### Abstract

We study the ergodic properties of a class of measures on for which, where is a collection of matrices. The measure 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 is an even integer the ergodic properties of are readily deduced from finite-dimensional Perron-Frobenius theory. We then consider an extension of this method to 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) |
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Pages (from-to) | 2219-2238 |

Number of pages | 20 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 40 |

Issue number | 8 |

DOIs | |

State | Published - Aug 1 2020 |

Externally published | Yes |

### Keywords

- Matrix equilibrium states
- Symbolic dynamics
- Thermodynamic formalism
- Weak Bernoulli property

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics