The weak Bernoulli property for matrix Gibbs states

TL;DR

This paper investigates the ergodic properties of matrix Gibbs states, establishing conditions under which they exhibit the weak Bernoulli property, using Perron-Frobenius theory and operator methods.

Contribution

It provides a sufficient condition for matrix Gibbs states to have the weak Bernoulli property, extending analysis to cases where the parameter t is an even integer or positive.

Findings

When t is an even integer, ergodic properties follow from finite-dimensional Perron-Frobenius theory.

Extension of methods to t>0 using operators on infinite-dimensional spaces.

Application of Bradley's general result to prove the main theorem.

Abstract

We study the ergodic properties of a class of measures on $\Sigma^{\mathbb{Z}}$ for which $\mu_{\mathcal{A},t}[x_{0}\cdots x_{n-1}]\approx e^{-nP}\left \|A_{x_{0}}\cdots A_{x_{n-1}}\right \| ^{t}$, where $\mathcal{A}=(A_{0}, \ldots , A_{M-1})$ is a collection of matrices. The measure $\mu_{\mathcal{A},t}$ is called a matrix Gibbs state. In particular we give a sufficient condition for a matrix Gibbs state to have the weak Bernoulli property. We employ a number of techniques to understand these measures including a novel approach based on Perron-Frobenius theory. We find that when $t$ is an even integer the ergodic properties of $\mu_{\mathcal{A} ,t}$ are readily deduced from finite dimensional Perron-Frobenius theory. We then consider an extension of this method to $t>0$ using operators on an infinite dimensional space. Finally we use a general result of Bradley to prove the main theorem.