In this paper, we introduce the concept of regular shift invariant set and prove that a compact system having a regular shift invariant set has an uncountable distributively chaotic set in which each point is recurrent but not weakly almost periodic, in particular, a continuous self-map of an interval having positive topological entropy has an uncountable distributively chaotic set in which each point is recurrent but not weakly almost periodic.

compact system, recurrent point, weakly almost periodic point, topological entropy, distributional chaos.