The weak Bernoulli property for matrix Gibbs states
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Bibliographic record
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.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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