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Expanding on average diffeomorphisms of surfaces: exponential mixing

Jonathan DeWitt, Dmitry Dolgopyat

Abstract

We show that the Bernoulli random dynamical system associated to a expanding on average tuple of volume preserving diffeomorphisms of a closed surface is exponentially mixing.

Expanding on average diffeomorphisms of surfaces: exponential mixing

Abstract

We show that the Bernoulli random dynamical system associated to a expanding on average tuple of volume preserving diffeomorphisms of a closed surface is exponentially mixing.

Paper Structure

This paper contains 60 sections, 4 theorems, 427 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Examples
  4. Relationship with other works
  5. Setting and basic definitions
  6. Random dynamics and skew products
  7. Stable subspaces
  8. Stable manifolds
  9. Norms
  10. Probability facts
  11. Outline of the paper
  12. Quenched and annealed properties
  13. Description of the key step
  14. Mixing in hyperbolic dynamics
  15. Estimates on the growth of vectors and temperedness
  16. ...and 45 more sections

Key Result

Theorem 1.1

(Quenched Exponential Mixing) Suppose that $M$ is a closed surface and that $(f_1,\ldots,f_m)$ is an expanding on average tuple of diffeomorphisms in $\operatorname{Diff}^2_{\operatorname{vol}}(M)$. Let $\beta\in (0,1)$ be a Hölder regularity. There exists $\eta>0$ such that for a.e. $\omega\in\Sigm where $f_{\sigma^j(\omega)}^i=f_{\omega_{j+i}}\cdots f_{\omega_{j+1}}$. Further, there exists $D_1>

Theorems & Definitions (109)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Conjecture 1.8
  • Conjecture 1.9
  • Definition 2.1
  • ...and 99 more