Some Iterative Properties of [Formula: see text]-Chaos in Non-Autonomous Discrete Systems

This paper is concerned with invariance [Formula: see text]-scrambled sets under iterations. The main results are an extension of the compound invariance of Li–Yorke chaos and distributional chaos. New definitions of [Formula: see text]-scrambled sets in non-autonomous discrete systems are given. For a positive integer k, the properties [Formula: see text] and [Formula: see text] of Furstenberg families are introduced. It is shown that, for any positive integer k, for any [Formula: see text] , Furstenberg family [Formula: see text] has properties [Formula: see text] and [Formula: see text] , where [Formula: see text] denotes the family of all infinite subsets of [Formula: see text] whose upper density is not less than s. Then, the following conclusion is obtained. D is an [Formula: see text]-scrambled set of [Formula: see text] if and only if D is an [Formula: see text]-scrambled set of [Formula: see text].