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On the dimension of limit sets on $\mathbb{P}(\mathbb{R}^3)$ via stationary measures: the theory and applications

Jialun Li, Wenyu Pan, Disheng Xu

Abstract

This paper investigates the (semi)group action of $\mathrm{SL}_3(\mathbb{R})$ on $\mathbb{P}(\mathbb{R}^3)$, a primary example of non-conformal, non-linear, and non-strictly contracting action. We study the Hausdorff dimension of a dynamically defined limit set in $\mathbb{P}(\mathbb{R}^3)$ and generalize the classical Patterson-Sullivan formula using the approach of stationary measures. The two main examples are Anosov representations in $\mathrm{SL}_3(\mathbb{R})$ and the Rauzy gasket. 1. For Anosov representations in $\mathrm{SL}_3(\mathbb{R})$, we establish a sharp lower bound for the dimension of their limit sets in $\mathbb{P}(\mathbb{R}^3)$. Coupled with the upper bound in Pozzetti-Sambarino-Wienhard, it shows that their Hausdorff dimensions equal the affinity exponents. The merit of our approach is that it works uniformly for all the components of irreducible Anosov representations in $\mathrm{SL}_3(\mathbb{R})$. As an application, it reveals a surprising dimension jump phenomenon in the Barbot component, which is a local generalization of Bowen's dimension rigidity result. 2. For the Rauzy gasket, we confirm a folklore conjecture about the Hausdorff dimension of the gasket and improve the numerical lower bound to $3/2$. These results originate from a dimension formula of stationary measures on $\mathbb{P}(\mathbb{R}^3)$. Let $ν$ be a probability measure on $\mathrm{SL}_3(\mathbb{R})$ whose support is finite and spans a Zariski dense subgroup. Let $μ$ be the associated stationary measure for the action on $\mathbb{P}(\mathbb{R}^3)$. Under the exponential separation condition on $ν$, we prove that the Hausdorff dimension of $μ$ equals its Lyapunov dimension, which extends Hochman-Solomyak and Bárány-Hochman-Rapaport to non-conformal and projective settings respectively.

On the dimension of limit sets on $\mathbb{P}(\mathbb{R}^3)$ via stationary measures: the theory and applications

Abstract

This paper investigates the (semi)group action of on , a primary example of non-conformal, non-linear, and non-strictly contracting action. We study the Hausdorff dimension of a dynamically defined limit set in and generalize the classical Patterson-Sullivan formula using the approach of stationary measures. The two main examples are Anosov representations in and the Rauzy gasket. 1. For Anosov representations in , we establish a sharp lower bound for the dimension of their limit sets in . Coupled with the upper bound in Pozzetti-Sambarino-Wienhard, it shows that their Hausdorff dimensions equal the affinity exponents. The merit of our approach is that it works uniformly for all the components of irreducible Anosov representations in . As an application, it reveals a surprising dimension jump phenomenon in the Barbot component, which is a local generalization of Bowen's dimension rigidity result. 2. For the Rauzy gasket, we confirm a folklore conjecture about the Hausdorff dimension of the gasket and improve the numerical lower bound to . These results originate from a dimension formula of stationary measures on . Let be a probability measure on whose support is finite and spans a Zariski dense subgroup. Let be the associated stationary measure for the action on . Under the exponential separation condition on , we prove that the Hausdorff dimension of equals its Lyapunov dimension, which extends Hochman-Solomyak and Bárány-Hochman-Rapaport to non-conformal and projective settings respectively.
Paper Structure (65 sections, 94 theorems, 451 equations, 5 figures, 1 table)

This paper contains 65 sections, 94 theorems, 451 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Hausdorff dimension of limit sets
  3. Anosov representation
  4. Dimension jump
  5. Rauzy gasket
  6. Dimension of stationary measures
  7. Hausdorff dimension and Lyapunov dimension of stationary measures
  8. Dimension of the projection of stationary measures
  9. Further questions
  10. Preliminaries
  11. Action of GL_n(R) on P(Rn)
  12. Projections in R3 and P(R3)
  13. Decomposition of SL_3(R) for the composition maps π_{V⊥}g
  14. Linearize the projection of L-action
  15. q-Adic partitions
  16. ...and 50 more sections

Key Result

Theorem 1.2

Let $\Gamma$ be a hyperbolic group and $\rho:\Gamma\rightarrow\rm{SL}_3(\mathbb{R})$ be an irreducible Anosov representation, then the Hausdorff dimension of the limit set $L(\rho(\Gamma))$ in $\mathbb{P}(\mathbb{R}^3)$ equals the affinity exponent $s_{\mathrm{A}}(\rho)$. Moreover, the exponent $s_{

Figures (5)

  • Figure 1: Figure in arnoux_rauzy_2013
  • Figure 2: $\pi_{{E_1}^\perp}\ell = h \circ \pi_{\ell^{-1}E_1,E_1^{\perp}}$
  • Figure 3: The $[\ell]$-attracting region
  • Figure 4: Case 1
  • Figure 5: Case 2

Theorems & Definitions (199)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Remark 2.2
  • ...and 189 more