The weak Bernoulli property for matrix Gibbs states

Abstract

We study the ergodic properties of a class of measures on$\unicode[STIX]{x1D6F4}^{\mathbb{Z}}$for which$\unicode[STIX]{x1D707}_{{\mathcal{A}},t}[x_{0}\cdots x_{n-1}]\approx e^{-nP}\Vert A_{x_{0}}\cdots A_{x_{n-1}}\Vert ^{t}$, where${\mathcal{A}}=(A_{0},\ldots ,A_{M-1})$is a collection of matrices. The measure$\unicode[STIX]{x1D707}_{{\mathcal{A}},t}$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$\unicode[STIX]{x1D707}_{{\mathcal{A}},t}$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.