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The weak Bernoulli property for matrix Gibbs states
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 40 / Issue 8 / August 2020
- Published online by Cambridge University Press:
- 18 December 2018, pp. 2219-2238
- Print publication:
- August 2020
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- Article
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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.
