un 2 00 5 Symplectic aspects of Aubry - Mather theory 1

On montre que les ensembles d'Aubry et de Mañé introduits par Mather en dynamique Lagrangienne sont des invariants symplectiques. On introduit pour ceci une barriere dans l'espace des phases. Ceci est aussi l'occasion d'´ ebaucher une théorie d'Aubry-Mather pour des Hamiltoniens non convexes. Abstract : We prove that the Aubry and Mañé sets introduced by Mather in Lagrangian dynamics are symplectic invariants. In order to do so, we introduce a barrier on phase space. This is also an occasion to suggest an Aubry Mather theory for non convex Hamiltonians. In Lagrangian dynamics, John Mather has defined several invariant sets, now called the Mather set, the Aubry set, and the Mañé set. These invariant sets provide obstructions to the existence of orbits wandering in phase space. Conversely, the existence of interesting orbits have been proved under some assumptions on the topology of these sets. Such results were first obtained by John Mather in [11], and then in several papers, see [1, 3, 4, 5, 16, 17] as well as recent unpublished works of John Mather. In order to apply these results on examples one has to understand the topology of the Aubry and Mañé set, which is a very difficult task. In many perturbative situations, averaging methods appear as a promising tool in that direction. In order to use these methods, one has to understand how the averaging transformations modify the Aubry-Mather sets. In the present paper, we answer this question and prove that the Mather set, the Aubry set and the Mañé set are symplectic invariants. In order to do so, we define a barrier on phase space, which is some symplectic analogue of the function called the Peierl's barrier by Mather in [11]. We then propose definitions of Aubry and Mañé sets for general Hamiltonian systems. We hope that these definitions may also serve as the starting point of an Aubry-Mather theory for some classes of non-convex Hamiltonians. We develop the first steps of such a theory. Several anterior works gave hints towards the symplectic nature of Aubry-Mather theory, see [2, 13, 14, 15] for example. These works prove the symplectic invariance of the α function of Mather, and one may consider that the symplectic invariance of the Aubry set is not a surprising result after them. However, the symplectic invariance of the Mañé set is, to my point of view, somewhat unexpected. It is possible that the geometric …