Invariant measures of the topological flow and measures at infinity on hyperbolic groups

TL;DR

This paper demonstrates that hyperbolic groups have a topological flow space that can be coded with a finite type subshift, leading to new results on measures, curves, and intersection numbers.

Contribution

It introduces a coding for the flow space of hyperbolic groups and applies it to derive regularity and uniqueness results previously established by others.

Findings

Regularity results for Manhattan curves

Uniqueness of measures of maximal Hausdorff dimension

Real analyticity of intersection numbers

Abstract

We show that for every non-elementary hyperbolic group, an associated topological flow space admits a coding based on a transitive subshift of finite type. Applications include regularity results for Manhattan curves, the uniqueness of measures of maximal Hausdorff dimension with potentials, and the real analyticity of intersection numbers for families of dominated representation, thus providing a direct proof of a result established by Bridgeman, Canary, Labourie and Sambarino in 2015.