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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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