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The key result of this article is key lemma: if a Jordan curve $\gamma$ is invariant by a given C 1+$\alpha$ -diffeomorphism f of a surface and if $\gamma$ carries an ergodic hyperbolic probability $\mu$, then $\mu$ is supported on a periodic orbit. From this Lemma we deduce three new results for the C 1+$\alpha$ symplectic twist maps f of the annulus: 1. if $\gamma$ is a loop at the boundary of an instability zone such that f |$\gamma$ has an irrational rotation number, then the convergence of any orbit to $\gamma$ is slower than exponential; 2. if $\mu$ is an invariant probability that is supported in an invariant curve $\gamma$ with an irrational rotation number, then $\gamma$ is C 1 $\mu$-almost everywhere; 3. we prove a part of the so-called "Greene criterion", introduced by J. M. Greene in [16] in 1978 and never proved: assume that (pn qn) is a sequence of rational numbers converging to an irrational number $\omega$; let (f k (x n)) 1$\le$k$\le$qn be a minimizing periodic orbit with rotation number pn qn and let us denote by R n its mean residue R n = |1/2 -- Tr(Df qn (x n))/4

International audience; 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. Some topics of the part of R. Potrie will be useful for this part. Many thanks to E. Maderna and L. Rifford for the invitation to give the mini-course and to R. Potrie for accepting to share the course with me.

International audience; We prove the following rigidity result for the Tonelli Hamiltonians. Let T * M be the cotangent bundle of a closed manifold M endowed with its usual symplectic form. Let (F_n) be a sequence of Tonelli Hamiltonians that C^0 converges on the compact subsets to a Tonelli Hamiltonian F. Let (G_n) be a sequence of Hamiltonians that that C^0 converges on the compact subsets to a Hamiltonian G. We assume that the sequence of the Poisson brackets ({F_n , G_n }) C^0-converges on the compact subsets to a C^1 function H. Then H = {F, G}.