Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows

Abstract

We prove that the geodesic flow on a locally CAT(-1) metric space which is compact, or more generally convex cocompact with non-elementary fundamental group, can be coded by a suspension flow over an irreducible shift of finite type with Hölder roof function. This is achieved by showing that the geodesic flow is a metric Anosov flow, and obtaining Hölder regularity of return times for a special class of geometrically constructed local cross-sections to the flow. We obtain a number of strong results on the dynamics of the flow with respect to equilibrium measures for Hölder potentials. In particular, we prove that the Bowen-Margulis measure is Bernoulli except for the exceptional case that all closed orbit periods are integer multiples of a common constant. We show that our techniques also extend to the geodesic flow associated to a projective Anosov representation, which verifies that the full power of symbolic dynamics is available in that setting.

v4: 28 pages, 4 figures. Final version, to appear in Journal de l'Ecole polytechnique

References in corpus (5)

• Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings
• Flows and joins of metric spaces
• Flat strips, Bowen-Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces
• (Non)Invariance of dynamical quantities for orbit equivalent flows
• The weak specification property for geodesic flows on CAT(-1) spaces

Cited by in corpus (2)

• Topological flows for hyperbolic groups
• The Martin boundary of an extension by a hyperbolic group