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On sensitive sets in topological dynamics. (English) Zbl 1153.37322
Summary: Notions of sensitive sets ($S$-sets) and regionally proximal sets ($Q$-sets) are introduced. It is shown that a transitive system is sensitive if and only if there is an $S$-set with Card$(S) \geqslant 2$, and for a transitive system each $S$-set is a $Q$-set. Moreover, the converse holds when $(X, T)$ is minimal. It turns out that each transitive $(X, T)$ has a maximal almost equicontinuous factor. According to the cardinalities of the $S$-sets, transitive systems are divided into several classes. Characterizations and examples are given for this classification both in minimal and transitive non-minimal settings. It is proved that for a transitive system any entropy set is an $S$-set, and consequently, a transitive system which has no uncountable $S$-sets has zero topological entropy. Moreover, it is shown that a transitive, non-minimal system with dense set of minimal points has an infinite $S$-set, and there exists a Devaney chaotic system which has no uncountable $S$-set. Finally, a non-minimal sensitive $E$-system is constructed such that each of its $S$-set has cardinality at most 4.

MSC:
37B40Topological entropy
37B05Transformations and group actions with special properties
37B10Symbolic dynamics
54H20Topological dynamics
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