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

作者：Ricks Russell^*

来源：Ergodic Theory and Dynamical Systems, 2017, 37(3): 939-970.

DOI：10.1017/etds.2015.78

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 non-positively 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 derivative(infinity) 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.

• 出版日期2017-5