Weakly mixing operators on topological vector spaces
Let \(X\) be a topological space and let \(T\) be a continuous map from \(X\) into itself. The dynamical system \((X,T)\) is called weakly mixing if, for any four nonempty open sets \(U_1,\, U_2,\, V_1,\;V_2\subset X\), there exists some \(n\in\mathbb{N}\) such that \(T^n(U_1)\cap V_1\not=\emptyset\) and \(T^n(U_2)\cap V_2\not=\emptyset\). In the paper under review, the authors review known characterizations of weakly dynamical systems. In the second part of the paper, the authors obtain additional characterizations of, and sufficient conditions for, the (weak) mixing property in the case of linear dynamical systems, that is, dynamical systems \(T: X\to X\) given by continuous linear operators on a topological vector space \(X\). They end the paper collecting some open problems on the weak mixing property.
scientific article; zbMATH DE number 5925099
scientific article