Dense invariant open distributionally scrambled sets and closed distributionally scrambled sets

It is known that in a compact dynamical system, the whole space can be a Li–Yorke scrambled set, but this does not hold for distributional chaos. In this paper we prove that the complement of a distributionally scrambled set must be an infinite set. Then we give an example of an uncountable dense invariant open extremal distributionally scrambled set which is the complement of a countable infinite set. This presents one kind of the “largest” (from the topological point of view) distributionally scrambled set in a compact dynamical system. Moreover, we construct an uncountable closed distributionally scrambled set.