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Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces

Michael Chow, Pratyush Sarkar

Abstract

Let $G$ be a connected semisimple real algebraic group and $Γ< G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We prove local mixing of the one-parameter diagonal flow $\{\exp(t\mathsf{v}) : t \in \mathbb R\}$ on $Γ\backslash G$ for any interior direction $\mathsf{v}$ of the limit cone of $Γ$ with respect to the Bowen--Margulis--Sullivan measure associated to $\mathsf{v}$. More generally, we allow a class of deviations to this flow along a direction $\mathsf{u}$ in some fixed subspace transverse to $\mathsf{v}$. We also obtain a uniform bound for the correlation function which decays exponentially in $\|\mathsf{u}\|^2$. The precise form of the result is required for several applications such as the asymptotic formula for the decay of matrix coefficients in $L^2(Γ\backslash G)$ proved by Edwards--Lee--Oh.

Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces

Abstract

Let be a connected semisimple real algebraic group and be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We prove local mixing of the one-parameter diagonal flow on for any interior direction of the limit cone of with respect to the Bowen--Margulis--Sullivan measure associated to . More generally, we allow a class of deviations to this flow along a direction in some fixed subspace transverse to . We also obtain a uniform bound for the correlation function which decays exponentially in . The precise form of the result is required for several applications such as the asymptotic formula for the decay of matrix coefficients in proved by Edwards--Lee--Oh.

Paper Structure

This paper contains 24 sections, 40 theorems, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Limit set and limit cone
  4. Patterson--Sullivan measures
  5. Bowen--Margulis--Sullivan measures
  6. Anosov subgroups
  7. PS and BMS measures for Anosov subgroups
  8. The vector bundle and the translation flow
  9. Ergodic decomposition of the BMS measure
  10. Coding the translation flow
  11. Metric Anosov flows and Markov sections
  12. Markov section for the translation flow
  13. Symbolic dynamics
  14. Thermodynamics
  15. First return vector map and holonomy
  16. ...and 9 more sections

Key Result

Theorem \oldthetheorem

Let $\mathsf{v} \in \mathop{\mathrm{int}}\nolimits(\mathcal{L}_\Gamma)$. There exists $\kappa_\mathsf{v} > 0$ such that for all $\phi_1, \phi_2 \in C_{\mathrm{c}}(\Gamma \backslash G)$, we have

Figures (1)

  • Figure 1: The Markov property.

Theorems & Definitions (95)

  • Definition \oldthetheorem: Local mixing
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem: ELO22b
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem: Iwasawa cocycle
  • Definition \oldthetheorem: Busemann function
  • ...and 85 more