On the Weakest Version of Distributional Chaos

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[Formula: see text] which is a topological invariant and implies Li–Yorke chaos, similarly as types DC1 and DC2; but unlike them, strict DC[Formula: see text] systems must have zero topological entropy.