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Measurable boundary maps and Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary

Dongryul M. Kim, Andrew Zimmer

Abstract

In this paper we develop a theory for Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary. Previously, such a theory has been successfully developed for measures supported on the partial flag manifold associated to the Anosov condition, which coincides with the Furstenberg boundary only under the strongest Anosov condition, Borel Anosov. We establish existence, uniqueness, and ergodicity results for the measures on the Furstenberg boundary under arbitrary Anosov conditions; we show ergodicity of Bowen--Margulis--Sullivan measures on the homogeneous space; and we establish strict convexity results for the critical exponent associated to functionals on the entire Cartan subspace. Using this strict convexity, we establish an entropy rigidity result for Anosov groups with Lipschitz limit set. A key tool we develop is a new sufficient condition for the existence of a measurable boundary map associated to a Zariski dense representation. This result not only applies to Anosov groups, but also to transverse groups, mapping class groups, and discrete subgroups of the isometry groups of Gromov hyperbolic spaces.

Measurable boundary maps and Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary

Abstract

In this paper we develop a theory for Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary. Previously, such a theory has been successfully developed for measures supported on the partial flag manifold associated to the Anosov condition, which coincides with the Furstenberg boundary only under the strongest Anosov condition, Borel Anosov. We establish existence, uniqueness, and ergodicity results for the measures on the Furstenberg boundary under arbitrary Anosov conditions; we show ergodicity of Bowen--Margulis--Sullivan measures on the homogeneous space; and we establish strict convexity results for the critical exponent associated to functionals on the entire Cartan subspace. Using this strict convexity, we establish an entropy rigidity result for Anosov groups with Lipschitz limit set. A key tool we develop is a new sufficient condition for the existence of a measurable boundary map associated to a Zariski dense representation. This result not only applies to Anosov groups, but also to transverse groups, mapping class groups, and discrete subgroups of the isometry groups of Gromov hyperbolic spaces.

Paper Structure

This paper contains 49 sections, 53 theorems, 227 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Measurable boundary maps
  3. Existence and uniqueness of PS-measures
  4. Double ergodicity and Bowen--Margulis--Sullivan measures
  5. Strict convexity of entropy
  6. Entropy rigidity
  7. Outline of Paper
  8. Background
  9. Patterson--Sullivan systems
  10. Boundary rigidity
  11. Properties of Patterson--Sullivan systems
  12. Notations for semisimple Lie groups
  13. Cartan and Jordan projections
  14. Parabolic subgroups and flag manifolds
  15. The special linear group
  16. ...and 34 more sections

Key Result

Theorem 1.2

Suppose $(M, \Gamma, \sigma, \mu)$ is a well-behaved Patterson--Sullivan system with respect to a hierarchy $\mathscr{H} = \{ \mathscr{H} (R) \subset \Gamma : R \ge 0\}$, the $\mathop{\mathrm{\mathcal{H}}}\nolimits$-conical limit set has full $\mu$-measure, and the $\Gamma$-action on $(M,\mu)$ is am

Figures (1)

  • Figure 1: Slice of $\mathfrak{a}^+$ along its unit sphere, when $\mathop{\mathrm{rank}}\nolimits \mathop{\mathrm{\mathsf{G}}}\nolimits = 3$ and $\# \theta = 2$. $\mathop{\mathrm{\mathcal{L}}}\nolimits(\Gamma) \subset \mathfrak{a}^+$ denotes the asymptotic cone of $\kappa(\Gamma)$, and there is a tangent direction for $\delta^{\phi}(\Gamma) \cdot \phi$ on $\mathop{\mathrm{\mathcal{L}}}\nolimits(\Gamma) \cap \mathfrak{a}_{\theta}$.

Theorems & Definitions (92)

  • Definition 1.1
  • Theorem 1.2: see Theorem \ref{['thm:boundary map']} below
  • Remark 1.3
  • Theorem 1.4: PS-measures on the Furstenberg boundary
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8: Ergodicity on homogeneous spaces
  • Remark 1.9
  • Example 1.10
  • ...and 82 more