proof of Poincaré recurrence theorem 2

Let {Un:n} be a basis of open sets for X, and for each n define


From theorem 1 we know that μ(Un)=0. Let N=nUn. Then μ(N)=0. We assert that if xX-N then x is recurrent. In fact, given a neighborhoodMathworldPlanetmathPlanetmath U of x, there is a basic neighborhood Un such that xUnU, and since xN we have that xUn-Un which by definition of Un means that there exists n1 such that fn(x)UnU; thus x is recurrent.

Title proof of Poincaré recurrence theorem 2
Canonical name ProofOfPoincareRecurrenceTheorem2
Date of creation 2013-03-22 14:29:58
Last modified on 2013-03-22 14:29:58
Owner Koro (127)
Last modified by Koro (127)
Numerical id 5
Author Koro (127)
Entry type Proof
Classification msc 37A05
Classification msc 37B20