Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T04:52:10.041Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

[1]Aaronson, J. & Keane, M.. Deterministic random walks and returns to zero. J. London Math. Soc. (in the press).Google Scholar
[2]Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429–460.Google Scholar
[3]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer Lecture Notes in Math. no. 470. Springer: Berli, 1975.Google Scholar
[4]Bowen, R.. Hausdorff dimension of quasi-circles I.H.E.S. Publ. Math. 50 (1979), 11–25.Google Scholar
[5]Gottschalk, W. H. & Hedland, G. A.. Topological Dynamics. Amer. Math. Soc. Coll. Publ. no. 36 (1955).Google Scholar
[6]Lyons, T. & McKean, H.. Winding of the plane Brownian motion. Advances in Math. (in the press).Google Scholar
[7]Morse, M.. Symbolic Dynamics (notes by R. Oldenburger). I.A.S.: Princeton, 1966.Google Scholar
[8]Patterson, S. J.. The limit set of a Fuchsian group. Acta. Math. 136 (1976), 241–273.CrossRefGoogle Scholar
[9]Series, C.. Symbolic dynamics for geodesic flows. Acta. Math. (in the press).Google Scholar
[10]Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. I.H.E.S. Publ. Math. 50 (1979), 171–202.Google Scholar