Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces

TL;DR

This paper constructs a Bowen-Margulis measure for rank one CAT(0) spaces, analyzes its properties, and characterizes the conditions under which the geodesic flow is mixing, extending known results to a broader class of spaces.

Contribution

It introduces a new construction of Bowen-Margulis measures for CAT(0) spaces and characterizes mixing conditions, including a novel structural result about geodesics and flat strips.

Findings

Almost no geodesic bounds a flat strip of positive width under the measure.

Almost every boundary point is isolated in the Tits metric under Patterson-Sullivan measure.

Finite Bowen-Margulis measure is not mixing only for certain tree structures with edge lengths in multiples of c.

Abstract

Let $X$ be a proper, geodesically complete CAT(0) space under a proper, non-elementary, isometric action by a group $\Gamma$ with a rank one element. We construct a generalized Bowen-Margulis measure on the space of unit-speed parametrized geodesics of $X$ modulo the $\Gamma$-action. Although the construction of Bowen-Margulis measures for rank one nonpositively curved manifolds and for CAT(-1) spaces is well-known, the construction for CAT(0) spaces hinges on establishing a new structural result of independent interest: Almost no geodesic, under the Bowen-Margulis measure, bounds a flat strip of any positive width. We also show that almost every point in $\partial_\infty X$, under the Patterson-Sullivan measure, is isolated in the Tits metric. (For these results we assume the Bowen-Margulis measure is finite, as it is in the cocompact case). Finally, we precisely characterize mixing when $X$ has full limit set: A finite Bowen-Margulis measure is not mixing under the geodesic flow precisely when $X$ is a tree with all edge lengths in $c \mathbb Z$ for some $c > 0$. This characterization is new, even in the setting of CAT(-1) spaces. More general (technical) versions of these results are also stated in the paper.