Furstenberg family and chaos

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.