Multiple Recurrence Theorem

Let (X,,μ) be a probability space, and let Ti:XX be measure-preserving transformationsPlanetmathPlanetmath, for i between 1 and q. Assume that all the transformations Ti commute. If EX is a positive measure set μ(E)>0, then, there exists n such that


In other words there exist a certain time n such that the subset of E for which all elements return to E simultaneously for all transformations Ti is a subset of E with positive measure. Observe that the theorem may be applied again to the set G=ET1-n(E)Tq-n(E), obtaining the existence of m such that


so that


So we may conclude that, when E has positive measure, there are infinite times for which there is a simultaneous return for a subset of E with positive measure.

As a corollary, since the powers T,T2Tq of a transformation T commute, we have that, for E with positive measure there exists n such that


As a consequence of the multiple recurrence theorem one may prove Szemerédi’s Theorem about arithmetic progressions.

Title Multiple Recurrence Theorem
Canonical name MultipleRecurrenceTheorem
Date of creation 2015-03-20 0:29:34
Last modified on 2015-03-20 0:29:34
Owner Filipe (28191)
Last modified by Filipe (28191)
Numerical id 1
Author Filipe (28191)
Entry type Theorem
Synonym Poincaré Multiple Recurrence Theorem; Fürstenberg Recurrence theorem
Related topic Poincaré Recurrence Theorem