Counting closed geodesics on rank one manifolds without focal points

TL;DR

This paper establishes asymptotic estimates for the number of free-homotopy classes containing closed geodesics on rank one manifolds without focal points, linking growth rate to topological entropy, and proves the Bernoulli property of the maximal entropy measure.

Contribution

It provides the first Margulis-type asymptotic formula for closed geodesics on rank one manifolds without focal points and proves the Bernoulli property of the measure of maximal entropy.

Findings

Asymptotic growth rate of closed geodesics matches exponential with rate h

Established the Bernoulli property for the measure of maximal entropy

Connected geometric dynamics with statistical properties

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.