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

A. M. Stepin ; A. M. Eremenko

Sbornik. Mathematics, Tome 195 (2004) no. 12, pp. 1795-1808

Résumé

The problem of the inclusion in a flow is considered for a measure-preserving transformation. It is shown that if a transformation $T$ has a simple spectrum, then the set of flows including $T$ – 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.