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 [BCLS15], which verifies that the full power of symbolic dynamics is available in that setting.
Highlights
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
A priori, orbit semiequivalence is too weak a relationship to preserve any interesting dynamical properties [GM10, KT19], and it is not known how to improve this construction of symbolic dynamics to a semi-conjugacy
In [CLT19], we used this weak symbolic description to prove that these geodesic flows are expansive flows with the weak specification property, and explored the consequences of this characterization
Summary
The boundary at infinity of a CAT(−1) space is the set of equivalence classes of geodesic rays, where two rays c, d : [0, ∞) → X are equivalent if they remain a bounded distance apart, i.e., if dX (c(t), d(t)) is bounded in t When X is convex cocompact, we need to restrict the geodesics we study so that the phase space for our flow is compact. It is easy to check that e−2|s−t|/e−2|s| e2t, which completes the proof It follows that the flow (gt) is Lipschitz, using Lemma 2.5 and the fact that dGX (gsx, gtx) = |s − t| for all x, and all s, t with |s − t| sufficiently small. We have that dGX (gtc, gtc ) (1 + K)e−tdGX (c, c )
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