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.

References in corpus (3)

• Equilibrium states for self-products of flows and the mixing properties of rank 1 geodesic flows
• Properties of equilibrium states for geodesic flows over manifolds without focal points
• Counting closed geodesics on rank one manifolds