Distributionally scrambled invariant sets in a compact metric space

The paper solves a question posed by Oprocha on the existence of invariant distributionally chaotic scrambled sets. We show, among other things, that a continuous map ff acting on compact metric space (X,d)(X,d) with a weak specification property, fixed point, and infinitely many mutually distinct periods has a dense Mycielski (i.e., cc dense set of type FσFσ) invariant distributionally scrambled set. As a consequence, we describe a class of maps with a distributionally scrambled invariant set of full Lebesgue measure in the case when XX is a kk-dimensional cube.