On the Distribution of Periodic Orbits

Let f : M → M be a C-map on a smooth Riemannian manifold M and let Λ ⊂ M be a compact f -invariant locally maximal set. In this paper we obtain several results concerning the distribution of the periodic orbits of f |Λ. These results are non-invertible and, in particular, non-uniformly hyperbolic versions of well-known results by Bowen, Ruelle, and others in the case of hyperbolic diffeomorphisms. We show that the topological pressure Ptop(φ) can be computed by the values of the potential φ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is wellapproximated by expanding periodic orbits. Moreover, we prove that certain equilibrium states are Bowen measures. Finally, we derive a large deviation result for the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.