Poincaré recurrence theorem

Let $(X,𝒮,\mu )$ be a probability space and let $f:X\to X$be a measure preserving transformation.

Theorem 1.

For any $E\mathrm{\in }\mathrm{S}$,the set of those points $x$ of $E$ such that $f^n\mathit{}\mathrm{(}x\mathrm{)}\mathrm{\notin }E$ for all$n\mathrm{>}\mathrm{0}$ has zero measure. That is, almost every point of $E$ returns to$E$. In fact, almost every point returns infinitely often; i.e.

$$\mu \left(\{x\in E:\text{ there exists }N\text{ such that }{f}^n(x)\notin E\text{ for all }n>N\}\right)=0.$$

The following is a topological version of this theorem:

Theorem 2.

If $X$ is a second countable Hausdorff space and$\mathrm{S}$ contains the Borel sigma-algebra, then theset of recurrent points of $f$ has full measure. That is, almost everypoint is recurrent.