2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99775We study Lagrangian systems on a closed manifold. We link the differentiability of Mather's beta-function with the topological complexity of the complement of the Aubry set. As a consequence, when the dimension of the manifold is less than or equal to two, the differentiability of the beta-function at a given homology class is forced by the irrationality of the homology class. As an application we prove the two-dimensional case of a conjecture by Ricardo Mane.17 pages, 2nd versionDynamical Systems37J40, 37J45, 37J50text