The weak Bernoulli property for matrix Gibbs states

We study the ergodic properties of a class of measures on ${\mathrm{\Sigma }}^{\mathbb{Z}}$ 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 $\mathcal{A}=(A_0,\dots ,A_{M-1})$ 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.