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The two main results of this paper concern the regularity of the invariant foliation of a C0-integrable symplectic twist diffeomorphisms of the 2-dimensional annulus, namely that $\bullet$ the generating function of such a foliation is C1 ; $\bullet$ the foliation is H{\"o}lder with exponent 1/2. We also characterize foliations by graphs that are straightenable via a symplectic homeomorphism and prove that every symplectic homeomorphism that leaves invariant all the leaves of a straightenable foliation has Arnol'd-Liouville coordinates, in which the Dynamics restricted to the leaves is conjugated to a rotation. We deduce that every Lipschitz integrable symplectic twist diffeomorphisms of the 2-dimensional annulus has Arnol'd-Liouville coordinates and then provide examples of 'strange' Lipschitz foliations in smooth curves that cannot be straightened by a symplectic homeomorphism and cannot be invariant by a symplectic twist diffeomorphism.This article is a part of another preprint of the authors, entitled On the transversal dependence of weak K.A.M. solutions for symplectic twist maps, after rewriting ant adding of the H{\"o}lder part., Comment: arXiv admin note: substantial text overlap with arXiv:1809.02372

In this article we study the fragility of Lagrangian periodic tori for symplectic twist maps of the $2d$-dimensional annulus and prove a rigidity result for completely integrable ones. More specifically, we consider $1$-parameter families of symplectic twist maps $(f_\varepsilon)_{\varepsilon\in \mathbb{R}}$, obtained by perturbing the generating function of an analytic map $f$ by a family of potentials $\{\varepsilon G\}_{\varepsilon\in \mathbb{R}}$. Firstly, for an analytic $G$ and for $(m,n)\in \mathbb{Z}\times \mathbb{N}^*$ with $m$ and $n$ coprime, we investigate the topological structure of the set of $\varepsilon\in \mathbb{R}$ for which $f_\varepsilon$ admits a Lagrangian periodic torus of rotation vector $(m,n)$. In particular we prove that, under a suitable non-degeneracy condition on $f$, this set consists of at most finitely many points. Then, we exploit this to deduce a rigidity result for integrable symplectic twist maps, in the case of deformations produced by a $C^2$ potential. Our analysis, which holds in any dimension, is based on a thorough investigation of the geometric and dynamical properties of Lagrangian periodic tori, which we believe is of its own interest., 37 pages; the title has changed and the structure of the paper has been modified, following the referees' suggestions. This final version has been accepted on "Advances in Mathematics"

We study invariant manifolds of conformal symplectic dynamical systems on a symplectic manifold (M, $\omega$) of dimension $\ge$4. This class of systems is the 1-dimensional extension of symplectic dynamical systems for which the symplectic form is transformed colinearly to itself. In this context, we first examine how the $\omega$-isotropy of an invariant manifold N relates to the entropy of the dynamics it carries. Central to our study is Yomdin's inequality, and a refinement obtained using that the local entropies have no effect transversally to the characteristic foliation of N. When (M, $\omega$) is exact and N is isotropic, we also show that N must be exact for some choice of the primitive of $\omega$, under the condition that the dynamics acts trivially on the cohomology of degree 1 of N. The conclusion partially extends to the case when N has a relatively compact one-sided orbit. We eventually prove the uniqueness of invariant submanifolds N when M is a cotangent bundle, provided that the dynamics is isotopic to the identity among Hamiltonian diffeomorphisms. In the case of the cotangent bundle of the torus, a theorem of Shelukhin allows us to conclude that N is unique even among submanifolds with compact orbits.

We study invariant manifolds of conformal symplectic dynamical systems on a symplectic manifold (M, ω) of dimension ≥4. This class of systems is the 1-dimensional extension of symplectic dynamical systems for which the symplectic form is transformed colinearly to itself. In this context, we first examine how the ω-isotropy of an invariant manifold N relates to the entropy of the dynamics it carries. Central to our study is Yomdin's inequality, and a refinement obtained using that the local entropies have no effect transversally to the characteristic foliation of N. When (M, ω) is exact and N is isotropic, we also show that N must be exact for some choice of the primitive of ω, under the condition that the dynamics acts trivially on the cohomology of degree 1 of N. The conclusion partially extends to the case when N has a relatively compact one-sided orbit. We eventually prove the uniqueness of invariant submanifolds N when M is a cotangent bundle, provided that the dynamics is isotopic to the identity among Hamiltonian diffeomorphisms. In the case of the cotangent bundle of the torus, a theorem of Shelukhin allows us to conclude that N is unique even among submanifolds with compact orbits.; On étudie les sous-variétés invariantes des dynamiques conformément symplectiques d'une variétés symplectique (M, ω) de dimension ≥4. Cette classe de systèmes est une extension 1-dimensionnelle des systèmes dynamiques symplectiques pour laquelle la forme symplectique est multipliée par une constante. Dans ce contexte, on s'intéresse tout d'abord à la relation entre la ω-isotropie d'une variété invariante N et l'entropie de la dynamique dessus. L'inégalité de Yomdin est un argument central de notre étude et un de ses raffinements obtenu en utilisant que les entropies locales n'ont pas d'effet transversalement au feuilletage caractéristique de N. Quand (M, ω) est exact et N est isotrope, on montre aussi que N est exacte pour une certaine primitive de ω, sous la condition que la dynamiques agit trivialement sur la cohomologie de degré 1 de N. Cette conclusion s'étend partiellement au cas où N a une orbite relativement compacte. On prouve finalement l'unicité de la variété invariante N quand M est un fibré cotangent, pourvu que la dynamique soit hamiltoniennement isotope à l'identité. Dans le cas du fibré cotangent d'un tore, un théorème de Shelukhin nous permet de conclure que N est unique même parmi les sous-variétés d'orbites relativement compactes.

