Checking ergodicity of some geodesic flows with infinite Gibbs measure

Abstract

Abstract This paper concerns a problem which arose from a paper of Sullivan. Let Γ be a discrete group of isometries of hyperbolic space H d +1 . We study the question of when the geodesic flow on the unit tangent bundle UT ( H d +1 /Γ) of H d +1 /Γ is ergodic with respect to certain natural measures. As a consequence, we study the question of when Γ is of divergence type. Ergodicity when the non-wandering set of UT ( H d +1 /Γ) is compact is already known from the theory of symbolic dynamics, due to Bowen, or from Sullivan's work. For such a Γ, we consider a subgroup Γ 1 of Γ with Γ/Γ 1 ≅ℤ υ and prove the geodesic flow on UT ( H d +1 /Γ 1 ) is ergodic (with respect to one of these natural measures) if and only if υ ≤ 2.