Spectral invariants towards a Non-convex Aubry-Mather theory

TL;DR

This paper extends Mather's theory to non-convex Hamiltonians using spectral invariants, showing the preservation of key relations and applying results to invariant measures in KAM theory.

Contribution

It introduces a non-convex Mather functional via spectral invariants, preserving fundamental relations and enabling new applications in KAM theory.

Findings

Relation between invariant measures and Mather's subdifferential is preserved in non-convex case

Extension of Mather's functional to non-convex Hamiltonians using spectral invariants

Application to existence of invariant measures with large rotation vectors after KAM tori disappear

Abstract

Aubry-Mather is traditionally concerned with Tonelli Hamiltonian (convex and super-linear). In \cite{Vi,MVZ}, Mather's $\alpha$ function is recovered from the homogenization of symplectic capacities. This allows the authors to extend the Mather functional to non convex cases. This article shows that the relation between invariant measures and the subdifferential of Mather's functional (which is the foundational statement of Mather) is preserved in the non convex case. We give applications in the context of the classical KAM theory to the existence of invariant measures with large rotation vector after the possible disappearance of some KAM tori.