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

### Volume 230 / 2015

#### 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 (resp. $2$-fold distally simple) if and only if $T_1$ is $2$-fold quasi-simple (resp. $2$-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.