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Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigidity

Dongryul M. Kim, Andrew Zimmer

Abstract

In this paper we introduce Patterson--Sullivan systems, which consist of a group action on a compact metrizable space and a quasi-invariant measure which behaves like a classical Patterson--Sullivan measure. For such systems we prove a generalization of Tukia's measurable boundary rigidity theorem. We then apply this generalization to (1) study the singularity conjecture for Patterson--Sullivan measures (or, conformal densities) and stationary measures of random walks on isometry groups of Gromov hyperbolic spaces, mapping class groups, and discrete subgroups of semisimple Lie groups; (2) prove versions of Tukia's theorem for word hyperbolic groups, Teichmüller spaces, and higher rank symmetric spaces; and (3) in a companion paper prove an entropy rigidity result for Anosov groups with Lipschitz limit sets.

Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigidity

Abstract

In this paper we introduce Patterson--Sullivan systems, which consist of a group action on a compact metrizable space and a quasi-invariant measure which behaves like a classical Patterson--Sullivan measure. For such systems we prove a generalization of Tukia's measurable boundary rigidity theorem. We then apply this generalization to (1) study the singularity conjecture for Patterson--Sullivan measures (or, conformal densities) and stationary measures of random walks on isometry groups of Gromov hyperbolic spaces, mapping class groups, and discrete subgroups of semisimple Lie groups; (2) prove versions of Tukia's theorem for word hyperbolic groups, Teichmüller spaces, and higher rank symmetric spaces; and (3) in a companion paper prove an entropy rigidity result for Anosov groups with Lipschitz limit sets.

Paper Structure

This paper contains 61 sections, 71 theorems, 318 equations.

Table of Contents

  1. Introduction
  2. Random walks
  3. Random walks on Gromov hyperbolic spaces
  4. Random walks on mapping class groups and Teichmüller spaces
  5. Random walks on discrete subgroups of Lie groups
  6. Tukia's measurable boundary rigidity theorem
  7. Tukia's theorem for word hyperbolic groups
  8. Tukia's theorem for Teichmüller spaces
  9. Tukia's theorem in higher rank
  10. Entropy rigidity for Anosov groups with Lipschitz limit sets
  11. Patterson--Sullivan systems
  12. Preliminaries
  13. Possibly ambiguous notation/terminology
  14. The Hausdorff distance
  15. Relatively hyperbolic groups
  16. ...and 46 more sections

Key Result

Theorem 1.1

Tukia1989 For $i=1,2$ let $\Gamma_i < \mathop{\mathrm{\mathsf{Isom}}}\nolimits(\mathop{\mathrm{\mathbb{H}}}\nolimits^{n_i})$ be a Zariski dense discrete subgroup and let $\mu_i$ be a Patterson--Sullivan measure for $\Gamma_i$ of dimension $\delta_i$. Suppose If the measures $f_* \mu_1$ and $\mu_2$ are not singular, then $n_1=n_2$ and $\rho$ extends to an isomorphism $\mathop{\mathrm{\mathsf{Isom}

Theorems & Definitions (139)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.4: corollary of Theorem \ref{['thm.singularity harm GH']} and Corollary \ref{['cor.singularity hitting PS symmetric space intro']}
  • Theorem 1.5: corollary of Theorem \ref{['thm:random walks GH in intro']}
  • Remark 1.6
  • Corollary 1.7: see Corollary \ref{['cor.singularity hitting PS symmetric space']} below
  • Remark 1.8
  • Theorem 1.9: see Theorem \ref{['thm:random walks GH']} below
  • Conjecture 1.10: Kaimanovich--Masur KM_MCG
  • Theorem 1.11: see Corollary \ref{['cor.singularity hitting PS MCG multitwist']} below
  • ...and 129 more