Proof of the $C^2$ Mañé's conjecture on surfaces

Abstract

We prove that $C^2$ generic hyperbolic Mañé sets contain a periodic periodic orbit. In dimension 2, adding a result by Contreras, Figalli, Rifford, which states that $C^2$ generic Mañé sets are hyperbolic; we obtain Mañé'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 Mañé set of the Lagrangian $L+f$ is a hyperbolic periodic orbit.