 Poincaré recurrence theorem

In a finite measure-preserving system, almost every point of every measurable set $A$ returns to $A$ infinitely often. The points of $A$ that never return form a wandering set: all its inverse images are pairwise disjoint and have equal measure, so that set must be null.

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.