Some ergodic properties of metrics on hyperbolic groups

Abstract

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