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Extremal ergodic measures and the finiteness property of matrix semigroups


Authors: Xiongping Dai, Yu Huang and MingQing Xiao
Journal: Proc. Amer. Math. Soc. 141 (2013), 393-401
MSC (2010): Primary 15B52; Secondary 15A30, 15A18
DOI: https://doi.org/10.1090/S0002-9939-2012-11330-9
Published electronically: June 1, 2012
MathSciNet review: 2996944
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Abstract: Let ${\boldsymbol {S}}=\{S_1,\ldots ,S_K\}$ be a finite set of complex $d\times d$ matrices and $\varSigma _{\!K}^+$ be the compact space of all one-sided infinite sequences $i_{\boldsymbol {\cdot }}\colon \mathbb {N}\rightarrow \{1,\dotsc ,K\}$. An ergodic probability $\mu _*$ of the Markov shift $\theta \colon \varSigma _{\!K}^+\rightarrow \varSigma _{\!K}^+;\ i_{\boldsymbol {\cdot }}\mapsto i_{\boldsymbol {\cdot }+1}$, is called “extremal” for ${\boldsymbol {S}}$ if ${\rho }({\boldsymbol {S}})=\lim _{n\to \infty }\sqrt [n]{\left \|S_{i_1}\cdots S_{i_n}\right \|}$ holds for $\mu _*$-a.e. $i_{\boldsymbol {\cdot }}\in \varSigma _{\!K}^+$, where $\rho ({\boldsymbol {S}})$ denotes the generalized/joint spectral radius of ${\boldsymbol {S}}$. Using the extremal norm and the Kingman subadditive ergodic theorem, it is shown that ${\boldsymbol {S}}$ has the spectral finiteness property (i.e. $\rho ({\boldsymbol {S}})=\sqrt [n]{\rho (S_{i_1}\cdots S_{i_n})}$ for some finite-length word $(i_1,\ldots ,i_n)$) if and only if for some extremal measure $\mu _*$ of ${\boldsymbol {S}}$, it has at least one periodic density point $i_{\boldsymbol {\cdot }}\in \varSigma _{\!K}^+$.


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Additional Information

Xiongping Dai
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
MR Author ID: 609395
Email: xpdai@nju.edu.cn

Yu Huang
Affiliation: Department of Mathematics, Zhongshan (Sun Yat-Sen) University, Guangzhou 510275, People’s Republic of China
MR Author ID: 197768
Email: stshyu@mail.sysu.edu.cn

MingQing Xiao
Affiliation: Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901-4408
Email: mxiao@math.siu.edu

Keywords: The finiteness property, joint/generalized spectral radius, extremal probability, random product of matrices
Received by editor(s): June 10, 2011
Received by editor(s) in revised form: June 16, 2011, and June 27, 2011
Published electronically: June 1, 2012
Additional Notes: This project was supported partly by National Natural Science Foundation of China (Nos. 11071112 and 11071263) and in part by NSF DMS-0605181, 1021203, of the United States.
Communicated by: Bryna Kra
Article copyright: © Copyright 2012 American Mathematical Society