Symplectic aspects of Aubry-Mather theory

Abstract

We prove that the so-called Aubry and Mane sets introduced by John Mather in Lagrangian dynamics are symplectic invariants. In order to do so, we introduce a barrier in phase space, and propose definitions of Aubry and Mane sets for non-convex Hamiltonian systems. On montre que les ensembles dits d'Aubry et de Mane introduits par John Mather en dynamique Lagrangienne sont des invariants symplectiques. Pour ceci on introduit une barriere sur l'espace des phases, et on definit des ensembles d'Aubry et de Mather pour des systemes Hamiltoniens non convexes.

Cited by in corpus (6)

• Symplectic topology of Mañé's critical values
• Weak KAM for commuting Hamiltonians
• Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms
• Locally maximizing orbits for the non-standard generating function of convex billiards and applications
• Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry-Mather theory
• Asymptotic trajectories of KAM torus