Non-weakly almost periodic recurrent points and distributionally scrambled sets on ∑₂ × S¹

Let R_r0 , R_r1 : S¹ → S¹ be irrational rotations and define ƒ : ∑₂ × S¹ → ∑₂ × S¹ by

ƒ(x, t) = (σ(x), R_rx1 (t)), 

for x = x₁x₂⋯∈ ∑₂, t ∈ S¹, where ∑₂ = {0, 1}^N, S¹ is the unit circle, σ : ∑₂ → ∑₂ is a shift, and r₀ and r₁ are rotational angles. In this paper, it is proved that the system (∑₂ × S¹, ƒ) has an uncountable distributionally ε-scrambled set for any 0 < ε ≤ diam Σ₂ × S¹ = 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: 
(1) each distributionally scrambled set of ƒ is not invariant; 
(2) the system (Σ₂ × S¹, ƒ) is Li–Yorke sensitive.