Multiple Recurrence Theorem
In other words there exist a certain time such that the subset of for which all elements return to simultaneously for all transformations is a subset of with positive measure. Observe that the theorem may be applied again to the set , obtaining the existence of such that
So we may conclude that, when has positive measure, there are infinite times for which there is a simultaneous return for a subset of with positive measure.
As a corollary, since the powers of a transformation commute, we have that, for with positive measure there exists such that
As a consequence of the multiple recurrence theorem one may prove SzemerÃ©di’s Theorem about arithmetic progressions.
|Title||Multiple Recurrence Theorem|
|Date of creation||2015-03-20 0:29:34|
|Last modified on||2015-03-20 0:29:34|
|Last modified by||Filipe (28191)|
|Synonym||PoincarÃ© Multiple Recurrence Theorem; FÃ¼rstenberg Recurrence theorem|
|Related topic||PoincarÃ© Recurrence Theorem|