Mather theory and symplectic rigidity

Mads R. Bisgaard

Published 2018-04-27Version 1

Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow of a Hamiltonian H on a symplectic manifold. These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures and we construct an example showing that their support can be extremely unstable when H fails to be convex, even for nearly integrable H. Parts of these results extend work by Viterbo, and Vichery. Using ideas due to Entov-Polterovich we also detect interesting invariant measures by studying a generalization of the symplectic shape of sublevel sets of H. This approach differs from the first one in that it works also for "flexible" symplectic manifolds. We present applications to Hamiltonian systems on R^2n and twisted cotangent bundles.