On the number of Mather measures of Lagrangian systems

TL;DR

This paper advances understanding of Mather measures in Lagrangian systems by applying convex analysis to infinite-dimensional problems, providing refined estimates on their generic number.

Contribution

It introduces a new approach using convex analysis and rectifiable sets to improve estimates on the number of Mather measures, solving a conjecture by John Mather.

Findings

Finer estimates on the generic number of Mather measures.

Application of convex analysis to infinite-dimensional optimization problems.

Introduction of rectifiable sets of finite codimension in Banach spaces.

Abstract

In 1996, Ricardo Ricardo Ma\~n\'e discovered that Mather measures are in fact the minimizers of a "universal" infinite dimensional linear programming problem. This fundamental result has many applications, one of the most important is to the estimates of the generic number of Mather measures. Ma\~n\'e obtained the first estimation of that sort by using finite dimensional approximations. Recently, we were able with Gonzalo Contreras to use this method of finite dimensional approximation in order to solve a conjecture of John Mather concerning the generic number of Mather measures for families of Lagrangian systems. In the present paper we obtain finer results in that direction by applying directly some classical tools of convex analysis to the infinite dimensional problem. We use a notion of countably rectifiable sets of finite codimension in Banach (and Frechet) spaces which may deserve independent interest.