Mather theory and symplectic rigidity

TL;DR

This paper introduces new invariant measures in symplectic topology that generalize Mather measures, analyzes their stability, and applies these concepts to Hamiltonian systems, extending previous work and providing new insights into symplectic rigidity.

Contribution

It develops a framework for invariant measures associated with Hamiltonian flows on symplectic manifolds, extending Mather theory beyond the Tonelli case and exploring their properties and applications.

Findings

Existence of invariant measures generalizing Mather measures.

Support of these measures can be highly unstable without convexity.

Applications to Hamiltonian systems on Euclidean and twisted cotangent bundles.

Abstract

Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $\phi_H$ of a Hamiltonian $H\in C^{\infty}(M)$ on a symplectic manifold $(M,\omega)$. 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 for $\phi_H$ 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 $(M,\omega)$ in which every compact subset can be displaced. We present applications to Hamiltonian systems on $\mathbb{R}^{2n}$ and twisted cotangent bundles.