Mixing of the Mineyev flow, orbital counting and Poincar\'e series for strongly hyperbolic metrics

Abstract

We obtain orbital counting results for the class of strongly hyperbolic metrics on hyperbolic groups. To achieve this we combine ergodic theoretic techniques involving the Mineyev topological flow and symbolic dynamics. Our results apply to the Green metric associated to an admissible, finitely supported, symmetric random walk and to the Mineyev hat metric. We also describe the domain of analyticity for the Poincar\'e series associated to these metrics, prove mixing results for the Mineyev topological flow and obtain correlation asymptotics for pairs of metrics.