather sets. Besides describing their structure and their dynamical significance, we shall analyze their attracting/repelling properties, as well as their noteworthy role in driving the asymptotic dynamics of the system.
We consider the model of a point-vortex under a periodic perturbation and give sufficient conditions for the existence of generalized quasi-periodic solutions with rotation number. The proof uses Aubry-Mather theory to obtain the existence of a family of minimal orbits of the Poincaré map associated to the system.
In this paper we study minimax Aubry-Mather measures and its main properties. We consider first the discrete time problem and then the continuous time case. In the discrete time problem we establish existence, study some of the main properties using duality theory and present some examples. In the continuous time case, we establish both existence and non-existence results. First we give some examples that show that in continuous time stationary minimax Mather measures are either trivial or fail to exist. A more natural definition in continuous time are $T$-periodic minimax Mather measures. We …
If we perturb a completely integrable Hamiltonian system with two degrees of freedom, the perturbed flow might display, on every energy level, invariant sets that are laminations over Aubry-Mather sets of a Poincaré section of the flow. Each one of these laminations carry a unique invariant probability measure for the flow on which mixing is impossible in this low dimensional frame. We prove that if the Aubry-Mather set has exactly one orbit of gaps and is hyperbolic then the special flow over it with any smooth ceiling function will be conjugate to a suspension with a constant ceiling functio…
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.
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…
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.
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…
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.
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 …
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.
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…
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…
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.
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.
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.
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.
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…
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…
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…
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…
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…
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…
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…
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…
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})$…
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…
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…
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.
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…
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…
Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $ϕ_H$ of a Hamiltonian $H\in C^{\infty}(M)$ on a symplectic manifold $(M,ω)$. These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures and we construct an example showing that their support can be extremely unstable when $H$ fails to be convex, even for nearly integrable $H$. Parts of these results extend work by Viterbo and Vichery. Using ideas due to Entov-Po…
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.
