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

Stephen Cantrell, Ryokichi Tanaka

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.

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

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.
Paper Structure (28 sections, 36 theorems, 347 equations, 3 figures)

This paper contains 28 sections, 36 theorems, 347 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic groups
  4. Patterson-Sullivan construction with tempered potentials
  5. Invariant measures on the boundary square
  6. Topological flows
  7. Topological flows: construction
  8. Measures and ergodicity
  9. Local intersection numbers
  10. Symbolic coding
  11. Automatic structures and shift spaces
  12. Suspension flows
  13. Thermodynamic formalism
  14. An enhanced coding
  15. Applications through the enhanced coding
  16. ...and 13 more sections

Key Result

Theorem 1.1

For every non-elementary hyperbolic group $\Gamma$, let ${\mathcal{F}}_\kappa$ be the topological flow space associated with a Hölder continuous cocycle $\kappa$ defined by a strongly hyperbolic metric. There exists a topologically transitive subshift of finite type $(\Sigma_0, \sigma)$ with a posit

Figures (3)

  • Figure 1: A part of a strictly invariant family of multicones of index $1$ for $(\Gamma, S)$, where $\Gamma$ is a genus two surface group and $S$ is the translation generators. The arrows indicate the destinations of intervals after translation by a single generator $s \in S$ (bottom), and an expanded picture shows a small portion of the disk bounded by the thick arc (top).
  • Figure 2: A Patterson-Sullivan measure
  • Figure 3: The harmonic measure

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5: Proposition 2.7 in CT
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 36 more