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