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Analogues of Auslander–Yorke theorems for multi-sensitivity

Published online by Cambridge University Press:  22 September 2016

WEN HUANG ,
SERGIĬ KOLYADA  and
GUOHUA ZHANG
WEN HUANG
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China email wenh@mail.ustc.edu.cn
SERGIĬ KOLYADA
Affiliation:
Institute of Mathematics, NASU, Tereshchenkivs’ka 3, 01601 Kyiv, Ukraine email skolyada@imath.kiev.ua
GUOHUA ZHANG
Affiliation:
School of Mathematical Sciences and LMNS, Fudan University and Shanghai Center for Mathematical Sciences, Shanghai 200433, China email chiaths.zhang@gmail.com
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Abstract

We study multi-sensitivity and thick sensitivity for continuous surjective selfmaps on compact metric spaces. Our main result states that a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor. This is an analog of the Auslander–Yorke dichotomy theorem: a minimal system is either sensitive or equicontinuous. Furthermore, we introduce the concept of a syndetically equicontinuous point, and we prove that a transitive system is either thickly sensitive or contains syndetically equicontinuous points, which is a refinement of another well-known result of Akin, Auslander and Berg.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 2 , April 2018 , pp. 651 - 665
Copyright
© Cambridge University Press, 2016 

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