Poincaré recurrence theorem

Let T preserve a finite measure and let A be measurable. Inside A consider B, the points that never visit A again at any positive time. The preimages of B under distinct nonnegative iterates are pairwise disjoint. Otherwise one trajectory would encounter B twice, contradicting the first encountered point's promise never to return even to A. Preservation makes every such preimage have the same measure as B. Finitely much total measure cannot contain infinitely many disjoint sets of the same positive measure, forcing B to be null.

A point of A that returns only finitely many times has a last visit to A, and that last visited state lies in B. Therefore all such starting points belong to a countable union of preimages of B, which is null by preservation. This establishes infinitely many returns for almost every point of A. The argument uses preimages throughout, so it remains valid for noninvertible preserving transformations. A null exceptional set may remain and cannot simply be removed from the theorem's wording.

In a second-countable metric space, apply recurrence to a countable basis of neighborhoods and discard the countable union of exceptional sets. Almost every point then returns arbitrarily close to itself infinitely often. This topological interpretation depends on a suitable countable basis; the measure-theoretic event statement is the primary theorem. It does not say a trajectory retraces an entire macroscopic state exactly, nor that all initial points recur or share a common return schedule.

An infinite invariant measure invalidates the disjoint-set contradiction. Translation by one on the integers preserves counting measure, yet the singleton containing zero never returns to itself. Its measure is finite and positive, showing that finite measure of A alone is insufficient for the general theorem. Infinite-measure systems can still be recurrent under additional conservativity assumptions, but recurrence no longer follows merely from preservation. The finite-ambient-measure hypothesis should therefore be checked before invoking the familiar recurrence conclusion.

Define the first return time to a positive-probability event as the smallest positive iterate landing in it. For an ergodic invertible probability-preserving system, Kac's return-time formula says that the integral of this time over the event is one. Equivalently its average under the conditional probability on that event is the reciprocal of the event's probability. This quantitative result is stronger than recurrence and uses the ergodic hypothesis in this stated form. We invoke the formula rather than deriving its full return-tower proof here.

If the system is not ergodic, an event may meet only one invariant component, so using reciprocal mass in the whole space can give the wrong answer. The return tower then fills only the accessible part of the space. A finite permutation makes this visible without measure-theoretic technicalities: the return length depends on the cycle reached from the event, while the probability law may assign mass to other cycles that never interact with it. An exact return calculation is a useful check on the formula's hypotheses.