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, cocompact, and isometric action; further assume X admits a rank one axis. Using the Patterson-Sullivan measure on the boundary, we construct a generalized Bowen-Margulis measure on the space of geodesics in X. This measure has full support and is invariant under the geodesic flow.

Our construction of the Bowen-Margulis measure 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 the boundary of X (under the Patterson-Sullivan measure) is isolated in the Tits metric.

Finally, we identify precisely which geodesically complete, cocompact rank one CAT(0) spaces are mixing. That is, we prove that the Bowen-Margulis measure is mixing under the geodesic flow unless X is a tree with all edge lengths in some discrete subgroup of the reals.