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

TL;DR

This paper develops orbital counting results for strongly hyperbolic metrics on hyperbolic groups using ergodic theory, symbolic dynamics, and Mineyev flows, and analyzes the Poincaré series and mixing properties.

Contribution

It introduces new techniques combining Mineyev flows and symbolic dynamics to study orbital counting and Poincaré series for strongly hyperbolic metrics on hyperbolic groups.

Findings

Orbital counting results for strongly hyperbolic metrics.

Description of the domain of analyticity for Poincaré series.

Proven mixing and correlation asymptotics for Mineyev flows.

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.