Poincaré–Birkhoff fixed-point theorem

E157608

fixed-point theorem mathematical theorem result in dynamical systems result in topology

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.

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All labels observed (3)

Label	Occurrences
Poincaré–Birkhoff fixed-point theorem canonical	1
Poincaré–Birkhoff theorem	1
Poincaré–Birkhoff twist theorem	1

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 ⓘ

Referenced by (3)

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

Henri Poincaré → notableWork → Poincaré–Birkhoff fixed-point theorem ⓘ

Poincaré–Birkhoff fixed-point theorem → alsoKnownAs → Poincaré–Birkhoff fixed-point theorem ⓘ

this entity surface form: Poincaré–Birkhoff theorem

Poincaré–Birkhoff fixed-point theorem → alsoKnownAs → Poincaré–Birkhoff fixed-point theorem ⓘ

this entity surface form: Poincaré–Birkhoff twist theorem