Proof of the $C^2$ Ma\~n\'e's conjecture on surfaces

TL;DR

This paper proves that for generic $C^2$ perturbations on surfaces, Mañé sets are hyperbolic periodic orbits, confirming Mañé's conjecture in the $C^2$ topology for two-dimensional systems.

Contribution

It establishes that $C^2$ generic Mañé sets on surfaces are hyperbolic periodic orbits, confirming Mañé's conjecture in the $C^2$ topology for surfaces.

Findings

$C^2$ generic Mañé sets contain a periodic orbit

On surfaces, $C^2$ generic Mañé sets are hyperbolic

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

Abstract

We prove that $C^2$ generic hyperbolic Ma\~n\'e sets contain a periodic periodic orbit. In dimension 2, adding a result by Contreras, Figalli, Rifford, which states that $C^2$ generic Ma\~n\'e sets are hyperbolic; we obtain Ma\~n\'e's Conjecture for surfaces in the $C^2$ topology: Given a Tonelli Lagrangian $L$ on a compact surface $M$ there is a $C^2$ open and dense set of functions $f:M\to\mathbb{R}$ such that the Ma\~n\'e set of the Lagrangian $L+f$ is a hyperbolic periodic orbit.