In
symplectic topology
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the H ...
and
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s, Poincaré–Birkhoff theorem (also known as Poincaré–Birkhoff fixed point theorem and Poincaré's last geometric theorem) states that every area-preserving, orientation-preserving
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
of an
annulus
Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to:
Human anatomy
* ''Anulus fibrosus disci intervertebralis'', spinal structure
* Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
that rotates the two boundaries in opposite directions has at least two
fixed points.
History
The Poincaré–Birkhoff theorem was discovered by
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
, who published it in a 1912 paper titled "Sur un théorème de géométrie", and proved it for some special cases. The general case was proved by
George D. Birkhoff
George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and durin ...
in his 1913 paper titled "Proof of Poincaré's geometric theorem".
[Poincaré last theorem. ''Encyclopedia of Mathematics''. URL: http://www.encyclopediaofmath.org/index.php?title=Poincar%C3%A9_last_theorem&oldid=23480]
References
Further reading
* M. Brown; W. D. Neumann. "Proof of the Poincaré-Birkhoff fixed-point theorem". ''Michigan Math. J.'' Vol. 24, 1977, p. 21–31.
* P. Le Calvez; J. Wang. "Some remarks on the Poincaré–Birkhoff theorem". ''Proc. Amer. Math. Soc.'' Vol. 138, No.2, 2010, p. 703–715.
* J. Franks. "Generalizations of the Poincaré-Birkhoff Theorem", ''Annals of Mathematics'', Second Series, Vol. 128, No. 1 (Jul., 1988), pp. 139–151.
{{DEFAULTSORT:Poincare-Birkhoff Theorem
Symplectic topology
Dynamical systems
Fixed-point theorems