Equidistribution results for geodesic flows

Using the works of Mane [On the topological entropy of the geodesic flows. J. Differential Geom. 45 (1989), 74-93] and Paternain [Topological pressure for geodesic flows. Ann. Sci. Ec. Norm. Super. (4) 33 (2000), 121-138] we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a C-infinity Riemannian metric. We prove large-deviation lower and upper bounds and a contraction principle for the geodesic flow in the space of probability measures of the unit tangent bundle. We deduce a way of approximating equilibrium states for continuous potentials.