On $n$-scrambled tuples and distributional chaos in a sequence

TL;DR

This paper investigates the relationship between n-scrambled tuples and distributional chaos in dynamical systems, extending previous work on chaos types and constructing examples with specific chaotic properties.

Contribution

It extends the understanding of chaos by analyzing n-scrambled tuples and their attraction properties, and constructs systems with precise distributional chaos levels.

Findings

Established relations between n-scrambled tuples and attraction properties

Extended previous chaos results to sequence-based distributional chaos

Constructed a system that is n-distributionally chaotic but not (n+1)-chaotic

Abstract

The main aim of the present paper is to study relations between $n$-scrambled tuples and their attraction-adherence properties with respect to various sequences of integers. This extends previous research on relations between chaos in the sense of Li and Yorke and distributional chaos with respect to a given sequence. Moreover, we construct a system which is $n$-distributionally chaotic but not $(n+1)$-chaotic.