2018/06/30 by MARK PIRAINO, Mark Piraino · 1 citation
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Theoretical and Computational Physics #math.DS
paper · pdf · doi:10.1017/etds.2018.129
published as Ergod. Th. Dynam. Sys. 40 (2020) 2219-2238 · V2: Complete rewrite of section 3, new decay of correlations result
arxiv created 2018/10/10 · openalex publication_date 2018/12/18 · arxiv updated 2020/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the ergodic properties of a class of measures on Σℤ for which μA,t[x0⋯ xn-1]≈ e-nP ‖A_x0⋯ A_xn-1 ‖ t, 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.