On the Weakest Version of Distributional Chaos

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 DC2 1 2 which is a topological invariant and implies Li-Yorke chaos, similarly as types DC1 and DC2; but unlike them, strict DC2 1 2 systems must have zero topological entropy. 2000 Mathematics Subject Classification. Primary 37D45; 37B40.