Abstract
This paper is concerned with invariance -scrambled sets under iterations. The main results are an extension of the compound invariance of Li–Yorke chaos and distributional chaos. New definitions of -scrambled sets in non-autonomous discrete systems are given. For a positive integer k, the properties and of Furstenberg families are introduced. It is shown that, for any positive integer k, for any , Furstenberg family has properties and , where denotes the family of all infinite subsets of whose upper density is not less than s. Then, the following conclusion is obtained. D is an -scrambled set of if and only if D is an -scrambled set of .
Highlights
Chaotic properties of a dynamical system have been extensively discussed since the introduction of the term chaos by Li and Yorke in 1975 [1] and Devaney in 1989 [2]
Theories related to scrambled sets are discussed extensively
The aim of this paper is to investigate the (F1, F2 )-scrambled sets of f 1,∞
Summary
Chaotic properties of a dynamical system have been extensively discussed since the introduction of the term chaos by Li and Yorke in 1975 [1] and Devaney in 1989 [2]. F ⊂ P is called a Furstenberg family if it is hereditary upwards, i.e., F1 ⊂ F2 and F1 ∈ F imply F2 ∈ F . Furstenberg family F is positive shift-invariant if F + i ∈ F for every F ∈ F and any i ∈ Z+ .
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