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

This paper is concerned with invariance ( F 1 , F 2 ) -scrambled sets under iterations. The main results are an extension of the compound invariance of Li–Yorke chaos and distributional chaos. New definitions of ( F 1 , F 2 ) -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 , f 1 , ∞ ) if and only if D is an ( M ¯ ( s ) , M ¯ ( t ) ) -scrambled set of ( X , f 1 , ∞ [ m ] ) .