In this article, we prove different results concerning the regularity of the C 0 -Lagrangian invariant graphs of the Tonelli flows. For example :• in dimension 2 and in the autonomous generic case, we prove that such a graph is in fact C 1 on some set with (Lebesgue) full measure; • under certain dynamical additional hypothesis, we prove that these graphs are C 1 . Résumé.Dans cet article, on démontre différents résultats concernant la régularité des graphes C 0 -lagrangiens invariants par des flots de Tonelli. Par exemple :• en dimension 2, dans le cas autonome et générique, on montre que ces graphes sont de classe C 1 sur un ensemble de mesure (de Lebesque) pleine ;• sous certaines hypothèses concernant la dynamique restreinte, on montre que ces graphes sont de classe C 1 .

International audienceNous montrons que C1-génériquement les difféomorphismes symplectiques d'une variété compacte sont transitifs. Sur les variétés symplectiques non-compactes, nous montrons que les orbites génériques d'un difféomorphisme générique ne sont pas bornées. L'outil principal est un lemme de connexion pour les pseudo-orbites des difféomorphismes symplectiques. Pour cela nous donnons des conditions générales sur un espace de difféomorphismes, en particulier au voisinage des orbites périodiques, qui permettent d'obtenir le lemme de connexion pour les pseudo-orbites. Abstract: We prove that C1-generic symplectic diffeomorphisms of a compact manifold are transitive. On non-compact manifolds, we show that generic orbits of generic diffeomorphisms are not bounded. The main tool is a connecting lemma for pseudo-orbits of symplectic diffeomorphisms. In the proof, we give general conditions on diffeomorphisms spaces, in particular at the neighborhood of the periodic orbits, which imply the connecting lemma for pseudo-orbits

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The manifold M being closed and connected, we prove that every submanifold of T * M that is Hamiltonianly isotopic to the zero-section and that is invariant by a Tonelli flow is a graph.

We prove that if (M,\omega) is a connected and compact four-dimensional symplectic manifold, there exist three open sets U_1, U_2, U_3 of {\rm Diff}^1_{\omega}(M) (for the C^1 topology) such that: U_1\cup U_2\cup U_3 is dense in {\rm Diff}^1_{\omega}(M);f\in U_1 if and only if f is Anosov and transitive;f\in U_2 if and only if f is partially hyperbolic; andf\in U_3 if and only if f has a stable completely elliptic periodic point.

We consider locally minimizing measures for the conservative twist maps of the ddimensional annulus or for the Tonelli Hamiltonian flows defined on a cotangent bundle T * M . For weakly hyperbolic such measures (i.e. measures with no zero Lyapunov exponents), we prove that the mean distance/angle between the stable and the unstable Oseledet's bundles gives an upper bound of the sum of the positive Lyapunov exponents and a lower bound of the smallest positive Lyapunov exponent. Some more precise results are proved too. * ANR Project BLANC07-3 187245, Hamilton-Jacobi and Weak KAM Theory † ANR DynNonHyp

Let L : T M → R be a Tonelli Lagrangian function (with M compact and connected and dim M ≥ 2). The tiered Aubry set (resp. Mañé set) A T (L) (resp. N T (L)) is the union of the Aubry sets (resp. Mañé sets) A(L + λ) (resp. N (L + λ)) for λ closed 1-form. Then :