Abstract
Let $L$ be an autonomous Tonelli Lagrangian on a closed manifold of dimension two. Let $\mathcal{C}$ be the set of cohomology classes whose Mather set consists of periodic orbits, none of which is a fixed point. Then for almost all $c$ in $\mathcal{C}$, the Aubry set of $c$ equals the Mather set of $c$.