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

Abstract

AbstractWe 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.