Semi-discrete approximation of Mañé, Aubry and Mather sets

Abstract

We study a variational discretization in time of Aubry--Mather theory for Tonelli Lagrangians on the flat torus. We define discrete Mañé, Aubry and Mather sets, establish their compactness and inclusion properties, and investigate their convergence as the time step tends to zero. We prove upper-limit results for the Mañé sets and, when the continuous Aubry set has finitely many static classes, for the Aubry sets. We obtain full Kuratowski and Hausdorff convergence of the Aubry sets when this set is a single equilibrium or periodic orbit, and, under additional assumptions, a finite union of equilibria. Every weak limit of discrete Mather measures is a continuous Mather measure. For equilibrium Aubry sets, we give sufficient conditions for exact preservation of Mather sets and measure families at small time steps. A mechanical example illustrates these results and exhibits exact preservation of the Aubry set as well.