The Aubry set for periodic Lagrangians on the circle

For periodic convex Lagrangians on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{S}^1 $$\end{document}, we show that, generically, in the sense of Mañé, there exists a dense open set of cohomology classes such that the Aubry set of these Lagrangians is a hyperbolic periodic orbit. This allows us to prove Mañé’s conjecture on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{S}^1 $$\end{document}.