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

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.