A simple map with no prime factors
Let \((X,{\mathcal F}, \mu)\) be a probability space and \(T:X \to X\) be an invertible ergodic measure-preserving map. A self-joining of the dynamical system \((X,{\mathcal F}, \mu, T)\) is a measure \(\lambda\) on \({\mathcal F}\bigotimes{\mathcal F}\) which is \(T\times T\) invariant and has both projections equal to \(\mu\). The map \(T\) is said to be simple (of order 2) if every ergodic self-joining of \(T\) is either \(\mu\times\mu\) or an off-diagonal measure \(\mu_{S}\), i.e., \(\mu_{S}(A\times B)=\mu(A\cap S^{-n}B)\) for some invertible measure-preserving map \(S\) commuting with \(T\). It is known [\textit{W. A. Veech}, Monatsh. Math. 94, 335-341 (1982; Zbl 0499.28016)] that if the map \(T\) is simple then it is a group extension of any of its non-trivial factors. On the other hand, if \(T\) is simple then it is either weakly mixing or has a discrete spectrum. Many examples of weakly mixing systems are now known, and all of them are either themselves prime or have prime factors. This leads to the question, posed by \textit{A. del Junco} and \textit{D. Rudolph} [Ergodic Theory Dyn. Syst. 7, 531-557 (1987; Zbl 0646.60010)], must a weakly mixing simple map have a non-trivial prime factor? The main result of the present paper is the negative answer to this question. It is constructed an action of the countable discrete group \(Z\otimes G\), where \(G=\bigotimes_{i=1}^{\infty}Z_{2}\), for which the map \(T\) generating the \(Z\)-sub-action is weakly mixing and simple, with centralizer coinciding with the full \(Z\otimes G\) action. Some generalizations of this construction are discussed as well.
scientific article; zbMATH DE number 1251452
scientific article