  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^1(M,\mathbb{R})$. This implies that, for any $ c \in H^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.

Dynamical Systems