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Exponential mixing of frame flows for convex cocompact locally symmetric spaces

Michael Chow, Pratyush Sarkar

Abstract

Let $G$ be a connected center-free simple real algebraic group of rank one and $Γ< G$ be a Zariski dense torsion-free convex cocompact subgroup. We prove that the frame flow on $Γ\backslash G$, i.e., the right translation action of a one-parameter subgroup $\{a_t\}_{t \in \mathbb R} < G$ of semisimple elements, is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The key step is proving suitable generalizations of the local non-integrability condition and the non-concentration property which are essential for Dolgopyat's method. This generalizes the work of Sarkar-Winter for $G = \operatorname{SO}(n, 1)^\circ$ and also strengthens the mixing result of Winter in the convex cocompact case.

Exponential mixing of frame flows for convex cocompact locally symmetric spaces

Abstract

Let be a connected center-free simple real algebraic group of rank one and be a Zariski dense torsion-free convex cocompact subgroup. We prove that the frame flow on , i.e., the right translation action of a one-parameter subgroup of semisimple elements, is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The key step is proving suitable generalizations of the local non-integrability condition and the non-concentration property which are essential for Dolgopyat's method. This generalizes the work of Sarkar-Winter for and also strengthens the mixing result of Winter in the convex cocompact case.
Paper Structure (22 sections, 26 theorems, 99 equations, 1 table)

This paper contains 22 sections, 26 theorems, 99 equations, 1 table.

Table of Contents

  1. Introduction
  2. Outline of the proof of \ref{['thm:TheoremExponentialMixingOfFrameFlow']}
  3. Organization of the paper
  4. Preliminaries
  5. Lie theory
  6. Convex cocompact subgroups
  7. Patterson--Sullivan density
  8. Bowen--Margulis--Sullivan measure
  9. Markov section
  10. Symbolic dynamics
  11. Thermodynamics
  12. Holonomy
  13. Extending to smooth maps
  14. Transfer operators with holonomy, their spectral bounds, and the proof of \ref{['thm:TheoremExponentialMixingOfFrameFlow']}
  15. Transfer operators
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\alpha \in (0, 1]$. There exist $\eta_\alpha > 0$ and $C > 0$ (independent of $\alpha$) such that for all $\phi, \psi \in C_{\mathrm{c}}^{0,\alpha}(\Gamma \backslash G)$ and $t > 0$, we have

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Corollary 2.2
  • Definition 2.3: Convex cocompact subgroup
  • Definition 2.4: Markov section
  • Definition 2.5: Cylinder
  • Definition 2.6: Pressure
  • Definition 2.7: Holonomy
  • Lemma 2.8
  • ...and 32 more