Poincaré recurrence theorem

In a measure-preserving system with finite total measure, almost every point in any measurable set returns to that set infinitely often. In metric spaces, this yields arbitrarily close returns to the initial state.

Measure-preserving transformation: A measurable map that leaves a measure unchanged under inverse images. The theorem applies when repeated iterations preserve the measure of every measurable set.
Measure theory: The mathematical study of measures, integration, and measurable sets. The theorem's assumptions and its almost-everywhere conclusion are expressed in measure-theoretic terms.
Henri Poincaré: A French mathematician whose work founded major parts of topology and qualitative dynamical systems. The recurrence theorem is named for his work on the long-term behavior of mechanical systems.
Poincaré recurrence time: The time required for a system to return close to a previous state at a chosen resolution. The theorem promises recurrence without making the recurrence time practically short.
Birkhoff ergodic theorem: A theorem stating that time averages of an integrable observable converge almost everywhere in measure-preserving systems. Unlike recurrence, it gives information about long-run averages, not merely repeated visits.
Invariant measure: A measure unchanged by the action of a transformation or dynamical evolution. Invariance keeps the measure of a region constant across every iterate.
Dynamical system: A rule describing how a state evolves over time, often through repeated iteration or a flow. The theorem describes long-term behavior under repeated evolution.
Les Méthodes nouvelles de la mécanique céleste: Henri Poincaré's three-volume work on celestial mechanics, published from 1892 to 1899. Poincaré discussed recurrence in the context of conservative mechanical systems in this work.
Recurrence plot: A graphical representation marking pairs of times when a trajectory visits nearby states. Recurrence plots turn the theorem's repeated near-returns into a tool for analyzing observed trajectories.
Ergodicity: A property of a measure-preserving system in which invariant sets have measure zero or full measure. Recurrence does not require ergodicity or imply that trajectories explore the whole space.

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