The equivalence relationship between Li-Yorke $\delta$-chaos and distributional $\delta$-chaos in a sequence

TL;DR

This paper establishes the equivalence between Li-Yorke $ ext{delta}$-chaos and distributional $ ext{delta}$-chaos in sequences, providing new insights into chaotic dynamics and their interrelations.

Contribution

It proves the equivalence of Li-Yorke and distributional chaos in sequences and explores conditions under which chaos types coincide or imply each other.

Findings

Distributional $ ext{delta}$-scramble pairs form a $G_ ext{delta}$ set.

Li-Yorke $ ext{delta}$-chaos is equivalent to distributional $ ext{delta}$-chaos in sequences.

Certain transitive systems imply distributional chaos in sequences.

Abstract

In this paper, we discuss the relationship between Li-Yorke chaos and distributional chaos in a sequence. We point out the set of all distributional $\delta$-scramble pairs in the sequence $Q$ is a $G_\delta$ set, and prove that Li-Yorke $\delta$-chaos is equivalent to distributional $\delta$-chaos in a sequence, a uniformly chaotic set is a distributional scramble set in some sequence and a class of transitive system implies distributional chaos in a sequence.