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arXiv · 1808.04395

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

David Constantine·Jean-François Lafont·Daniel J. Thompson
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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.

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David Constantine, Jean-François Lafont, Daniel J. Thompson. 2018-08-13. Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows. https://doi.org/10.5802/jep.115

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