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

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.

v2: 26 pages, 1 figure. Theorems stated in much more generality (in particular, the cocompactness hypothesis was removed almost everywhere), also a number of proofs dropped. This is the July 2015 version that was accepted for publication in Ergodic Theory and Dynamical Systems. v1: 39 pages, 1 figure

References in corpus (1)

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

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• Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces
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• A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries
• On the geodesic flow on CAT(0) spaces