Some ergodic properties of metrics on hyperbolic groups

2017/07/07 by Uri Bader, Bader, Uri, Alex Furman +1 · 2 citations
Mathematics · #37Axx #37Dxx #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1707.02020

openalex publication_date 2017/07/07 · openalex created_date 2022/09/14 · openalex updated_date 2026/07/28

Abstract

Let Γ be a non-elementary Gromov-hyperbolic group, and ∂ 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 ∂⁽²⁾Γ, and its square [ν×ν] on ∂⁽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 (∂Γ,ν) and on (∂⁽²⁾Γ,[ν×ν]). We also prove some ergodic theorems for Γ-actions guided by the geometry of (Γ,d).

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• Random Trees in Hyperbolic FPP
• Uniqueness of the measure of maximal entropy for geodesic flows on coarse hyperbolic manifolds without conjugate points

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