Mixing of the Mineyev flow, orbital counting and Poincaré series for strongly hyperbolic metrics

TL;DR

This work extends orbital counting to strongly hyperbolic metrics on hyperbolic groups, including Green and Mineyev hat metrics, by fusing Mineyev flow dynamics with Cannon coding and thermodynamic formalism. It establishes that the Poincaré series $\eta(s)=\sum_{x}{e^{-s d(o,x)}}$ extends meromorphically with a simple pole at $s=v_d$ when the length spectrum is non-arithmetic, and derives precise counting with possible $O(T^{-\kappa})$ errors under a badly approximable ratio condition. The Mineyev flow is shown to be weakly mixing in the non-arithmetic case, with rapid mixing under certain diophantine-type assumptions, and the framework yields correlation asymptotics for pairs of metrics, including an asymptotic $\frac{C e^{\alpha T}}{\sqrt{T}}$ term under independence. Applications to CAT$(-1)$-spaces and Anosov representations yield refined counting with error terms and domain-of-analyticity results for the associated Poincaré series, highlighting the broad reach of the approach for geometric group theory and dynamical systems.

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 Poincaré series associated to these metrics, prove mixing results for the Mineyev topological flow and obtain correlation asymptotics for pairs of metrics.

Theorems & Definitions (70)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Corollary 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10