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

TL;DR

This paper establishes a symbolic coding for geodesic flows on CAT(-1) spaces and other Anosov flows, demonstrating Bernoulli properties and extending techniques to projective Anosov representations.

Contribution

It introduces a new symbolic coding method for geodesic flows on CAT(-1) spaces and extends the approach to projective Anosov representations, showing Bernoulli measures and H"older regularity.

Findings

Geodesic flow is a metric Anosov flow with H"older regularity.

Bowen-Margulis measure is Bernoulli except for a specific periodicity case.

Techniques extend to flows from projective Anosov representations.

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 H\"older roof function. This is achieved by showing that the geodesic flow is a metric Anosov flow, and obtaining H\"older 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 H\"older 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.