Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces
| Autor: | Russell Ricks |
|---|---|
| Rok vydání: | 2015 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics Mathematics::Dynamical Systems Geodesic General Mathematics Modulo Dynamical Systems (math.DS) STRIPS 01 natural sciences law.invention Mathematics::Group Theory Mathematics - Geometric Topology law 0103 physical sciences FOS: Mathematics Geodesic flow Mathematics::Metric Geometry Ergodic theory Mathematics - Dynamical Systems 0101 mathematics Mathematics Applied Mathematics 010102 general mathematics Geometric Topology (math.GT) 16. Peace & justice Differential Geometry (math.DG) Mathematics::Differential Geometry 010307 mathematical physics Limit set |
| Zdroj: | Ergodic Theory and Dynamical Systems. 37:939-970 |
| ISSN: | 1469-4417 0143-3857 |
| DOI: | 10.1017/etds.2015.78 |
| Popis: | 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. Comment: 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 |
| Databáze: | OpenAIRE |
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