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Checking ergodicity of some geodesic flows with infinite Gibbs measure

Published online by Cambridge University Press:  19 September 2008

Mary Rees
Mary Rees*
Affiliation:
From the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France
*
†Dr Mary Rees, Institute des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France.
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Abstract

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This paper concerns a problem which arose from a paper of Sullivan. Let Γ be a discrete group of isometries of hyperbolic space Hd+1. We study the question of when the geodesic flow on the unit tangent bundle UT (Hd+1/Γ) of Hd+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 (Hd+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 Γ1 of Γ with Γ/Γ1 ≅ℤυ and prove the geodesic flow on UT (Hd+1/Γ1) is ergodic (with respect to one of these natural measures) if and only if υ ≤ 2.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 1 , Issue 1 , March 1981 , pp. 107 - 133
Copyright
Copyright © Cambridge University Press 1981

References

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