Abstract

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

Highlights

  • Introduction and preliminariesA discrete dynamical system is a pair (X, g), where X is a compact metric space and g : X −→ X is a continuous map

  • The set of Li–Yorke pairs of modulus δ is denoted by LY(g, δ) and the set of Li– Yorke pairs by LY(g)

  • This paper considers the following dynamical system

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Summary

Introduction

Introduction and preliminariesA discrete dynamical system (briefly, dynamical system) is a pair (X, g), where X is a compact metric space and g : X −→ X is a continuous map. According to Schweizer and Smítal [9], a dynamical system (X, g) is distributionally εchaotic for some ε > 0 if there exists an uncountable subset S ⊂ X such that for any pair of distinct points x, y ∈ S, one has that Fx∗,y(t, g) = 1 for all t > 0 and Fx,y(ε, g) = 0.

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Conclusion

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