On the weakest version of distributional chaos

TL;DR

This paper refines the understanding of distributional chaos type 3, showing it is a very weak and unstable form of chaos that can lack Li-Yorke pairs, but a strengthened version DC2.5 is a topological invariant implying Li-Yorke chaos.

Contribution

The paper corrects previous results on distributional chaos type 3 and introduces a stronger, topologically invariant version DC2.5 that guarantees Li-Yorke chaos.

Findings

DC3 chaos can be unstable and lack Li-Yorke pairs.

DC2.5 is a topological invariant implying Li-Yorke chaos.

Strict DC2.5 systems have zero topological entropy.

Abstract

The aim of the paper is to correct and improve some results concerning distributional chaos of type 3. We show that in a general compact metric space, distributional chaos of type 3, denoted DC3, even when assuming the existence of an uncountable scrambled set, is a very weak form of chaos. In particular, (i) the chaos can be unstable (it can be destroyed by conjugacy), and (ii) such an unstable system may contain no Li-Yorke pair. However, the definition can be strengthened to get DC$2\frac{1}{2}$ which is a topological invariant and implies Li-Yorke chaos, similarly as types DC1 and DC2; but unlike them, strict DC$2\frac{1}{2}$ systems must have zero topological entropy.