On n-scrambled sets

Abstract

This paper investigates n-scrambled sets in the sense of distribution and Li-Yorke, focusing on the equivalence of n-scrambled sets and the existence of uncountable n-scrambled sets. For the equivalence, n-scrambled sets can imply m-scrambled sets in both senses for any 2≤m<n, but their inverses require certain conditions to hold. A method is provided for constructing an uncountable n-scrambled set in the distribution and Li-Yorke senses, which also verifies the equivalence between n-scrambled sets. For the existence, it is shown that distributionally n-scrambled tuples can be derived from the weak specification property (WSP), a 1-periodic and a non-1-periodic point. And the Li-Yorke (n+1)-scrambled tuple can be characterized by an n-scrambled tuple utilizing the SP. Finally, the uncountable Li-Yorke n-scrambled set is established based on the weak mixing, and the uncountable mean Li-Yorke n-scrambled set is induced by the shadowing and Devaney chaos.