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 -