Birkhoff Recurrence Theorem

Let $T:X\to X$ be a continuous tranformation in a compact metric space $X$. Then, there exists some point $x\in X$ that is recurrent to $T$, that is, there exists a sequence ${(n_k)}_k$ such that $T^{n_k}(x)\to x$ when $k\to \mathrm{\infty }$.

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 measures 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 generalization of Birkhoff recurrence theorem for multiple commuting transformations, 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