Mixing of the Mineyev flow, orbital counting and Poincaré 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, to the Mineyev hat metric and to Hilbert length functions associated to Anosov representations. We also describe the domain of analyticity for the Poincaré series associated to these metrics, prove mixing results for the Mineyev topological flow and obtain correlation asymptotics for pairs of metrics.