Electronic Research Announcements in Mathematical Sciences (ERA-MS)

Realization of joint spectral radius via Ergodic theory

Pages: 22 - 30, Volume 18, 2011      doi:10.3934/era.2011.18.22

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Xiongping Dai - Department of Mathematics, Nanjing University, Nanjing, 210093, China (email)
Yu Huang - Department of Mathematics, Zhongshan University, Guangzhou 510275, China (email)
Mingqing Xiao - Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, United States (email)

Abstract: Based on the classic multiplicative ergodic theorem and the semi-uniform subadditive ergodic theorem, we show that there always exists at least one ergodic Borel probability measure such that the joint spectral radius of a finite set of square matrices of the same size can be realized almost everywhere with respect to this Borel probability measure. The existence of at least one ergodic Borel probability measure, in the context of the joint spectral radius problem, is obtained in a general setting.

Keywords:  The finiteness conjecture, random product of matrices, joint spectral radius.
Mathematics Subject Classification:  Primary 15A18, 37N40; Secondary 15A69, 65F15.

Received: August 2010;      Revised: March 2011;      Available Online: June 2011.