The Poincaré recurrence theorem is a fundamental result in the field of dynamical systems and ergodic theory, named after the French mathematician Henri Poincaré. The theorem essentially states that in a closed system where the dynamics are governed by deterministic laws and the system is confined to a finite volume, a system will eventually return to a state very close to its initial conditions after a sufficient amount of time.
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In a finite measure-preserving system, for each measurable set , almost every point of returns to infinitely often. The exceptional null set may depend on . To prove the claim, let consist of points that never return at a positive time. The sets , , are pairwise disjoint: a point in two of them would give a future return from one point of to another point of . They have equal measure by invariance, so finiteness forces .
A point of with only finitely many future visits has a last visit, lying in . Such points lie in the countable union , which is null. This completes the infinite-return assertion without requiring invertibility. The theorem supplies no universal return-time bound. On the infinite counting-measure system on , a singleton never returns, showing why finite total measure matters.