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Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization

Dongryul M. Kim, Hee Oh

TL;DR

This work extends classical finiteness, uniqueness of maximal entropy, and strong mixing results for geodesic flows to higher-rank settings by studying relatively Anosov subgroups Γ of a semisimple real group G. It builds a framework of fibered Bowen–Margulis–Sullivan measures m^{BMS}_ψ on base spaces Ω_ψ parameterized by tangent ψ to the growth indicator ψ_Γ^θ, and proves finite base mass and strong mixing for the corresponding one-dimensional flow φ_t. A coarse reparameterization from Groves–Manning cusp spaces to Ω_ψ, with a cocycle t(σ,s) and uniformly bounded fibers, yields exponential expansion along unstable foliations, enabling a thickness–thickness decomposition and entropy analysis. The key results are: finiteness of m_ψ, the base entropy maximizing property with h_{m_ψ}({φ_t}) = δ_ψ = 1, and the uniqueness of the maximal-entropy measure for relatively Anosov Γ; together with a robust reparameterization and expansion framework, these provide a higher-rank analogue of Sullivan–Otal–Péigné and Babillot-mixing phenomena.

Abstract

For a geometrically finite Kleinian group $Γ$, the Bowen-Margulis-Sullivan measure is finite and is the unique measure of maximal entropy for the geodesic flow, as shown by Sullivan and Otal-Peigné respectively. Moreover, it is strongly mixing by a result of Babillot. We obtain a higher-rank analogue of this theorem. Given a relatively Anosov subgroup $Γ$ of a semisimple real algebraic group, there is a family of flow spaces parameterized by linear forms tangent to the growth indicator. We construct a reparameterization of each flow space by the geodesic flow on the Groves-Manning space of $Γ$ which exhibits exponential expansion along unstable foliations. Using this reparameterization, we prove that the Bowen-Margulis-Sullivan measure of each flow space is finite and is the unique measure of maximal entropy. Moreover, it is strongly mixing.

Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization

TL;DR

This work extends classical finiteness, uniqueness of maximal entropy, and strong mixing results for geodesic flows to higher-rank settings by studying relatively Anosov subgroups Γ of a semisimple real group G. It builds a framework of fibered Bowen–Margulis–Sullivan measures m^{BMS}_ψ on base spaces Ω_ψ parameterized by tangent ψ to the growth indicator ψ_Γ^θ, and proves finite base mass and strong mixing for the corresponding one-dimensional flow φ_t. A coarse reparameterization from Groves–Manning cusp spaces to Ω_ψ, with a cocycle t(σ,s) and uniformly bounded fibers, yields exponential expansion along unstable foliations, enabling a thickness–thickness decomposition and entropy analysis. The key results are: finiteness of m_ψ, the base entropy maximizing property with h_{m_ψ}({φ_t}) = δ_ψ = 1, and the uniqueness of the maximal-entropy measure for relatively Anosov Γ; together with a robust reparameterization and expansion framework, these provide a higher-rank analogue of Sullivan–Otal–Péigné and Babillot-mixing phenomena.

Abstract

For a geometrically finite Kleinian group , the Bowen-Margulis-Sullivan measure is finite and is the unique measure of maximal entropy for the geodesic flow, as shown by Sullivan and Otal-Peigné respectively. Moreover, it is strongly mixing by a result of Babillot. We obtain a higher-rank analogue of this theorem. Given a relatively Anosov subgroup of a semisimple real algebraic group, there is a family of flow spaces parameterized by linear forms tangent to the growth indicator. We construct a reparameterization of each flow space by the geodesic flow on the Groves-Manning space of which exhibits exponential expansion along unstable foliations. Using this reparameterization, we prove that the Bowen-Margulis-Sullivan measure of each flow space is finite and is the unique measure of maximal entropy. Moreover, it is strongly mixing.
Paper Structure (10 sections, 46 theorems, 292 equations, 2 figures)

This paper contains 10 sections, 46 theorems, 292 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vector bundle structure of the non-wandering set $\Omega_\Gamma$
  4. Strong mixing for transverse groups with finite BMS measure
  5. Relatively Anosov groups
  6. Reparameterization for relatively Anosov groups
  7. Uniformity of fibers of reparameterization
  8. Exponential expansion on unstable foliations
  9. Finiteness of Bowen-Margulis-Sullivan measures
  10. Unique measure of maximal entropy

Key Result

Theorem 1.1

Let $\Gamma$ be a relatively Anosov subgroup of $G$. For any $(\Gamma, \theta)$-proper linear form $\psi \in \mathfrak a_{\theta}^*$ tangent to the growth indicator of $\Gamma$, the BMS measure $m_\psi$ is finite: Moreover, the system $(\Omega_\psi, m_\psi, \phi_t)$ is strongly mixing.

Figures (2)

  • Figure 1: Two possible configurations of $\sigma \in \mathcal{G}_P$ in Case 1 depending on whether $T_{\sigma_0}^+ = \infty$ or not. Only the first item in Case 1 applies to the left figure.
  • Figure 2: Two possible configurations of $\sigma \in \mathcal{G}_P$ in Case 2 depending on whether $T_{\tilde{\sigma}_0}^- = - \infty$ or not. Only the first item in Case 2 applies to the left figure.

Theorems & Definitions (72)

  • Theorem 1.1: Finiteness and mixing
  • Theorem 1.2: Unique measure of maximal entropy
  • Remark 1.3
  • Theorem 1.4: Reparameterization
  • Remark 1.5
  • Definition 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Remark 3.2
  • Theorem 4.1
  • ...and 62 more