Abstract
A Furstenberg family \(\mathcal{F}\) is a family, consisting of some subsets of the set of positive integers, which is hereditary upwards, i.e. A ⊂ B and A ∈ \(\mathcal{F}\) imply B ∈ \(\mathcal{F}\). 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 \(\mathcal{F}\), the definition of \(\mathcal{F}\)-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 \(\mathcal{F}\)-scrambled pairs corresponding respectively to suitable Furstenberg family \(\mathcal{F}\). In the present paper we explore the basic properties of the set of \(\mathcal{F}\)-scrambled pairs of a system. The generically \(\mathcal{F}\)-chaotic system and the generically strongly \(\mathcal{F}\)-chaotic system are defined. A criterion for a generically strongly \(\mathcal{F}\)-chaotic system is showed.
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