Some ergodic properties of metrics on hyperbolic groups

Abstract

Let $\hat{I}“$ be a non-elementary Gromov-hyperbolic group, and $\partial \hat{I}“$ denote its Gromov boundary. We consider $\hat{I}“$-invariant proper $\hat{I}´$-hyperbolic, quasi-convex metric $d$ on $\hat{I}“$, and the associated Patterson-Sullivan measure class $[\hat{I}½]$ on ${\partial }^{(2)}\hat{I}“$, and its square $[\hat{I}½\times \hat{I}½]$ on ${\partial }^{(2)}\hat{I}“$ -- the space of distinct pairs of points on the boundary. We construct an analogue of a geodesic flow to study ergodicity properties of the $\hat{I}“$-actions on $(\partial \hat{I}“,\hat{I}½)$ and on $({\partial }^{(2)}\hat{I}“,[\hat{I}½\times \hat{I}½])$. We also prove some ergodic theorems for $\hat{I}“$-actions guided by the geometry of $(\hat{I}“,d)$.

This expended version includes some new results, and more details of the constructions and the proofs