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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Extremal ergodic measures and the finiteness property of matrix semigroups
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by Xiongping Dai, Yu Huang and MingQing Xiao PDF
Proc. Amer. Math. Soc. 141 (2013), 393-401 Request permission

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
  • 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
  • © Copyright 2012 American Mathematical Society
  • 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
  • MathSciNet review: 2996944