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When is a dynamical system mean sensitive?

Published online by Cambridge University Press:  06 November 2017

FELIPE GARCÍA-RAMOS
Affiliation:
Instituto de Fisica, Universidad Autonoma de San Luis Potosi, Manuel Nava 6, SLP, Mexico78290 email felipegra@yahoo.com Catedras CONACyT, Av. Insurgentes Sur 1582, Del. Benito Juárez, Mexico City, Mexico03940
JIE LI
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, PR China email jiel0516@mail.ustc.edu.cn Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, PR China
RUIFENG ZHANG
Affiliation:
School of Mathematics, Hefei University of Technology, Hefei, Anhui, 230009, PR China email rfzhang@mail.ustc.edu.cn

Abstract

This article is devoted to studying which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things, we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand, we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive. As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.

Type
Original Article
Copyright
© Cambridge University Press, 2017 

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