The centralizer of a rank-one flow
Given a Lebesgue probability space \((X,\chi,\mu)\) and a measurable flow \(\phi=\{\phi^ t: t\in\mathbb{R}\}\) of measure-preserving transformations of \(X\) define the centralizer of \(\phi\) as the set of all measure-preserving transformations of \(X\) commuting with each \(\phi^ t\), \(t\in\mathbb{R}\). Extending the notion of rank-one to flows and modifying some coding arguments the author has proved that King's closure theorem [\textit{J. King}, Ergodic Theory Dyn. Syst. 6, 363-384 (1986; Zbl 0581.47003)] has a full analogon for flows: if \(\phi\) is a rank-one flow then its centralizer coincides with the weak closure of \(\{\phi^ t: t\in\mathbb{R}\}\). (The weak topology is generated by the weak convergence: a sequence \((T_ i: i\in\mathbb{N})\) of measure-preserving transformations of \(X\) converges weakly to \(S\) if and only if \(\mu(T^{-1}_ i(A)\Delta S^{-1}(A))\to 0\) for each \(A\in\chi\)).
scientific article; zbMATH DE number 474988