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Anisotropic spaces for the bilateral shift

Mateus Marra, Daniel Smania

TL;DR

The paper develops anisotropic Besov-type Banach spaces tailored to a symbolic bilateral shift and its skew-product structure, enabling a quasicompact transfer operator with a spectral gap. By constructing Gibbs states via cohomology to unilateral potentials and proving precise action on atoms, the authors show the unique 1-eigenvector is the Gibbs state and obtain exponential decay of correlations for Hölder observables and a broad class of measures. A key contribution is the rigorous framework that handles distributions (not just functions), which is essential for non-smooth or non-absolutely continuous invariant measures in this invertible, hyperbolic setting. The results provide a robust method to derive statistical properties (mixing, SRB-type behavior, and decay rates) for bilateral shifts, with potential applications to more general non-smooth hyperbolic systems and symbolic dynamics.

Abstract

Given two Hölder potentials $φ_+$ and $ψ_-$ for the unilateral shift, we define anisotropic Banach spaces of distributions on the bilateral shift space with a finite alphabet. On these spaces, the transfer operator for the bilateral shift is quasicompact with a spectral gap, and the unique Gibbs state associated with $φ_+$ spans its $1$-eigenspace. This result allows us to establish exponential decay of correlations for Hölder observables and a wide range of measures on the bilateral shift space.

Anisotropic spaces for the bilateral shift

TL;DR

The paper develops anisotropic Besov-type Banach spaces tailored to a symbolic bilateral shift and its skew-product structure, enabling a quasicompact transfer operator with a spectral gap. By constructing Gibbs states via cohomology to unilateral potentials and proving precise action on atoms, the authors show the unique 1-eigenvector is the Gibbs state and obtain exponential decay of correlations for Hölder observables and a broad class of measures. A key contribution is the rigorous framework that handles distributions (not just functions), which is essential for non-smooth or non-absolutely continuous invariant measures in this invertible, hyperbolic setting. The results provide a robust method to derive statistical properties (mixing, SRB-type behavior, and decay rates) for bilateral shifts, with potential applications to more general non-smooth hyperbolic systems and symbolic dynamics.

Abstract

Given two Hölder potentials and for the unilateral shift, we define anisotropic Banach spaces of distributions on the bilateral shift space with a finite alphabet. On these spaces, the transfer operator for the bilateral shift is quasicompact with a spectral gap, and the unique Gibbs state associated with spans its -eigenspace. This result allows us to establish exponential decay of correlations for Hölder observables and a wide range of measures on the bilateral shift space.

Paper Structure

This paper contains 29 sections, 23 theorems, 204 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bilateral shift and Gibbs states
  3. Shifts
  4. Bilateral shift as a skew-product
  5. Cylinders and words
  6. Topology and metric
  7. Gibbs state for the bilateral shift
  8. Gibbs states for unilateral shifts
  9. Constructing Gibbs states
  10. Product structure of Gibbs states and SRB measures
  11. Do you want to understand the past or the future? You need to choose
  12. Main results: Spectral gap and Exponential decay of correlations
  13. Do we really need distributions?
  14. Measure space with a good grid
  15. Banach Spaces
  16. ...and 14 more sections

Key Result

Proposition 2.8.5

Let $\psi_+\colon I^+\rightarrow \mathbb{R}$ and $\psi_-\colon I^-\rightarrow \mathbb{R}$ be Hölder continuous functions with zero topological pressure. Let $m=m_+^{\psi_+}\times m_-^{\psi_-}$. Recall that $\phi$, $\phi_+$ and $\phi_-$ are cohomologous to each other, so $m_+^{\phi_+}\times m_-^{\phi_-}$ is in case $C$.

Figures (1)

  • Figure 1: An atom $a_Q$.

Theorems & Definitions (43)

  • Remark 2.6.2
  • Definition 2.8.4
  • Proposition 2.8.5
  • Theorem A: Spectral Gap for Anisotropic Banach Spaces
  • Theorem B: Exponential Decay of Correlations I
  • Theorem C: Exponential Decay of Correlations II
  • Proposition 3.1.6
  • proof
  • Definition 4.0.8
  • Remark 4.0.9
  • ...and 33 more