Metastability in the Furstenberg–Zimmer tower

Jeremy Avigad¹ ; Henry Towsner²

¹Department of Philosophy and Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213, U.S.A.
²Department of Mathematics University of California Los Angeles, CA 90095-1555, U.S.A.

Fundamenta Mathematicae, Tome 210 (2010) no. 3, pp. 243-268

Résumé

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.

Détail

DOI : 10.4064/fm210-3-2

Mots-clés : according furstenberg zimmer structure theorem every measure preserving system has maximal distal factor weak mixing relative factor furstenberg katznelson structural analysis measure preserving systems provide perspicuous proof szemer dis theorem beleznay foreman showed general transfinite construction maximal distal factor separable measure preserving system extend arbitrarily far countable ordinals here furstenberg katznelson proof does require full strength maximal distal factor sense proof only depends combinatorial weakening its properties combinatorially weaker property obtains fairly low transfinite construction namely omega omega omega level