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The weak Bernoulli property for matrix Gibbs states

Part of: Ergodic theory

Published online by Cambridge University Press:  18 December 2018

MARK PIRAINO Open the ORCID record for MARK PIRAINO [Opens in a new window]
MARK PIRAINO*
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Canada email mpiraino@uvic.ca
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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.

Keywords

symbolic dynamicsthermodynamic formalismweak Bernoulli propertymatrix equilibrium states

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 8 , August 2020 , pp. 2219 - 2238
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Copyright
© Cambridge University Press, 2018

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