Differentiability of Mather's average action and integrability on closed surfaces

TL;DR

This paper investigates the conditions under which Mather's $eta$-function is differentiable on closed surfaces and explores how this relates to the integrability of the dynamical system.

Contribution

It provides new insights into the link between the differentiability of Mather's $eta$-function and the integrability of Hamiltonian systems on closed surfaces.

Findings

Differentiability of Mather's $eta$-function characterizes integrability.

Results connect geometric properties of surfaces with dynamical behavior.

New criteria for integrability based on $eta$-function differentiability.

Abstract

In this article we study the differentiability of Mather's $\beta$-function on closed surfaces and its relation to the integrability of the system.