Coalescence of circle extensions of measure-preserving transformations

Abstract

AbstractWe prove that for each ergodic automorphism T:(X, ℬ, μ)→(X, ℬ, μ) for which we can find an element S∈C(T) such that the corresponding Z2-action (S, T) on (X, ℬ, μ) is free, there exists a circle valued cocycle φ such that the group extension Tφ is ergodic but is not coalescent. In particular, the existence of such a cocycle is proved for all ergodic rigid automorphisms. As a corollary, in the class of ergodic transformations of [0,1) × [0,1) given byfor each irrational α we find φ such that Tφ is not coalescent. In some special cases the group law of the centralizer is given.