Given a finite irreducible set of real $d \times d$ matrices $A_1,\ldots,A_M$ and a real parameter $s>0$, there exists a unique shift-invariant equilibrium state associated to $(A_1,\ldots,A_M,s)$. In this article we characterise the ergodic properties of such equilibrium states in terms of the algebraic properties of the semigroup generated by the associated matrices. We completely characterise when the equilibrium state has zero entropy, when it gives distinct Lyapunov exponents to the natural cocycle generated by $A_1,\ldots,A_M$, and when it is a Bernoulli measure. We also give a general sufficient condition for the equilibrium state to be mixing, and give an example where the equilibrium state is ergodic but not totally ergodic. Connections with a class of measures investigated by S. Kusuoka are explored in an appendix.
给定一个从有限型$X$的移位到sofic移位$Y$的因子码$π$，$Y$上的遍历测度$ν$，以及$X$上的具有可和变差的函数$V$，证明了$X$上的遍历测度投影到$ν$并使$h(μ) + \int V dμ$在光纤$π^{-1}(ν)$中的所有测度中最大化的个数的不变上界.如果完全支持$ν$，则此界限是$π$的类度。这推广了$V=0$特殊情况下的一个结果。
Optimal Bias Potentials via Ergodic Optimal Control and Generator Learning
Otto-von-Guericke Universität（奥托·冯·格里克大学）；Max-Planck-Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；Brandenburg University of Technology Cottbus-Senftenberg（勃兰登堡工业大学科特布斯-森夫滕贝格分校）；Martin-Luther-Universität Halle-Wittenberg（马丁·路德·哈雷-维滕贝格大学）；LAAS-CNRS, Université de Toulouse（图卢兹大学法国国家科学研究中心实验室）
