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 ProofOfPoincareRecurrenceTheorem1 2013-03-22 14:29:56 2013-03-22 14:29:56 Koro (127) Koro (127) 5 Koro (127) Proof msc 37A05 msc 37B20