Closed geodesics on compact symmetric spaces of higher rank

Abstract

In this article, we consider a compact symmetric space $M$ of higher rank. 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 asymptotic estimates: \[\#P(t)=\frac{e^{ht}}{ht}(1+O(e^{-ut}))\] for some $u>0$, where $h$ is the topological entropy of the geodesic flow.