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: $$ \begin{align*}\lim_{t\to \infty}\#P(t)/\frac{e^{ht}}{ht}=1\end{align*} $$ where h is the topological entropy of the geodesic flow. We also show that the unique measure of maximal entropy of the geodesic flow has the Bernoulli property.