the Poincare recurrence theorem
/ pwan-kah-RAY /
Poincare recurrence is the first and most striking fact of measure-preserving dynamics: in any finite-measure system, almost every point of any region you choose will, simply by the passage of time, return to that region — and in fact return infinitely often. Long before averages converge, the dynamics already refuses to forget where it has been.
Precisely, let (Omega, F, P, T) be measure-preserving (P a probability, or any finite measure) and let A in F with P(A) > 0. Then for P-almost every x in A there are infinitely many n >= 1 with T^n x in A. The proof is a one-line pigeonhole on measure: let B be the points of A that never return. The sets B, T^(-1) B, T^(-2) B, ... are pairwise disjoint (if T^(-i) B and T^(-j) B met, a point would return), all have measure P(B), and live inside a space of total measure 1; infinitely many disjoint sets of equal positive measure is impossible, so P(B) = 0. Iterating the argument upgrades "returns once" to "returns infinitely often".
The theorem is the source of the famous recurrence paradox: an isolated gas, modelled as measure-preserving Hamiltonian flow on a finite-energy shell, must eventually return arbitrarily close to its initial configuration — seemingly contradicting the second law of thermodynamics. The resolutions are honest and important. First, finiteness of the measure is essential: simple translation on the line (Lebesgue measure, infinite) has no recurrence at all. Second, recurrence says nothing about how long you wait; the Kac lemma gives the expected return time to A as 1/P(A), which for a macroscopic gas is astronomically larger than the age of the universe. Recurrence is a statement about eventual return, not about practical reversibility.
Two gases mixed in a box: model the molecules as a measure-preserving flow. Poincare guarantees the configuration will eventually return arbitrarily close to "all gas A on the left". Kac's lemma says the expected wait is 1/P(that region), where P of such a tiny region is like 10^(-10^23) — recurrence in principle, never in practice.
Recurrence is certain; the waiting time (Kac: 1/P(A)) is what saves the second law of thermodynamics.
Finiteness of the measure cannot be dropped. On an infinite-measure space (e.g. a transient random walk, or translation on R) almost every point may leave A forever; recurrence is a phenomenon of finite total mass.