Aubry sets vs Mather sets in two degrees of freedom

TL;DR

This paper investigates the relationship between Aubry and Mather sets for autonomous Tonelli Lagrangians on closed surfaces, showing they almost always coincide when the Mather set contains only periodic orbits.

Contribution

It provides a clarification of the conditions under which Aubry and Mather sets are essentially the same in the context of two-dimensional systems.

Findings

Aubry and Mather sets almost always coincide when Mather set contains only periodic orbits.

The study focuses on autonomous Tonelli Lagrangians on closed surfaces.

The relationship between these sets is clarified in the specific case of non-fixed periodic orbits.

Abstract

We study autonomous Tonelli Lagrangians on closed surfaces. We aim to clarify the relationship between the Aubry set and the Mather set, when the latter consists of periodic orbits which are not fixed points. Our main result says that in that case the Aubry set and the Mather set almost always coincide.