We study the ergodic properties of a class of measures on ΣZ for which µA,t[x0 · · · xn−1] ≈ e−nP Ax0 · · · Axn−1t, where A = (A0, . . ., AM −1) is a collection of matrices. The measure µA,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 µA,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.
|Original language||English (US)|
|State||Published - Jun 13 2018|
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