Extremal ergodic measures and the finiteness property of matrix semigroups

Let $\bS=\{S_1,...,S_K\}$ be a finite set of complex $d\times d$ matrices and $\varSigma_{K}^+$ the compact space of all one-sided infinite sequences $i_{\bcdot}\colon\mathbb{N}\rightarrow\{1,...,K\}$. An ergodic probability $μ_*$ of the Markov shift $θ\colon\varSigma_{K}^+\rightarrow\varSigma_{K}^+;\ i_{\bcdot}\mapsto i_{\bcdot+1}$, is called "extremal" for $\bS$, if $ρ(\bS)=\lim_{n\to\infty}\sqrt[n]{\norm{S_{i_1}...S_{i_n}}}$ holds for $μ_*$-a.e. $i_{\bcdot}\in\varSigma_{K}^+$, where $ρ(\bS)$ denotes the generalized/joint spectral radius of $\bS$. Using extremal norm and Kingman subadditive ergodic theorem, it is shown that $\bS$ has the spectral finiteness property (i.e. $ρ(\bS)=\sqrt[n]{ρ(S_{i_1}...S_{i_n})}$ for some finite-length word $(i_1,...,i_n)$) if and only if for some extremal measure $μ_*$ of $\bS$, it has at least one periodic density point $i_{\bcdot}\in\varSigma_{K}^+$.

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