Flat Strips, Bowen-Margulis Measures, and Mixing of the Geodesic Flow for Rank One CAT(0) Spaces.

CAT(0);geometry;dynamics;ergodic theory;Mathematics;Science;Mathematics

Ricks, Russell M.Ji, Lizhen ;

University of Michigan

关键词: CAT(0);    geometry;    dynamics;    ergodic theory;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/113535/rmricks_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.