# proof of Poincaré recurrence theorem 2

Let $\{U_{n}:n\in\mathbb{N}\}$ be a basis of open sets for $X$, and for each $n$ define

 $U_{n}^{\prime}=\{x\in U_{n}:\forall n\geq 1,\,f^{n}(x)\notin U_{n}\}.$

From theorem 1 we know that $\mu(U_{n}^{\prime})=0$. Let $N=\bigcup_{n\in\mathbb{N}}U_{n}^{\prime}.$ Then $\mu(N)=0$. We assert that if $x\in X-N$ then $x$ is recurrent. In fact, given a neighborhood $U$ of $x$, there is a basic neighborhood $U_{n}$ such that $x\subset U_{n}\subset U$, and since $x\notin N$ we have that $x\in U_{n}-U_{n}^{\prime}$ which by definition of $U_{n}^{\prime}$ means that there exists $n\geq 1$ such that $f^{n}(x)\in U_{n}\subset U$; thus $x$ is recurrent. $\Box$

Title proof of Poincaré recurrence theorem 2 ProofOfPoincareRecurrenceTheorem2 2013-03-22 14:29:58 2013-03-22 14:29:58 Koro (127) Koro (127) 5 Koro (127) Proof msc 37A05 msc 37B20