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

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.