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A remark on rapid mixing for hyperbolic flows

Daofei Zhang

TL;DR

The paper addresses rapid mixing for hyperbolic flows by weakening the Dolgopyat-type criterion from a positive box dimension on the temporal-distance range $ ext{range}(\\mathcal{T})$ to the existence of two values whose ratio is Diophantine. It develops a Dolgopyat-type estimate for transfer operators using Markov partitions and a detailed cancellation lemma under this Diophantine condition, yielding superpolynomial decay of correlations and a polynomial Prime Orbit Theorem error term. The results apply to suspension flows over hyperbolic diffeomorphisms and to perturbations of contact flows, showing the criterion's broader applicability even when the previous dimension condition is inapplicable or hard to verify. By bridging hyperbolic dynamics with a Diophantine-oscillation framework, the work enhances the toolkit for establishing rapid mixing in settings with complex temporal structure, with implications for limit theorems and orbit counting.

Abstract

We establish an improved criterion for rapid mixing of hyperbolic flows by weakening the requirement on the temporal distance function from positive box dimension to the existence of two values whose ratio is Diophantine. We also demonstrate the applicability of our results through explicit examples where the previous dimension condition were either too restrictive or computationally infeasible to verify.

A remark on rapid mixing for hyperbolic flows

TL;DR

The paper addresses rapid mixing for hyperbolic flows by weakening the Dolgopyat-type criterion from a positive box dimension on the temporal-distance range to the existence of two values whose ratio is Diophantine. It develops a Dolgopyat-type estimate for transfer operators using Markov partitions and a detailed cancellation lemma under this Diophantine condition, yielding superpolynomial decay of correlations and a polynomial Prime Orbit Theorem error term. The results apply to suspension flows over hyperbolic diffeomorphisms and to perturbations of contact flows, showing the criterion's broader applicability even when the previous dimension condition is inapplicable or hard to verify. By bridging hyperbolic dynamics with a Diophantine-oscillation framework, the work enhances the toolkit for establishing rapid mixing in settings with complex temporal structure, with implications for limit theorems and orbit counting.

Abstract

We establish an improved criterion for rapid mixing of hyperbolic flows by weakening the requirement on the temporal distance function from positive box dimension to the existence of two values whose ratio is Diophantine. We also demonstrate the applicability of our results through explicit examples where the previous dimension condition were either too restrictive or computationally infeasible to verify.
Paper Structure (13 sections, 20 theorems, 47 equations, 1 figure)

This paper contains 13 sections, 20 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Examples
  4. Outline of the paper
  5. Hyperbolic flows and symbolic dynamics
  6. Hyperbolic flows
  7. Symbolic dynamics
  8. A Dolgopyat type estimate
  9. Markov partitions
  10. Deducing the Diophantine condition
  11. Transfer operators and a Dolgopyat type estimate
  12. Proof of Proposition \ref{['Dolgopyat type estimate']}
  13. Proof of Lemma \ref{['Lemma 4.4']}

Key Result

Theorem 1.1

If $\emph{range}(\mathcal{T})$ has positive lower box dimension, then the hyperbolic flow $g_t$ 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 $M$.

Figures (1)

  • Figure 1.1: The temporal distance $\mathcal{T}(z_{1},z_{2})$ of $z_{1}$ and $z_{2}$ where $z_{4}=[z_{1},z_{2}]$

Theorems & Definitions (38)

  • Theorem 1.1: Dolgopyat Dol98b
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • proof
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 28 more