Poincaré recurrence theorem

Remarks

Let $B$ be the set of points of $A$ that never return to $A$, so $B = A \setminus \bigcup_{k \ge 1} T^{-k}(A)$. The sets $B, T^{-1}(B), T^{-2}(B), \dots$ are pairwise disjoint and all have measure $\mu(B)$. Since their total measure is at most $\mu(X) < \infty$, one gets $\mu(B) = 0$. Applying this to $T^m$ for each $m \ge 1$ shows that almost every point of $A$ returns infinitely often. The hypothesis $\mu(X) < \infty$ is needed: the translation $T(x) = x + 1$ on $\R$ preserves Lebesgue measure, but no point of $A = [0,1)$ ever returns to $A$. Example: for an irrational rotation of the circle and an arc $A$ of positive length, every point of $A$ comes back to $A$ infinitely often.

Going deeper

For a student

The theorem uses only finiteness of the measure and invariance, not any mixing property. It is the starting point of recurrence in ergodic theory, which leads to multiple recurrence and Furstenberg's proof of Szemerédi's theorem. For infinite measures, the maps for which the conclusion holds are called conservative, and the Mathlib form of the theorem is stated for conservative maps.