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We study Lagrangian systems on a closed manifoldM. We link the differentiability of Mather’sβ-function with the topological complexity of the complement of the Aubry set. As a consequence, whenM is a closed, orientable surface, the differentiability of theβ-function at a given homology class is forced by the irrationality of the homology class. This allows us to prove the two-dimensional case of a conjecture by Mañé.

On Aubry sets and Mather’s action functional