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

Joanna Kułaga-Przymus¹

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland and Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland

Fundamenta Mathematicae, Tome 230 (2015) no. 1, pp. 15-76.

Résumé

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.

Détail

DOI : 10.4064/fm230-1-2

Keywords: compare self joining embeddability properties particular prove measure preserving flow mathbb ergodic fold quasi simple resp fold distally simple only fold quasi simple resp fold distally simple furstenberg zimmer decomposition flow mathbb ergodic respect flow factor mathbb example fold quasi simple flow disjoint simple flows whose time one map simple describe classes flows flows minimal self joining property flows so called ratner property whose time one maps have unique embeddings measurable flows example fold simple flow whose time one map has embedding