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Ergodic dichotomy for subspace flows in higher rank

Dongryul M. Kim, Hee Oh, Yahui Wang

Abstract

In this paper, we study the ergodicity of a one-parameter diagonalizable subgroup of a connected semisimple real algebraic group $G$ acting on a homogeneous space or, more generally, a homogeneous-like space, equipped with a Bowen-Margulis-Sullivan type measure. These flow spaces are associated with Anosov subgroups of $G$, or more generally, with transverse subgroups of $G$. We obtain an ergodicity criterion similar to the Hopf-Tsuji-Sullivan dichotomy for the ergodicity of the geodesic flow on hyperbolic manifolds. In addition, we extend this criterion to the action of any connected diagonal subgroup of arbitrary dimension. Our criterion provides a codimension dichotomy on the ergodicity of a connected diagonalizable subgroup for general Anosov subgroups. This generalizes an earlier work by Burger-Landesberg-Lee-Oh for Borel Anosov subgroups.

Ergodic dichotomy for subspace flows in higher rank

Abstract

In this paper, we study the ergodicity of a one-parameter diagonalizable subgroup of a connected semisimple real algebraic group acting on a homogeneous space or, more generally, a homogeneous-like space, equipped with a Bowen-Margulis-Sullivan type measure. These flow spaces are associated with Anosov subgroups of , or more generally, with transverse subgroups of . We obtain an ergodicity criterion similar to the Hopf-Tsuji-Sullivan dichotomy for the ergodicity of the geodesic flow on hyperbolic manifolds. In addition, we extend this criterion to the action of any connected diagonal subgroup of arbitrary dimension. Our criterion provides a codimension dichotomy on the ergodicity of a connected diagonalizable subgroup for general Anosov subgroups. This generalizes an earlier work by Burger-Landesberg-Lee-Oh for Borel Anosov subgroups.
Paper Structure (10 sections, 58 theorems, 203 equations, 1 figure)

This paper contains 10 sections, 58 theorems, 203 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Continuity of shadows
  4. Growth indicators and conformal measures on $\mathcal{F}_\theta$
  5. Directional recurrence for transverse subgroups
  6. Bowen-Margulis-Sullivan measures
  7. Directional conical sets and Poincaré series
  8. Transitivity subgroup and ergodicity of directional flows
  9. Ergodic dichotomy for subspace flows
  10. Dichotomy theorems for Anosov subgroups

Key Result

Theorem 1.1

Let $\Gamma$ be a Zariski dense $\theta$-transverse subgroup of $G$. Fix a non-zero vector $u\in \mathfrak a_\theta^+$ and a $(\Gamma, \theta)$-proper linear form $\psi \in \mathfrak a_{\theta}^*$. Suppose that there exists a pair $(\nu, \nu_{\operatorname{i}})$ of $(\Gamma, \psi)$ and $(\Gamma, \ps The second case:

Figures (1)

  • Figure 1: Shadows

Theorems & Definitions (101)

  • Theorem 1.1: Ergodic dichotomy for directional flows
  • Remark 1.2
  • Theorem 1.3: Ergodic dichotomy for subspace flows
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: Codimension dichotomy
  • Corollary 1.7: $\theta$-rank dichotomy
  • Remark 1.8
  • Corollary 1.9
  • Lemma 2.1
  • ...and 91 more