Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative infinite mixing for non-compact skew products

Paolo Giulietti, Andy Hammerlindl, Davide Ravotti

Abstract

We consider skew products over subshifts of finite type in which the fibers are copies of the real line, and we study their mixing properties with respect to any infinite invariant measure given by the product of a Gibbs measure on the base and Lebesgue measure on the fibers. Assuming that the system is accessible, we prove a quantitative version of Krickeberg mixing for a class of observables which is dense in the space of continuous functions vanishing at infinity.

Quantitative infinite mixing for non-compact skew products

Abstract

We consider skew products over subshifts of finite type in which the fibers are copies of the real line, and we study their mixing properties with respect to any infinite invariant measure given by the product of a Gibbs measure on the base and Lebesgue measure on the fibers. Assuming that the system is accessible, we prove a quantitative version of Krickeberg mixing for a class of observables which is dense in the space of continuous functions vanishing at infinity.
Paper Structure (22 sections, 47 theorems, 190 equations)

This paper contains 22 sections, 47 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Statement of the main results
  4. Skew products and their accessibilities properties
  5. Krickeberg mixing
  6. The observables
  7. Asymptotics of correlations
  8. A technical result
  9. Transfer operators and smooth curves of Lipschitz functions
  10. Twisted transfer operators
  11. Smooth curves of Lipschitz functions
  12. Fourier transforms of the observables
  13. Fourier transforms
  14. Proof of \ref{['prop:FT_smooth_curves']}
  15. Rapid decay
  16. ...and 7 more sections

Key Result

Proposition 2.1

Let $f \colon X \to { \mathbb R}$ be a real-valued Lipschitz function. If the skew product $F\colon \Sigma \times { \mathbb R} \to \Sigma \times { \mathbb R}$ given by $F(x,t) = (\sigma x, t+f \circ \pi (x))$ is accessible, then $f$ has the collapsed accessibility property.

Theorems & Definitions (88)

  • Proposition 2.1
  • Theorem A
  • Theorem B
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 78 more