Birkhoff Recurrence Theorem

Let T:XX be a continuousPlanetmathPlanetmath tranformation in a compactPlanetmathPlanetmath metric space X. Then, there exists some point xX that is recurrent to T, that is, there exists a sequencePlanetmathPlanetmath (nk)k such that Tnk(x)x when k.

Several proofs of this theorem are available. It may be obtained from topological arguments together with Zorn’s lemma. It is also a consequence of Krylov-Bogolyubov theorem, or existence of invariant probability measuresMathworldPlanetmath theorem, which asserts that every continuous transformation in a compact metric space admits an invariant probability measure, and an application of Poincaré Recurrence theorem to that invariant probability measure yields Birkhoff Recurrence theorem.

There is also a generalizationPlanetmathPlanetmath of Birkhoff recurrence theorem for multiple commuting transformationsPlanetmathPlanetmath, known as Birkhoff Multiple Recurrence theorem.

Title Birkhoff Recurrence Theorem
Canonical name BirkhoffRecurrenceTheorem
Date of creation 2015-03-20 0:56:48
Last modified on 2015-03-20 0:56:48
Owner Filipe (28191)
Last modified by Filipe (28191)
Numerical id 2
Author Filipe (28191)
Entry type Theorem
Related topic Poincaré Recurrence Theorem