A simple map with no prime factors

Abstract

An ergodic measure-preserving transformationT of a probability space is said to be simple (of order 2) if every ergodic joining λ ofT with itself is eitherμ×μ or an off-diagonal measureμS, i.e.,μS(A×B)=μ(A∩S;−n;B) for some invertible, measure preservingS commuting withT. Veech proved that ifT is simple thenT is a group extension of any of its non-trivial factors. Here we construct an example of a weakly mixing simpleT which has no prime factors. This is achieved by constructing an action of the countable Abelian group ℤ⊕G, whereG=⊕i=1∞ ℤ2, such that the ℤ-subaction is simple and has centralizer coinciding with the full ℤ⊕G-action.