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

We introduce a natural equivalence relation on the space ${\text{sH}}_0$ 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 ${\text{sH}}_0\times X$. 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.