Poincaré–Birkhoff fixed-point theorem
E157608
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Poincaré–Birkhoff fixed-point theorem canonical | 1 |
| Poincaré–Birkhoff theorem | 1 |
| Poincaré–Birkhoff twist theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1358655 — 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–Birkhoff fixed-point theorem Context triple: [Henri Poincaré, notableWork, Poincaré–Birkhoff fixed-point theorem]
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A.
Poincaré–Bendixson theorem
The Poincaré–Bendixson theorem is a fundamental result in the qualitative theory of dynamical systems that characterizes the possible long-term behaviors of trajectories in two-dimensional continuous flows, ruling out chaotic dynamics in the plane.
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B.
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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C.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
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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.
Poincaré recurrence theorem
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.
- 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–Birkhoff fixed-point theorem Target entity description: The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
-
A.
Poincaré–Bendixson theorem
The Poincaré–Bendixson theorem is a fundamental result in the qualitative theory of dynamical systems that characterizes the possible long-term behaviors of trajectories in two-dimensional continuous flows, ruling out chaotic dynamics in the plane.
-
B.
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.
-
C.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
-
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.
Poincaré recurrence theorem
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.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
fixed-point theorem
ⓘ
mathematical theorem ⓘ result in dynamical systems ⓘ result in topology ⓘ |
| alsoKnownAs |
Poincaré–Birkhoff fixed-point theorem
ⓘ
surface form:
Poincaré–Birkhoff theorem
Poincaré–Birkhoff fixed-point theorem ⓘ
surface form:
Poincaré–Birkhoff twist theorem
|
| appliesTo |
area-preserving homeomorphisms of the annulus
ⓘ
area-preserving twist maps of the annulus ⓘ |
| assumption |
map has twist condition on the boundary circles
ⓘ
map is an orientation-preserving homeomorphism ⓘ map is area-preserving ⓘ map is defined on a closed annulus ⓘ map sends boundary components of the annulus to themselves ⓘ |
| conclusion | map has at least two fixed points in the annulus ⓘ |
| domain | closed annulus in the plane ⓘ |
| field |
Hamiltonian dynamics
ⓘ
dynamical systems ⓘ symplectic geometry ⓘ topology ⓘ |
| generalizationOf | earlier results on periodic orbits in annular regions ⓘ |
| guarantees | existence of at least two fixed points ⓘ |
| historicalOrigin | work of Henri Poincaré on celestial mechanics ⓘ |
| influenceOn |
Kolmogorov–Arnold–Moser theory
ⓘ
surface form:
KAM theory
modern symplectic topology ⓘ theory of twist maps ⓘ |
| involvesConcept |
area preservation
ⓘ
fixed point ⓘ orientation-preserving homeomorphism ⓘ twist condition ⓘ |
| namedAfter |
George David Birkhoff
ⓘ
Henri Poincaré ⓘ |
| provedBy | George David Birkhoff ⓘ |
| relatedTo |
Arnold conjecture
ⓘ
Brouwer fixed-point theorem ⓘ Hamiltonian diffeomorphism ⓘ annulus ⓘ area-preserving map ⓘ twist map ⓘ |
| usedIn |
study of periodic orbits of area-preserving maps
ⓘ
study of planar Hamiltonian systems ⓘ symplectic fixed-point theory ⓘ |
| yearProved | 1913 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Poincaré–Birkhoff fixed-point theorem Description of subject: The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Poincaré–Birkhoff theorem
this entity surface form:
Poincaré–Birkhoff twist theorem