Scrambled and distributionally scrambledn-tuples

Abstract

This article investigates the relation between the distributional chaos and the existence of a scrambled triple. We show that for a continuous mapping f acting on a compact metric space (X, d), the possession of an infinite extremal distributionally scrambled set is not sufficient for the existence of a scrambled triple. We also construct an invariant Mycielski set with an uncountable extremal distributionally scrambled set without any scrambled triple.

On ‘observable’ Li–Yorke tuples for interval maps

“…However, there are dynamical systems on the Cantor set such that each d-tuple of distinct points is LY, but no uncountable set with this property for (d + 1)-tuples exists [22]. In fact, the system need not have LY (d + 1)-tuples at all [14]. In many cases, e.g.…”