Counting closed geodesics on rank one manifolds without focal points

Abstract

In this article, we consider a closed rank one Riemannian manifold $M$ without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on $M$ with length at most $t$, and $\# P(t)$ its cardinality. We obtain the following Margulis-type asymptotic estimates: \[\lim_{t\to \infty}\#P(t)/\frac{e^{ht}}{ht}=1\] where $h$ is the topological entropy of the geodesic flow. In the appendix, we also show that the unique measure of maximal entropy of the geodesic flow has the Bernoulli property.