Some Iterative Properties of ( F 1 , F 2 ) -Chaos in Non-Autonomous Discrete Systems

This paper is concerned with invariance (F1,F2)-scrambled sets under iterations. The main results are an extension of the compound invariance of Li–Yorke chaos and distributional chaos. New definitions of (F1,F2)-scrambled sets in non-autonomous discrete systems are given. For a positive integer k, the properties P(k) and Q(k) of Furstenberg families are introduced. It is shown that, for any positive integer k, for any s ∈ [0, 1], Furstenberg family M(s) has properties P(k) and Q(k), where M(s) denotes the family of all infinite subsets of Z+ whose upper density is not less than s. Then, the following conclusion is obtained. D is an (M(s), M(t))-scrambled set of (X, f1,∞) if and only if D is an (M(s), M(t))-scrambled set of (X, f [m] 1,∞).