# Counting closed geodesics in a compact rank one locally CAT(0) space

## TL;DR

This paper establishes the asymptotic growth rate of the number of parallel classes of closed geodesics in a compact rank one locally CAT(0) space, linking it to the entropy of the geodesic flow.

## Contribution

It proves a precise asymptotic formula for counting closed geodesics in a broad class of non-positively curved spaces with rank one axes.

## Key findings

- The number of parallel classes of closed geodesics grows exponentially with rate determined by entropy.
- The asymptotic count matches the exponential growth rate e^{ht}/(ht).
- The result extends classical geodesic counting to non-manifold CAT(0) spaces.

## Abstract

Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. Assume $X$ is not homothetic to a metric graph with integer edge lengths. Let $P_t$ be the number of parallel classes of oriented closed geodesics of length $\le t$; then $\lim\limits_{t \to \infty} P_t / \frac{e^{ht}}{ht} = 1$, where $h$ is the entropy of the geodesic flow on the space $SX$ of parametrized unit-speed geodesics in $X$.