Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems

Assume that (Y,ρ) is a nontrivial complete metric space, and that (Y,g1,∞) is a time-varying discrete dynamical system (T-VDDS), which is given by sequences (gl)l=1∞ of continuous selfmaps gl:Y→Y. In this paper, for a given Furstenberg family G and a given T-VDDS (Y,g1,∞), G-scrambled pairs of points of the system (Y,g1,∞) (which contains the well-known scrambled pairs) are provided. Some properties of the set of G-scrambled pairs of a given T-VDDS (Y,g1,∞) are studied. Moreover, the generically G-chaotic T-VDDS and the generically strongly G-chaotic T-VDDS are defined. A sufficient condition for a given T-VDDS to be generically strongly G-chaotic is also presented.