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

Volume 230 / 2015

Joanna Kułaga-Przymus Fundamenta Mathematicae 230 (2015), 15-76 MSC: Primary 37A05, 37A10; Secondary 37A30, 37A35. DOI: 10.4064/fm230-1-2

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.

Authors

  • Joanna Kułaga-PrzymusInstitute 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
    e-mail

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