Furstenberg family and chaos

Furstenberg family ℱ is a family, consisting of some subsets of the set of positive integers, which is hereditary upwards, i.e. A ⊂ B and A ∈ ℱ imply B ∈ ℱ . For a given system (i.e., a pair of a complete metric space and a continuous self-map of the space) and for a Furstenberg family ℱ , the definition of ℱ -scrambled pairs of points in the space has been given, which brings the well-known scrambled pairs in Li-Yorke sense and the scrambled pairs in distribution sense to be ℱ -scrambled pairs corresponding respectively to suitable Furstenberg family ℱ . In the present paper we explore the basic properties of the set of ℱ -scrambled pairs of a system. The generically ℱ -chaotic system and the generically strongly ℱ -chaotic system are defined. A criterion for a generically strongly ℱ -chaotic system is showed.