On finite quotient Aubry set for generic geodesic flows

Abstract: We study the structure of the Mather and Aubry sets for the family of lagrangians given by the kinetic energy associated to a riemannian metric $g$ on a closed manifold $M$. In this case the Euler-Lagrange flow is the geodesic flow of $(M,g)$. We prove that there exists a residual subset $\mathcal G$ of the set of all conformal metrics to $g$, such that, if $\overline g \in \mathcal G$ then the corresponding geodesic flow has a finitely many ergodic c-minimizing measures, for each non-trivial cohomology class $c \in H<sup>1(M,\mathbb{R})$. This implies that, for any $c \in H<sup>1(M,\mathbb{R})$, the quotient Aubry set for the cohomology class c has a finite number of elements for this particular family of lagrangian systems.