TY - JOUR

T1 - Sufficient conditions under which a transitive system is chaotic

AU - Akin, E.

AU - Glasner, E.

AU - Huang, W.

AU - Shao, S.

AU - Ye, X.

PY - 2010/10

Y1 - 2010/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=79451475124&partnerID=8YFLogxK

U2 - 10.1017/S0143385709000753

DO - 10.1017/S0143385709000753

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AN - SCOPUS:79451475124

SN - 0143-3857

VL - 30

SP - 1277

EP - 1310

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 5

ER -