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Metastability in the Furstenberg-Zimmer tower

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Avigad J., Towsner H.

Fundamenta Mathematicae

2010

tom 210

nr 3

243-268

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 $ω^{ω^{ω}}$th level.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Avigad J.

1 Towsner H.

1 2010

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Metastability in the Furstenberg-Zimmer tower