Shifts, rotations and distributional chaos

Abstract

Let R r 0 , R r 1 : S 1 − → S 1 be rotations on the unit circle S 1 and define f : Σ 2 × S 1 − → Σ 2 × S 1 as f (x, t) = (σ (x), R r x 1 (t)), for x = x 1 x 2 • • • ∈ Σ 2 := {0, 1} N , t ∈ S 1 , where σ : Σ 2 − → Σ 2 is the shift, and r 0 and r 1 are rotational angles. It is first proved that the system (Σ 2 × S 1 , f) exhibits maximal distributional chaos for any r 0 , r 1 ∈ R (no assumption of r 0 , r 1 ∈ R \ Q), generalizing Theorem 1 in Wu and Chen (Topol. Appl. 162:91-99, 2014). It is also obtained that (Σ 2 × S 1 , f) is cofinitely sensitive and (M 1 ,M 1)-sensitive and that (Σ 2 × S 1 , f) is densely chaotic if and only if r 1-r 0 ∈ R \ Q.