Non-weakly almost periodic recurrent points and distributionally scrambled sets on

Let R r 0 , R r 1 : S 1 → S 1 be irrational rotations and define f : Σ 2 × S 1 → Σ 2 × S 1 by f ( x , t ) = ( σ ( x ) , R r x 1 ( t ) ) , for x = x 1 x 2 ⋯ ∈ Σ 2 , t ∈ S 1 , where Σ 2 = { 0 , 1 } N , S 1 is the unit circle, σ : Σ 2 → Σ 2 is a shift, and r 0 and r 1 are rotational angles. In this paper, it is proved that the system ( Σ 2 × S 1 , f ) has an uncountable distributionally ϵ-scrambled set for any 0 < ϵ ⩽ diam Σ 2 × S 1 = 1 in which each point is recurrent but is not weakly almost periodic. This is a positive answer to a question posed in Wang et al. (2003) [6]. Furthermore, the following results are obtained: ch distributionally scrambled set of f is not invariant; e system ( Σ 2 × S 1 , f ) is Li–Yorke sensitive.