Metastability in the Furstenberg-Zimmer tower

2009/02/02 by Jeremy Avigad, Avigad, Jeremy, Henry Towsner +1 · 1 citation
Mathematics · #03F03 #37A25 #37A45 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:03F03 #msc:37A25 #msc:37A45

paper · pdf · doi:10.48550/arxiv.0902.0356

arxiv created 2010/06/16 · arxiv updated 2010/06/17

Abstract

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^ωωth level.

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