The link between the shape of the irrational Aubry-Mather sets and their Lyapunov exponents

Arnaud, Marie-Claude

doi:10.4007/annals.2011.174.3.4

Cited by 18 publications

(18 citation statements)

References 27 publications

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“…The basic idea is based on the following dichotomy for Green bundles: either they are always transverse, in which case one gets hyperbolicity of the Aubry set; or the Green bundles coincide along a given orbit of the Aubry set, and in this latter case, elaborating on previous works by Arnaud [4,5], we show that the restriction to the projected Aubry set of any critical solution is C 2 along the projected orbit. This additional regularity property is not enough to apply the techniques which were introduced in [19,20], since there the authors had to require the existence of a critical solution which is C 1,1 in a neighborhood of the projected orbit and C 2 along it.…”

Section: Introductionmentioning

confidence: 67%

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Generic hyperbolicity of Aubry sets on surfaces

2014

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Given a Tonelli Hamiltonian of class C 2 on the cotangent bundle of a compact surface, we show that there is an open dense set of potentials in the C 2 topology for which the Aubry set is hyperbolic in its energy level. IntroductionLet M be a smooth compact Riemannian manifold without boundary of dimension n ≥ 2, and H : T * M → R be a Tonelli Hamiltonian of class C 2 . As shown by Mather [27], one can construct a compact invariant subset of T * M which enjoys several variational properties and has the distinguished feature of being a Lipschitz graph over M . This set, called the Aubry set associated to H and denoted byÃ(H), captures many important features of the Hamiltonian dynamics.Fathi [16] established a bridge between the Aubry-Mather theory and the properties of viscosity solutions/subsolutions of the critical Hamilton-Jacobi equation associated with H, giving rise to the weak KAM theory. The differentials of critical (viscosity) subsolutions are uniquely determined on the projection ofÃ(H) onto M (denoted by A(H)), and all critical subsolutions are indeed C 1,1 on the projected Aubry set A(H). We refer the reader to Section 2.1 below for a precise definition of the Aubry set and more details in weak KAM theory.A famous open problem concerning the structure ofÃ(H) is the so-called "Mañé conjecture" [25] which states that, for a generic Hamiltonian, the Aubry set is either a hyperbolic equilibrium or a hyperbolic periodic orbit. In [19,20], the second and third author obtained several results in the direction of proving the validity of the Mañé conjecture. However, all that results heavily rely on the assumption of the existence of a sufficiently smooth critical (sub-)solution. The goal of this paper is to combine some of the techniques developed in [19,20] Is it true that generically the Aubry set is hyperbolic? As mentioned by Herman at the beginning of [22, Section 6], the subject of the instabilities of Hamiltonian flows and the problem of topological stability "lacks any non-trivial result". Our main theorem solves in the affirmative Herman's problem on surfaces for the C 2 -topology.0 The second and third authors are supported by the program "Project ANR-07-BLAN-0361, Hamilton-Jacobi et théorie KAM faible". AF is also supported by NSF Grants DMS-0969962 and DSM-1262411. GC is supported by CONACYT Grant 178838. LR is also supported by the ANR project GCM, program "Blanche", project number NT09-504490. 1 Theorem 1.1. Let H : T * M → R be a Tonelli Hamiltonian of class C 2 , and assume that dim M = 2. Then there is an open dense set of potentials V ⊂ C 2 (M ) such that, for every V ∈ V, the Aubry set associated to the Hamiltonian H + V is hyperbolic in its energy level.The proof of Theorem 1.1 relies on the properties of Green bundles which can be associated with each orbit of the Aubry set. The basic idea is based on the following dichotomy for Green bundles: either they are always transverse, in which case one gets hyperbolicity of the Aubry set; or the Green bundles coincide along a given or...

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Section: Introductionmentioning

confidence: 67%

“…The present section is inspired by ideas and techniques developed by Arnaud in [4,5], and by the last two authors in [20]. Let S ⊂ R k be a compact set which has the origin as a cluster point.…”

Section: Paratingent Cones and Green Bundlesmentioning

confidence: 99%

Generic hyperbolicity of Aubry sets on surfaces

2014

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“…In the above theorem, the generalized Hessian of u at x x depends upon the Riemannian metric g. However, it is worth noticing that assumption (iii) does not depend on the metric g. Such an assumption is motivated by some recent results of Arnaud [3][4][5]. Let us also point out that, since the graph of du is invariant under the Hamiltonian flow in V, assumption (iii) implies that H ess g x u is a singleton for any x 2 O C .x x/.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Closing Aubry Sets I

Figalli

Rifford

2014

Comm Pure Appl Math

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Given a Tonelli Hamiltonian H W T M ! R of class C k , with k 2, we prove the following results: (1) Assume there exist a recurrent point of the projected Aubry set x x and a critical viscosity subsolution u such that u is a C 1 critical solution in an open neighborhood of the positive orbit of x x. Suppose further that u is "C 2 at x x". Then there exists a C k potential V W M ! R, small in C 2 -topology, for which the Aubry set of the new Hamiltonian H C V is either an equilibrium point or a periodic orbit. (2) If M is two dimensional, (1) holds replacing "C 1 critical solution + C 2 at x x" by "C 3 critical subsolution". These results can be considered as a first step through the attempt of proving the Mañé's conjecture in C 2 -topology. In a second paper [27], we will generalize (2) to arbitrary dimension. Moreover, such an extension will need the introduction of some new techniques, which will allow us to prove in [27] the Mañé's density conjecture in C 1 -topology. Our proofs are based on a combination of techniques coming from finite-dimensional control theory and Hamilton-Jacobi theory, together with some of the ideas that were used to prove C 1 -closing lemmas for dynamical systems.

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“…What the theory of [2] tells us is that x → df x in this case cannot be the the restriction of a C 1 section of T * T 2 . Already from Mather's theory [21] (or from Theorem 12) we know that it must be Lipschitz, and the question remains as to whether it is something in-between.…”

Section: Examplesmentioning

confidence: 99%

“…The theory developed in [2] can be adapted to analyze the regularity of supp µ. That theory is about maps on the annulus S 1 × R. To adapt it, take a smooth circle β ⊂ M transversal to restriction of the geodesic flow determined by supp µ, and look at the map φ : T β → T β determined by the first-return map of that flow.…”

Section: Examplesmentioning

confidence: 99%

Characterization of minimizable Lagrangian action functionals and a dual Mather theorem

Rios-Zertuche¹

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We show that a necessary and sufficient condition for a smooth function on the tangent bundle of a manifold to be a Lagrangian density whose action can be minimized is, roughly speaking, that it be the sum of a constant, a nonnegative function vanishing on the support of the minimizers, and an exact form.We show that this exact form corresponds to the differential of a Lipschitz function on the manifold that is differentiable on the projection of the support of the minimizers, and its derivative there is Lipschitz. This function generalizes the notion of subsolution of the Hamilton-Jacobi equation that appears in weak kam theory, and the Lipschitzity result allows for the recovery of Mather's celebrated 1991 result as a special case. We also show that our result is sharp with several examples.Finally, we apply the same type of reasoning to an example of a finite horizon Legendre problem in optimal control, and together with the Lipschitzity result we obtain the Hamilton-Jacobi-Bellman equation and the Maximum Principle.

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The link between the shape of the irrational Aubry-Mather sets and their Lyapunov exponents

Cited by 18 publications

References 27 publications

Generic hyperbolicity of Aubry sets on surfaces

Generic hyperbolicity of Aubry sets on surfaces

Closing Aubry Sets I

Characterization of minimizable Lagrangian action functionals and a dual Mather theorem

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