proof of Poincaré recurrence theorem 1

Let $A_{n}=\cup_{k=n}^{\infty}f^{-k}E$ . Clearly, $E\subset A_{0}$ and $A_{i}\subset A_{j}$ when $j\leq i$ . Also, $A_{i}=f^{j-i}A_{j}$ , so that $\mu(A_{i})=\mu(A_{j})$ for all $i,j\geq 0$ , by the $f$ -invariance of $\mu$ . Now for any $n>0$ we have $E-A_{n}\subset A_{0}-A_{n}$ , so that

\mu(E-A_{n})\leq\mu(A_{0}-A_{n})=\mu(A_{0})-\mu(A_{n})=0.

Hence $\mu(E-A_{n})=0$ for all $n>0$ , so that $\mu(E-\cap_{n=1}^{\infty}A_{n})=\mu(\cup_{n=1}^{\infty}E-A_{n})=0$ . But $E-\cap_{n=1}^{\infty}A_{n}$ is precisely the set of those $x\in E$ such that for some $n$ and for all $k>n$ we have $f^{k}(x)\notin E$ . $\Box$

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