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

Russell Ricks

doi:10.3934/dcds.2020266

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2021, Volume 41, Issue 2: 507-523. Doi: 10.3934/dcds.2020266

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

Russell Ricks

Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902, USA

Received: October 2019

Revised: April 2020

Early access: July 2020

Published: February 2021

The author would like to thank three anonymous referees, who all made helpful suggestions to improve the paper. The author was partially supported by NSF RTG 1045119

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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 Poincaré series.

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Mathematics Subject Classification: Primary:37D40, 37B40;Secondary:28D20.

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Figure 1. Shadows of $ p $ on $ \partial X $, from basepoints $ x \in X $ (left) and $ \xi \in \partial X $ (right)

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