On sensitive sets in topological dynamics

In this paper 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 ) ⩾ 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.