# The unique measure of maximal entropy for a compact rank one locally CAT(0) space

## TL;DR

This paper proves the uniqueness of the measure of maximal entropy for geodesic flow on certain compact rank one locally CAT(0) spaces, linking it to the critical exponent of the Poincaré series.

## Contribution

It establishes the uniqueness of the Bowen-Margulis measure as the measure of maximal entropy in this geometric setting.

## Key findings

- Bowen-Margulis measure is unique for the geodesic flow.
- Topological entropy equals the critical exponent of the Poincaré series.
- Results extend understanding of entropy in non-positive curvature spaces.

## Abstract

Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. We prove the Bowen-Margulis measure on the space of geodesics is the unique measure of maximal entropy for the geodesic flow, which has topological entropy equal to the critical exponent of the Poincar\'e series.