A dynamical condition for differentiability of Mather's average action

TL;DR

This paper establishes conditions under which Mather's average action function is differentiable across all relevant homology classes, linking this property to the structure of invariant Lagrangian graphs and flow integrability.

Contribution

It proves the differentiability of Mather's beta function on classes associated with Lipschitz Lagrangian graphs and explores its connection to Hamiltonian flow integrability.

Findings

Differentiability of beta function on certain homology classes.

Relationship between local differentiability and flow integrability.

Conditions involving Lipschitz Lagrangian graphs for differentiability.

Abstract

We prove the differentiability of $\beta $ of Mather function on all homology classes corresponding to rotation vectors of measures whose supports are contained in a Lipschitz Lagrangian absorbing graph, invariant by Tonelli Hamiltonians. We also show the relationship between local differentiability of $\beta $ and local integrability of the Hamiltonian flow.