Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

Pratyush Sarkar, Dale Winter

Abstract

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen-Margulis-Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

Abstract

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen-Margulis-Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

Paper Structure

This paper contains 30 sections, 38 theorems, 198 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline of the proof of \ref{['thm:TheoremExponentialMixingOfFrameFlow']}
  3. Organization of the paper
  4. Preliminaries
  5. Limit set
  6. Patterson--Sullivan density
  7. Bowen--Margulis--Sullivan measure
  8. Coding the geodesic flow
  9. Markov sections
  10. Symbolic dynamics
  11. Thermodynamics
  12. Holonomy and representation theory
  13. Transfer operators with holonomy and their spectral bounds
  14. Modified constructions using the smooth structure on $G$
  15. Transfer operators
  16. ...and 15 more sections

Key Result

Theorem \oldthetheorem

There exist $\eta > 0$, $C > 0$, and $r \in \mathbb N$ such that for all $\phi \in C_{\mathrm{c}}^r(\Gamma \backslash G, \mathbb R)$, $\psi \in C_{\mathrm{c}}^1(\Gamma \backslash G, \mathbb R)$, and $t > 0$, we have

Figures (2)

  • Figure 1: The Markov property.
  • Figure 2:

Theorems & Definitions (93)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark
  • Remark
  • Theorem \oldthetheorem
  • Remark
  • Definition \oldthetheorem: Limit set
  • Definition \oldthetheorem: Critical exponent
  • Remark
  • Definition \oldthetheorem: Convex cocompact
  • ...and 83 more