Poincaré recurrence
In a bounded Hamiltonian system, almost every trajectory returns arbitrarily close to its starting state — given enough time.
Definition
Henri Poincaré proved in 1890 that for a bounded Hamiltonian system (one whose phase-space motion stays in a region of finite volume), almost every trajectory will, given enough time, return arbitrarily close to any point it has previously visited. The proof follows directly from Liouville's theorem: phase-space volume is preserved under evolution, but the total accessible volume is finite, so trajectories must revisit themselves.
The recurrence times can be astronomically long — longer than the age of the universe for any macroscopic system — but the theorem is exact. It unsettled Boltzmann, who had argued that entropy always increases, and it hints at the statistical-mechanics paradox: why do we observe an arrow of time in a reversible system where every state must be revisited?
History
Proved by Henri Poincaré in 1890 in his prize-winning memoir on the three-body problem; clarified by Ernst Zermelo in 1896 in a sharp critique of Boltzmann's H-theorem.