Directions and equidistribution in homology for periodic orbits, Ergodic Theory Dyn

Abstract. We discuss the asymptotic distribution of the directions in homology of periodic orbits of Anosov flows. We obtaining a limiting measure which is either a Dirac measure on a single point or is fully supported. In the latter case, we relate the result to a more general equidistribution problem. 0. Introduction and Results Let M be a compact smooth Riemannian manifold and let φt:M →M be a transitive Anosov flow. Such a flow has a countable infinity of (prime) periodic orbits γ with length l(γ). Write π(T) = #{γ: l(γ) ≤ T}; then a classical result of Margulis [8],[9] states that π(T) ∼ eh(φ)T /h(φ)T, as T → +∞, where h(φ)> 0 is the topological entropy of φ