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Mean equicontinuity and mean sensitivity

Published online by Cambridge University Press:  04 August 2014

JIAN LI Open the ORCID record for JIAN LI [Opens in a new window] ,
SIMING TU  and
XIANGDONG YE
JIAN LI
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong 515063,PR China email lijian09@mail.ustc.edu.cn
SIMING TU
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China email tsming@mail.ustc.edu.cn, yexd@ustc.edu.cn
XIANGDONG YE
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China email tsming@mail.ustc.edu.cn, yexd@ustc.edu.cn
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Abstract

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the latter property must have zero entropy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 8 , December 2015 , pp. 2587 - 2612
Copyright
© Cambridge University Press, 2014 

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