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Sequence entropy tuples and mean sensitive tuples

Part of: Topological dynamics Ergodic theory

Published online by Cambridge University Press:  20 February 2023

JIE LI ,
CHUNLIN LIU Open the ORCID record for CHUNLIN LIU [Opens in a new window] ,
SIMING TU  and
TAO YU
JIE LI
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, P. R. China (e-mail: jiel0516@mail.ustc.edu.cn)
CHUNLIN LIU*
Affiliation:
CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
SIMING TU
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong 519082, P. R. China (e-mail: tusiming3@mail.sysu.edu.cn)
TAO YU
Affiliation:
Department of Mathematics, Shantou University, Shantou 515063, P. R. China (e-mail: ytnuo@mail.ustc.edu.cn)
*
e-mail: lcl666@mail.ustc.edu.cn
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Abstract

Using the idea of local entropy theory, we characterize the sequence entropy tuple via mean forms of the sensitive tuple in both topological and measure-theoretical senses. For the measure-theoretical sense, we show that for an ergodic measure-preserving system, the $\mu $-sequence entropy tuple, the $\mu $-mean sensitive tuple, and the $\mu $-sensitive in the mean tuple coincide, and give an example to show that the ergodicity condition is necessary. For the topological sense, we show that for a certain class of minimal systems, the mean sensitive tuple is the sequence entropy tuple.

Keywords

sequence entropy tuplesmean sensitive tuplessensitive in the mean tuples

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 1 , January 2024 , pp. 184 - 203
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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