# Metastability in the Furstenberg-Zimmer tower - Mathematics > Dynamical Systems

Abstract: According to the Furstenberg-Zimmer structure theorem, everymeasure-preserving system has a maximal distal factor, and is weak mixingrelative to that factor. Furstenberg and Katznelson used this structuralanalysis of measure-preserving systems to provide a perspicuous proof ofSzemer\-edi-s theorem. Beleznay and Foreman showed that, in general, thetransfinite construction of the maximal distal factor of a separablemeasure-preserving system can extend arbitrarily far into the countableordinals. Here we show that the Furstenberg-Katznelson proof does not requirethe full strength of the maximal distal factor, in the sense that the proofonly depends on a combinatorial weakening of its properties. We show that thiscombinatorially weaker property obtains fairly low in the transfiniteconstruction, namely, by the $\omega^{\omega^\omega}$th level.