proof of Poincaré recurrence theorem 1

Let An=k=nf-kE. Clearly, EA0 and AiAj when ji. Also, Ai=fj-iAj, so that μ(Ai)=μ(Aj) for all i,j0, by the f-invariance of μ. Now for any n>0 we have E-AnA0-An, so that


Hence μ(E-An)=0 for all n>0, so that μ(E-n=1An)=μ(n=1E-An)=0. But E-n=1An is precisely the set of those xE such that for some n and for all k>n we have fk(x)E.

Title proof of Poincaré recurrence theorem 1
Canonical name ProofOfPoincareRecurrenceTheorem1
Date of creation 2013-03-22 14:29:56
Last modified on 2013-03-22 14:29:56
Owner Koro (127)
Last modified by Koro (127)
Numerical id 5
Author Koro (127)
Entry type Proof
Classification msc 37A05
Classification msc 37B20