Generic Ma\~n\'e sets

TL;DR

This paper proves that generic hyperbolic Mañé sets contain periodic orbits and confirms Mañé's Conjecture for surfaces in the $C^2$ topology, advancing understanding of dynamical systems.

Contribution

It establishes the presence of periodic orbits in generic hyperbolic Mañé sets and verifies Mañé's Conjecture for surfaces within the $C^2$ topology.

Findings

Hyperbolic Mañé sets contain periodic orbits.

Ma e's Conjecture holds for surfaces in the $C^2$ topology.

Results apply to generic dynamical systems in the specified class.

Abstract

We prove that $C^2$ generic hyperbolic Ma\~n\'e sets contain a periodic orbit. In dimesion 2, adding a result with A. Figalli and L. Rifford, we obtain Ma\~n\'e's Conjecture for surfaces in the $C^2$ topology.