# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Metastability in the Furstenberg–Zimmer tower

### Tom 210 / 2010

Fundamenta Mathematicae 210 (2010), 243-268 MSC: 37A05, 03F03, 03E15. DOI: 10.4064/fm210-3-2

#### Streszczenie

According to the Furstenberg–Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemerédi's theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg–Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the $\omega^{\omega^\omega}$th level.

#### Autorzy

and Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, U.S.A.
e-mail
• Henry TowsnerDepartment of Mathematics
University of California
Los Angeles, CA 90095-1555, U.S.A.
e-mail

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