Non-unique inclusion in a flow and vast centralizer of a generic measure-preserving transformation

Abstract

The problem of the inclusion in a?flow is considered for a?measure-preserving transformation. It is shown that if a?transformation has a?simple spectrum, then the set of flows including? ? provided that it is not empty ? consists either of a?unique element or of infinitely many spectrally non-equivalent flows.It is proved that, generically, inclusions in a?flow are maximally non-unique in the following sense: the centralizer of a?generic transformation contains a?subgroup isomorphic to an infinite-dimensional torus. The corresponding proof is based on the so-called dynamical alternative, a?topological analogue of Fubini's theorem, a?fundamental fact from descriptive set theory about the almost openness of analytic sets, and Dougherty's lemma describing conditions ensuring that the image of a?separable metric space is a?second-category set.