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Sufficient conditions under which a transitive system is chaotic

Published online by Cambridge University Press:  04 November 2009

E. AKIN ,
E. GLASNER ,
W. HUANG ,
S. SHAO  and
X. YE
E. AKIN
Affiliation:
Mathematics Department, The City College, 137 Street and Convent Avenue, New York City, NY 10031, USA (email: ethanakin@earthlink.net)
E. GLASNER
Affiliation:
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel (email: glasner@math.tau.ac.il, eli.glasner@gmail.com)
W. HUANG
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China (email: wenh@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn)
S. SHAO
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China (email: wenh@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn)
X. YE
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China (email: wenh@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn)
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Abstract

Let (X,T) be a topologically transitive dynamical system. We show that if there is a subsystem (Y,T) of (X,T) such that (X×Y,T×T) is transitive, then (X,T) is strongly chaotic in the sense of Li and Yorke. We then show that many of the known sufficient conditions in the literature, as well as a few new results, are corollaries of this statement. In fact, the kind of chaotic behavior we deduce in these results is a much stronger variant of Li–Yorke chaos which we call uniform chaos. For minimal systems we show, among other results, that uniform chaos is preserved by extensions and that a minimal system which is not uniformly chaotic is PI.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 5 , October 2010 , pp. 1277 - 1310
Copyright
Copyright © Cambridge University Press 2009

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