Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rapid mixing and superpolynomial equidistribution for torus extensions of hyperbolic flows

Daofei Zhang

Abstract

In this paper, we study mixing rates for $\mathbb{T}^{d}$-extensions of hyperbolic flows. Given three closed orbits with their holonomies, we can relate them to a point in $\mathbb{R}^{d+1}$. We prove that the extension flow enjoys rapid mixing, if the associated point is an inhomogeneously Diophantine number. Under the same assumption, we also obtain the superpolynomial equidistribution, namely, a superpolynomial error term in the equidistribution of the holonomy around closed orbits. Lastly, we apply these results to a class of three-dimensional frame flows.

Rapid mixing and superpolynomial equidistribution for torus extensions of hyperbolic flows

Abstract

In this paper, we study mixing rates for -extensions of hyperbolic flows. Given three closed orbits with their holonomies, we can relate them to a point in . We prove that the extension flow enjoys rapid mixing, if the associated point is an inhomogeneously Diophantine number. Under the same assumption, we also obtain the superpolynomial equidistribution, namely, a superpolynomial error term in the equidistribution of the holonomy around closed orbits. Lastly, we apply these results to a class of three-dimensional frame flows.
Paper Structure (11 sections, 26 theorems, 73 equations)

This paper contains 11 sections, 26 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Examples
  4. Outline of the paper
  5. Basic definitions, Markov partitions and symbolic dynamics
  6. Definitions of hyperbolic flows and their group extensions
  7. Suspension flows of $\mathbb{T}^{d}$ skew products over two-sided subshifts
  8. Relating the extension flow to a symbolic flow
  9. A Dolgopyat type estimate
  10. Proof of Proposition \ref{['Dolgopyat type estimate']}
  11. Proof of Lemma \ref{['Lemma 3.4.4']}

Key Result

Theorem 1.1

If there exist two closed orbits $\tau_{1}, \tau_{2}$ with periods $\ell_{1}, \ell_{2}$ such that the associated number $\alpha:=\frac{\ell_{1}}{\ell_{2}}$ is a Diophantine number, then $g_{t}$ is rapidly mixing with respect to $\mu_{\Phi}$. Specifically, the quantity decays to zero faster than any polynomial rate as $t\to\infty$ for any smooth functions $E, F$ on $M$.

Theorems & Definitions (52)

  • Theorem 1.1: Dolgopyat Dol98b
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Lemma 1.5
  • Theorem 1.6
  • proof
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2
  • ...and 42 more