Metastability in the Furstenberg–Zimmer tower

Jeremy Avigad,Henry Towsner

doi:10.4064/fm210-3-2

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 Szemeredi’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 ωω ω th level.

Highlights

Let X = (X, B, μ, T ) be a measure-preserving system, that is, a finite measure space (X, B, μ) together with a measure-preserving transformation, T
A (T -invariant) factor Y of such a system is said to be distal if it is the last element of an increasing finite or transfinite sequence (Yα)α≤θ of factors, such that Y0 is the trivial factor, for each α < θ, Yα+1 is compact relative to Yα, and for each limit ordinal γ ≤ θ, Yγ is the limit of the preceding factors
Even for the original version of the theorem, the Furstenberg–Katznelson proof

Summary

Introduction

Furstenberg [7] proceeded to give an ergodic-theoretic proof of Szemeredi’s theorem that used only a finite sequence of compact extensions of the trivial factor. The Furstenberg–Zimmer structure theorem shows that any measurepreserving system X has a maximal distal factor, that is, a factor Y that is built up using a transfinite sequence of compact extensions; and that X is weak mixing relative to Y. Gk in L2(X ) such that mini≤k f − gi y < δ for almost every y in X Another factor Z ⊇ Y is said to be a compact extension of Y if every element of Z is a limit of functions that are almost periodic relative to Y.

Integration in

But this may be rewritten as