Poincaré recurrence theorem

E156189

mathematical theorem result in dynamical systems result in ergodic theory

The Poincaré recurrence theorem is a fundamental result in dynamical systems and ergodic theory stating that certain systems will, after a sufficiently long but finite time, return arbitrarily close to their initial state.

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All labels observed (3)

Label	Occurrences
Poincaré recurrence theorem canonical	6
Poincaré recurrence	4
In a measure-preserving dynamical system with finite total measure, almost every point of a measurable set returns arbitrarily close to its initial position infinitely often.	1

How this entity was disambiguated

This entity first appeared as the object of triple T1358647 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Poincaré recurrence theorem
Context triple: [Henri Poincaré, notableWork, Poincaré recurrence theorem]

A. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
B. Noether's theorem
Noether's theorem is a fundamental result in theoretical physics and mathematics that links continuous symmetries of a physical system to corresponding conservation laws, such as energy or momentum conservation.
C. H-theorem
The H-theorem is Boltzmann’s foundational result in statistical mechanics that explains the irreversible increase of entropy in a gas from time-reversible microscopic dynamics, providing a key link between mechanics and the second law of thermodynamics.
D. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
E. Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Poincaré recurrence theorem
Target entity description: The Poincaré recurrence theorem is a fundamental result in dynamical systems and ergodic theory stating that certain systems will, after a sufficiently long but finite time, return arbitrarily close to their initial state.

A. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
B. Noether's theorem
Noether's theorem is a fundamental result in theoretical physics and mathematics that links continuous symmetries of a physical system to corresponding conservation laws, such as energy or momentum conservation.
C. H-theorem
The H-theorem is Boltzmann’s foundational result in statistical mechanics that explains the irreversible increase of entropy in a gas from time-reversible microscopic dynamics, providing a key link between mechanics and the second law of thermodynamics.
D. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
E. Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in dynamical systems ⓘ result in ergodic theory ⓘ
appliesTo	Hamiltonian systems with bounded energy surface and finite phase-space volume ⓘ measure-preserving dynamical systems ⓘ
assumption	finite measure space ⓘ invertible transformation (in common formulations) ⓘ measure-preserving transformation ⓘ
conclusion	almost every point in a set of positive measure returns to that set infinitely many times ⓘ orbits are recurrent for almost all initial conditions ⓘ the system returns arbitrarily close to its initial state after sufficiently long but finite times, for almost all initial states ⓘ
conclusionOnSet	for almost every x in A, T^n(x) returns to A for infinitely many integers n > 0 ⓘ
conditionOnSet	set A must have positive measure ⓘ
contrastWith	irreversible macroscopic behavior in thermodynamics ⓘ
doesNotImply	exact periodicity of orbits ⓘ finite recurrence time uniform for all initial conditions ⓘ
field	Hamiltonian mechanics ⓘ dynamical systems ⓘ ergodic theory ⓘ measure theory ⓘ
formalSetting	measure space (X, Σ, μ) ⓘ measure-preserving transformation T: X → X ⓘ
historicalContext	introduced by Henri Poincaré in the late 19th century ⓘ
implies	in a finite measure, measure-preserving system, wandering sets have measure zero ⓘ typical trajectories revisit any neighborhood of their starting point infinitely often ⓘ
inspired	Zermelo's recurrence objection to Boltzmann ⓘ discussions of Loschmidt's paradox ⓘ
involves	infinite time evolution ⓘ invariant measure ⓘ iterates of a transformation ⓘ
language	often formulated using the concept of 'almost everywhere' ⓘ
mathematicalArea	probability theory ⓘ topological dynamics ⓘ
namedAfter	Henri Poincaré ⓘ
quantifier	holds for almost every point with respect to the invariant measure ⓘ
relatedConcept	Birkhoff ergodic theorem ⓘ Kac's lemma ⓘ Liouville's theorem in Hamiltonian mechanics ⓘ ergodicity ⓘ phase space volume preservation ⓘ recurrence in dynamical systems ⓘ
statement	In a measure-preserving dynamical system with finite total measure, almost every point of a measurable set returns arbitrarily close to its initial position infinitely often. ⓘ linked to: Poincaré recurrence theorem
typeOfRecurrence	metric recurrence (return arbitrarily close in phase space) ⓘ
usedIn	analysis of long-term behavior of orbits ⓘ foundations of thermodynamics ⓘ statistical mechanics ⓘ study of conservative dynamical systems ⓘ

How these facts were elicited

Referenced by (11)

Full triples — surface form annotated when it differs from this entity's canonical label.

Henri Poincaré → notableWork → Poincaré recurrence theorem ⓘ

H-theorem → relatedConcept → Poincaré recurrence theorem ⓘ

Kac ring model → relatedTo → Poincaré recurrence ⓘ