Closed geodesics on compact symmetric spaces of higher rank

TL;DR

This paper derives asymptotic estimates for the number of free-homotopy classes containing closed geodesics of bounded length on higher rank compact symmetric spaces, linking growth rate to topological entropy.

Contribution

It provides the first asymptotic formula for counting closed geodesics on higher rank symmetric spaces, extending classical results to this broader setting.

Findings

^{ht}/(ht) growth rate for closed geodesics

Asymptotic estimate with exponential accuracy

Connection between geodesic count and topological entropy

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.