On embeddability of automorphisms into measurable flows from the point of view of self-joining properties

Abstract

We compare self-joining- and embeddability properties. In particular, we prove that a measure preserving flow $(T_t{)}_{t\in \mathbb{R}}$ with $T_1$ ergodic is 2-fold quasi-simple (2-fold distally simple) if and only if $T_1$ is 2-fold quasi-simple (2-fold distally simple). We also show that the Furstenberg-Zimmer decomposition for a flow $(T_t{)}_{t\in \mathbb{R}}$ with $T_1$ ergodic with respect to any flow factor is the same for $(T_t{)}_{t\in \mathbb{R}}$ and for $T_1$. We give an example of a 2-fold quasi-simple flow disjoint from simple flows and whose time-one map is simple. We describe two classes of flows (flows with minimal self-joining property and flows with the so-called Ratner property) whose time-one maps have unique embeddings into measurable flows. We also give an example of a 2-fold simple flow whose time-one map has more than one embedding.

arXiv comment64 pages. This is an extended version of the paper accepted for publication in Fundamenta Mathematicae. In comparison to the arXiv v1, the paper has been shortened and rearranged, figures were removed, no other other substantial changes have been made