On local aspects of topological weak mixing, sequence entropy and chaos (Q2928253)

Mixing and entropy are two fundamental notions in qualitative theory of dynamical systems which describe the complexity of the dynamics. \textit{F. Blanchard} and \textit{W. Huang} [Discrete Contin. Dyn. Syst. 20, No. 2, 275--311 (2008; Zbl 1151.37019)] showed that dynamical systems with positive entropy have a weakly mixing subset. In [\textit{P. Oprocha} and \textit{G. Zhang}, Stud. Math. 202, No. 3, 261--288 (2011; Zbl 1217.37012)], the authors introduced the concept of weakly mixing sets of order \(n\) and constructed a minimal invertible dynamical system which contains a nontrivial weakly mixing set of order \(2\) but does not contain any nontrivial weakly mixing set of order \(3\).NEWLINENEWLINEIn the paper under review, the authors extend their previous work to the general case. They construct, for any \(n\geq 2\), a minimal topological dynamical system that admits perfect weakly mixing sets of order \(n\) for which any weakly mixing set of order \(n+1\) is a singleton. In addition, a wider investigation is made of the relationships between properties of weakly mixing sets and other notions in topological dynamics, including regional proximality, notions of equicontinuity, chaos, and sequence entropy.