Semi-Discrete Approximation of Aubry and Mather sets

TL;DR

This paper develops a semi-discrete approximation framework for Aubry and Mather sets in Tonelli Lagrangian systems on the torus, analyzing convergence and geometric preservation.

Contribution

It introduces discrete analogues of Aubry and Mather sets, proving their convergence under certain conditions, advancing structure-preserving approximation theory.

Findings

Semi-discrete variational framework captures ergodic constants and invariant geometry.

Proves upper Kuratowski limit inclusions for Aubry and Mather sets.

Establishes full convergence under hyperbolicity and genericity assumptions.

Abstract

We study the semi-discrete approximation of Aubry and Mather sets for Tonelli Lagrangians on the flat torus. Starting from the discrete Lax--Oleinik equation, we introduce natural discrete analogues of these sets and analyze their convergence, as the time step tends to zero, in the sense of Kuratowski. Our results show that the semi-discrete variational framework captures not only the ergodic constant, but also the minimizing invariant geometry of the continuous dynamics. In full generality, we prove upper Kuratowski limit inclusions for both the Aubry and Mather sets. For the Aubry set, we establish full convergence under a hyperbolicity assumption on the continuous Aubry set. For the Mather set, we prove full convergence under a genericity assumption ensuring that the Lagrangian admits finitely many ergodic Mather measures. This provides a first rigorous step toward a structure-preserving approximation theory for Aubry and Mather sets in the Tonelli setting, and clarifies how discrete variational models recover the central geometric objects of weak KAM and Aubry--Mather theory.