# Multiple Recurrence Theorem

Let $(X,\mathcal{B},\mu )$ be a probability space, and let ${T}_{i}:X\to X$ be measure-preserving transformations^{}, for $i$ between $1$ and $q$. Assume that all the transformations ${T}_{i}$ commute. If $E\subset X$ is a positive measure set $\mu (E)>0$, then, there exists $n\in \mathbb{N}$ such that

$$\mu (E\cap {T}_{1}^{-n}(E)\cap \mathrm{\cdots}\cap {T}_{q}^{-n}(E))>0$$ |

In other words there exist a certain time $n$ such that the subset of $E$ for which all elements return to $E$ simultaneously for all transformations ${T}_{i}$ is a subset of $E$ with positive measure. Observe that the theorem may be applied again to the set $G=E\cap {T}_{1}^{-n}(E)\cap \mathrm{\cdots}\cap {T}_{q}^{-n}(E)$, obtaining the existence of $m\in \mathbb{N}$ such that

$$\mu (G\cap {T}_{1}^{-m}(G)\cap \mathrm{\cdots}\cap {T}_{q}^{-m}(G))>0$$ |

so that

$$\mu (E\cap {T}_{1}^{-(m+n)}(E)\cap \mathrm{\cdots}\cap {T}_{q}^{-(m+n)}(E))\ge \mu (G\cap {T}_{1}^{-m}(G)\cap \mathrm{\cdots}\cap {T}_{q}^{-m}(G))>0$$ |

So we may conclude that, when $E$ has positive measure, there are infinite times for which there is a simultaneous return for a subset of $E$ with positive measure.

As a corollary, since the powers $T,{T}^{2}\mathrm{\cdots}{T}^{q}$ of a transformation $T$ commute, we have that, for $E$ with positive measure there exists $n\in \mathbb{N}$ such that

$$\mu (E\cap {T}^{-n}(E)\cap \mathrm{\cdots}\cap {T}^{-qn}(E))>0$$ |

As a consequence of the multiple recurrence theorem one may prove SzemerÃ©di’s Theorem about arithmetic progressions.

Title | Multiple Recurrence Theorem |
---|---|

Canonical name | MultipleRecurrenceTheorem |

Date of creation | 2015-03-20 0:29:34 |

Last modified on | 2015-03-20 0:29:34 |

Owner | Filipe (28191) |

Last modified by | Filipe (28191) |

Numerical id | 1 |

Author | Filipe (28191) |

Entry type | Theorem |

Synonym | PoincarÃ© Multiple Recurrence Theorem; FÃ¼rstenberg Recurrence theorem |

Related topic | PoincarÃ© Recurrence Theorem |