Anisotropic spaces for the bilateral shift

TL;DR

This paper introduces anisotropic Banach spaces tailored for the bilateral shift, demonstrating quasicompactness of the transfer operator, spectral gap, and exponential decay of correlations for H"older observables.

Contribution

It constructs new anisotropic spaces for the bilateral shift and proves spectral properties of the transfer operator, extending previous results from unilateral to bilateral shifts.

Findings

Transfer operator is quasicompact with a spectral gap.

Unique Gibbs state spans the 1-eigenspace.

Exponential decay of correlations for H"older observables.

Abstract

Given two H\"older potentials $\phi_+$ and $\psi_-$ 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 $\phi_+$ spans its $1$-eigenspace. This result allows us to establish exponential decay of correlations for H\"older observables and a wide range of measures on the bilateral shift space.