The two main results of this paper concern the regularity of the invariant foliation of a C0-integrable symplectic twist diffeomorphisms of the 2-dimensional annulus, namely that • the generating function of such a foliation is C1 ; • the foliation is Hölder with exponent 1/2. We also characterize foliations by graphs that are straightenable via a symplectic homeomorphism and prove that every symplectic homeomorphism that leaves invariant all the leaves of a straightenable foliation has Arnol'd-Liouville coordinates, in which the Dynamics restricted to the leaves is conjugated to a rotation. We deduce that every Lipschitz integrable symplectic twist diffeomorphisms of the 2-dimensional annulus has Arnol'd-Liouville coordinates and then provide examples of 'strange' Lipschitz foliations in smooth curves that cannot be straightened by a symplectic homeomorphism and cannot be invariant by a symplectic twist diffeomorphism.This article is a part of another preprint of the authors, entitled On the transversal dependence of weak K.A.M. solutions for symplectic twist maps, after rewriting ant adding of the Hölder part.; Les deux résultats principaux ce cet article concernent la régularité des feuilletages invariants par une application twist symplectique C0 intégrable, plus particulièrement on montre que:•la fonction génératrice d'un tel feuilletage est de classe C1 ; • le feuilletage est Holder d'exposant 1/2. On caractérise aussi les feuilletages en graphes qui sont redressables par un homéomorphisme symplectique et prouvons qu'un homéomorphisme symplectique qui laisse invariant les feuilles d'un feuilletage redressable admet des coordonnées d'Arnol'd-Liouville, dans lesquelles la dynamique restreinte aux feuilles est conjuguée à une rotation. On en déduit qu'un twist symplectique qui est Lipschitz intégrable admet des coordonnées d'Arnol'd-Liouville et ensuite donnons un exemple d'un feuilletage lipschitz "sauvage" en courbes lisses qui ne peut pas être redressé par un homéomorphisme symplectique et ne peut pas être laissé invariant par un twist symplectique.Cet article est la moitié d'un précédent preprint des auteurs, intitulé On the transversal dependence of weak K.A.M. solutions for symplectic twist maps, qu'on a réorganisé et auquel on a rajouté la partie Hölder.

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 present the proof of Berger and Turaev of Herman's positive entropy conjecture. In every neighbourhood of identity in the set of smooth symplectic diffeomorphisms of the 2-dimensional disc, there exists a diffeomorphism with positive metric entropy., in French. S\'eminaire Bourbaki, Nov 2019, Paris, France

International audience; We present the proof of Berger and Turaev of Herman's positive entropy conjecture. In every neighbourhood of identity in the set of smooth symplectic diffeomorphisms of the 2-dimensional disc, there exists a diffeomorphism with positive metric entropy.

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 $u(\theta, c)$ in two variables: the angle $\theta$ 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 example, we show that our choice is not the so-called discounted one (see \cite{DFIZ2}), that is sometimes discontinuous. We also provide examples of `strange' continuous foliations that cannot be straightened by a symplectic homeomorphism., Comment: 43 pages, 3 figures

In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to determine the Dynamics on each Lagrangian torus and that the C 1 regularity of the foliation by invariant Lagrangian tori is crucial to prove the continuity of Arnol'd-Liouville coordinates. We also explore various notions of C 0 and Lipschitz integrability.

Let M be a closed and connected manifold, H : T * M × R/Z → R a Tonelli 1-periodic Hamiltonian and L ⊂ T * M a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if L is invariant by the time-one map of H, then L is a graph over M. An interesting consequence in the autonomous case is that in this case, L is invariant by all the time t maps of the Hamiltonian flow of H.

The globally positive diffeomorphisms of the 2n-dimensional annulus are important because they represent what happens close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite. For these globally positive diffeomorphisms, an Aubry-Mather theory was developed by Garibaldi \& Thieullen that provides the existence of some minimizing measures. Using the two Green bundles G- and G+ that can be defined along the support of these minimizing measures, we will prove that there is a deep link between: -the angle between G- and G+ along the support of the considered measure m; -the size of the smallest positive Lyapunov exponent of m; -the tangent cone to the support of m., 35 pages

We prove that all the Tonelli Hamiltonians defined on the cotangent bundle $T^*\T^n$ of the $n$-dimensional torus that have no conjugate points are $C^0$ integrable, i.e. $T^*\T^n$ is $C^0$ foliated by a family $\Fc$ of invariant $C^0$ Lagrangian graphs. Assuming that the Hamiltonian is $C^\infty$, we prove that there exists a $G_\delta$ subset $\Gc$ of $\Fc$ such that the dynamics restricted to every element of $\Gc$ is strictly ergodic. Moreover, we prove that the Lyapunov exponents of every $C^0$ integrable Tonelli Hamiltonian are zero and deduce that the metric and topological entropies vanish., Comment: 37 pages

The manifold M being compact and connected and H being a Tonelli Hamiltonian such that the cotangent bundle of M is equal to the dual tiered Mane set, we prove that there is a partition of the cotangent bundle of M into invariant C0 Lagrangian graphs. Moreover, among these graphs, those that are C1 cover a residual subset of this cotangent bundle The dynamic restricted to each of these sets is non wandering.

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.