arXiv 1107.01232011-07-01math.DSmath.RA

极值遍历测度与矩阵半群的有限性

Extremal ergodic measures and the finiteness property of matrix semigroups

Xiongping Dai, Yu Huang, Mingqing Xiao

备注：: 9 pages; accepted by Proceedings of the AMS

摘要：

设$\bS=\{S_1,...,S_K\}$是复$d\times d$矩阵的有限集，$\varSigma_{K}^+$是所有单边无穷序列$i_{\bcdot}\colon\mathbb{N}\rightarrow\{1,...,K\}$的紧空间.马尔可夫位移$θ\colon\varSigma_{K}^+\rightarrow\varSigma_{K}^+;\ i_{\bcdot}\mapsto i_{\bcdot+1}$的遍历概率$μ_*$称为$\bS$的“极值”，如果$ρ(\bS)=\lim_{n\to\infty}\sqrt[n]{\norm{S_{i_1}...S_{i_n}}}$对$μ_*$成立-a. e. $i_{\bcdot}\in\varSigma_{K}^+$，其中$ρ(\bS)$表示$\bS$的广义/联合谱半径。利用极值范数和Kingman次可加遍历定理，证明了$\bS$具有谱有限性（即对有限长字$(i_1,...,i_n)$具有$ρ(\bS)=\sqrt[n]{ρ(S_{i_1}...S_{i_n})}$）当且仅当对$\bS$的某个极值测度$μ_*$，它至少有一个周期密度点$i_{\bcdot}\in\varSigma_{K}^+$ .

英文摘要：

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 $i_{\bcdot}\in\varSigma_{K}^+$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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