Poincaré recurrence theorem
E156189
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.
All labels observed (3)
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]
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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.
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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.
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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.
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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.
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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 |
Poincaré recurrence theorem
self-linksurface differs
ⓘ
surface form:
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.
|
| 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
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Instruction
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Input
Subject: Poincaré recurrence theorem Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Poincaré recurrence
this entity surface form:
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.
this entity surface form:
Poincaré recurrence