Shifts, rotations and distributional chaos

Let Rr0,Rr1:S1⟶S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{r_{0}}, R_{r_{1}}: \mathbb{S}^{1}\longrightarrow \mathbb{S} ^{1}$\end{document} be rotations on the unit circle S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{S}^{1}$\end{document} and define f:Σ2×S1⟶Σ2×S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f: \varSigma _{2}\times \mathbb{S}^{1}\longrightarrow \varSigma _{2}\times \mathbb{S}^{1}$\end{document} as f(x,t)=(σ(x),Rrx1(t)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f(x, t)=\bigl(\sigma (x), R_{r_{x_{1}}}(t)\bigr), $$\end{document} for x=x1x2⋯∈Σ2:={0,1}N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x=x_{1}x_{2}\cdots \in \varSigma _{2}:=\{0, 1\}^{\mathbb{N}}$\end{document}, t∈S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\in \mathbb{S}^{1}$\end{document}, where σ:Σ2⟶Σ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma: \varSigma _{2}\longrightarrow \varSigma _{2}$\end{document} is the shift, and r0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{0}$\end{document} and r1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{1}$\end{document} are rotational angles. It is first proved that the system (Σ2×S1,f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\varSigma _{2}\times \mathbb{S}^{1}, f)$\end{document} exhibits maximal distributional chaos for any r0,r1∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{0}, r_{1}\in \mathbb{R}$\end{document} (no assumption of r0,r1∈R∖Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{0}, r_{1}\in \mathbb{R}\setminus \mathbb{Q}$\end{document}), generalizing Theorem 1 in Wu and Chen (Topol. Appl. 162:91–99, 2014). It is also obtained that (Σ2×S1,f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\varSigma _{2}\times \mathbb{S}^{1}, f)$\end{document} is cofinitely sensitive and (Mˆ1,Mˆ1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\hat{\mathscr{M}} ^{1}, \hat{\mathscr{M}}^{1})$\end{document}-sensitive and that (Σ2×S1,f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\varSigma _{2}\times \mathbb{S}^{1}, f)$\end{document} is densely chaotic if and only if r1−r0∈R∖Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r_{1}-r_{0} \in \mathbb{R}\setminus \mathbb{Q}$\end{document}.