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.

单词	BirkhoffRecurrenceTheorem
释义	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.
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