# 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 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 |
---|---|

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 | PoincarÃ© Recurrence Theorem |