Poincare recurrence theorem

Statement

If $T$ preserves a probability measure and $E$ is measurable, then almost every $x\in E$ has $T^nx\in E$ for infinitely many positive integers $n$ . This is the Poincaré recurrence theorem; ergodicity and invertibility are not required.

Proof

Let $B\subseteq E$ be the points that never return to $E$ at positive times. The sets $T^{-n}B$ , $n\geq0$ , are pairwise disjoint and have equal measure. Finiteness of the total measure forces $\mu(B)=0$ . A point of $E$ with only finitely many returns eventually reaches $B$ , so all such points lie in the null set $\bigcup_{n\geq0}T^{-n}B$ .

Scope

Recurrence does not assert a period or a uniform return-time bound. Translation on the real line with infinite Lebesgue measure shows why finiteness of the measure matters.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §2.6.