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

Let X be a proper, geodesically complete CAT(0) space under a proper, non-elementary, isometric action by a group Γ with a rank one element. We construct a generalized Bowen-Margulis measure on the space of unit-speed parametrized geodesics of X modulo the Γ-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 ∂∞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 cZ 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.