Metastability in the Furstenberg-Zimmer tower

Jeremy Avigad, and Henry Towsner. "Metastability in the Furstenberg-Zimmer tower." Fundamenta Mathematicae 210.3 (2010): 243-268. <http://eudml.org/doc/283345>.

@article{JeremyAvigad2010,
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 $ω^\{ω^\{ω\}\}$th level.},
author = {Jeremy Avigad, Henry Towsner},
journal = {Fundamenta Mathematicae},
keywords = {measure-preserving systems; maximal distal factor; metastability; ergodic theory; mathematical logic},
language = {eng},
number = {3},
pages = {243-268},
title = {Metastability in the Furstenberg-Zimmer tower},
url = {http://eudml.org/doc/283345},
volume = {210},
year = {2010},
}

TY - JOUR
AU - Jeremy Avigad
AU - Henry Towsner
TI - Metastability in the Furstenberg-Zimmer tower
JO - Fundamenta Mathematicae
PY - 2010
VL - 210
IS - 3
SP - 243
EP - 268
AB - 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.
LA - eng
KW - measure-preserving systems; maximal distal factor; metastability; ergodic theory; mathematical logic
UR - http://eudml.org/doc/283345
ER -