Generically Mane set supports uniquely ergodic measure for residual cohomology class

TL;DR

This paper proves that for a generic class of Tonelli Lagrangians, there exists a large residual set of cohomology classes where the associated invariant measures are uniquely ergodic, unifying several Mather sets.

Contribution

It establishes the existence of a residual set of cohomology classes with uniquely ergodic measures for generic Tonelli Lagrangians, linking Mather, Aubry, and Mañé sets.

Findings

Residual set of cohomology classes with uniquely ergodic measures

Equality of Mather, Aubry, and Mañé sets for these classes

Supports on a uniquely ergodic measure

Abstract

In this paper, we proved that for generic Tonelli Lagrangian, there always exists a residual set $\mathcal{G}\subset H^1(M,\mathbb{R})$ such that \[ \widetilde{\mathcal{M}}(c)=\widetilde{\mathcal{A}}(c)=\widetilde{\mathcal{N}}(c),\quad \forall c\in\mathcal{G} \] with $\widetilde{\mathcal{M}}(c)$ supports on a uniquely ergodic measure.