Sufficient conditions under which a transitive system is chaotic

E. Akin*, E. Glasner, W. Huang, S. Shao, X. Ye

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)1277-1310
Number of pages34
JournalErgodic Theory and Dynamical Systems
Volume30
Issue number5
DOIs
StatePublished - Oct 2010

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