On ℱ-sensitive pairs

In the present paper, we define sensitive pairs via Furstenberg families and discuss the relation of three definitions: sensitivity, ℱ-sensitivity and ℱ-sensitive pairs, see Theorem 1. For transitive systems, we give some sufficient conditions to ensure the existence of ℱ-sensitive pairs. In particular, each non-minimal E system (M system, P system) has positive lower density (ℱs, ℱr resp.)-sensitive pairs almost everywhere. Moreover, each non-minimal M system is ℱts-sensitive. Finally, by some examples we show that: (1) ℱ-sensitivity can not imply the existence of ℱ-sensitive pairs. That means there exists an ℱ-sensitive system, which has no ℱ-sensitive pairs. (2) There is no immediate relation between the existence of sensitive pairs and Li-Yorke chaos, i.e., there exists a system (X,f) without Li-Yorke scrambled pairs, which has κℬ-sensitive pairs almost everywhere. (3) If the system (G,f) is sensitive, where G is a finite graph, then it has κℬ-sensitive pairs almost everywhere.