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

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

2007-12-26

arxiv.org/abs/0712.4158v2

Mathematics

Dynamical Systems

37 pages. This new version corrects several typos including one in the statement of theorem 1.5

Scientific paper

We introduce a natural equivalence relation on the space $\sH_0$ 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 $\eta$ is such a measure and $G$ acts on a space $(X,\mu)$ by p.m.p. transformations then $\eta \times \mu$ is virtually ergodic with respect to a natural equivalence relation on $\sH_0\times X$. 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.

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