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Invariant measures on the space of horofunctions of a word hyperbolic group

Published online by Cambridge University Press:  21 May 2009

LEWIS BOWEN
LEWIS BOWEN*
Affiliation:
Mathematics Department, University of Hawaii, Mānoa, Hawaii (email: [email protected])
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Abstract

We introduce a natural equivalence relation on the space ℋ0 of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen–Marcus theorem. Furthermore, if η is such a measure and G acts on a probability space (X,μ) by measure-preserving transformations then η×μ is virtually ergodic with respect to a natural equivalence relation on ℋ0×X. This is comparable to a special case of the Howe–Moore theorem. These results are applied to prove a new ergodic theorem for spherical averages in the case of a word hyperbolic group acting on a finite space.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010 , pp. 97 - 129
Copyright
Copyright © Cambridge University Press 2009

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