# Birkhoff Recurrence Theorem

Let $T:X\rightarrow X$ 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 $T^{n_{k}}(x)\rightarrow x$ when $k\rightarrow\infty$.

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 PoincarÃ© 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 BirkhoffRecurrenceTheorem 2015-03-20 0:56:48 2015-03-20 0:56:48 Filipe (28191) Filipe (28191) 2 Filipe (28191) Theorem PoincarÃ© Recurrence Theorem