Symbolic dynamics for geodesic flows, hyperbolic measures and periodic orbits

In the first part we study two different methods for coding geodesics on the modular surface. There exist a geometric code, obtained by recording oriented excursions into the cusp of the modular surface, and an arithmetic code obtained by using the boundary expansions of the end points of the geodesic at infinity and a certain “reduction theory”. We identify a class of bi-infinite sequences that are realized as geometric codes, describe it as the maximal topological 1-step countable Markov chain, and show that the set of all geometric codes is not a finite-step Markov chain. We present three arithmetic methods for coding oriented geodesics on the modular surface using various continued fraction expansions and show that the space of admissible coding sequences for each coding is a one-step topological Markov chain with countable alphabet. We also present conditions under which these arithmetic codes coincide with the geometric code.;In the second part we present two qualitative results about the asymptotic growth and distribution of periodic orbits in nonuniformly hyperbolic dynamical systems. We prove that for a C2 diffeomorphism on a compact Riemannian manifold preserving a hyperbolic ergodic measure, there exists a hyperbolic periodic point such that the closure of its unstable (stable) manifold has positive measure. We also show that for a surface diffeomorphism preserving an ergodic hyperbolic measure which is not locally maximal, there exist multiplicatively many periodic orbits equidistributed with respect to this measure.