Poincaré–Birkhoff theorem

A fixed-point theorem for area-preserving twist maps of an annulus. Under suitable boundary-rotation conditions, such a map has at least two fixed points.

Area-preserving map: A map that preserves area, or a corresponding measure, when transforming a space. Area preservation is the structural constraint on the annulus map in the theorem.
Henri Poincaré: French mathematician who founded major parts of topology, dynamical systems, and celestial mechanics. His work on periodic orbits in celestial mechanics supplied the theorem’s original setting.
Symplectic geometry: The study of manifolds equipped with a closed, nondegenerate differential 2-form. Area preservation on a surface is the simplest setting for symplectic structure.
Celestial mechanics: The study of the motions of celestial bodies under gravitational forces. Periodic-orbit questions in this field helped motivate the theorem.
Brouwer fixed-point theorem: A theorem stating that every continuous map from a closed ball to itself has a fixed point. Brouwer’s result uses a ball and self-mapping condition rather than annular twist and rotation.
Twist map: A map of an annulus in which angular displacement varies monotonically with radial position. The twist condition makes different annular levels rotate by different amounts.
George David Birkhoff: American mathematician known for contributions to dynamical systems, ergodic theory, and mathematical analysis. He proved the annulus fixed-point result now bearing his name.
Hamiltonian mechanics: A formulation of classical mechanics in terms of a Hamiltonian function and symplectic evolution. Its time evolution produces area-preserving maps and motivates applications of the theorem.
Periodic orbit of a Hamiltonian system: A closed trajectory generated by Hamiltonian dynamics after a finite elapsed time. The theorem can guarantee such trajectories through fixed points of return maps.
Lefschetz fixed-point theorem: A theorem relating the number of fixed points of a map to its induced action on homology. Lefschetz theory uses homological data, unlike the annulus theorem’s twist and boundary rotations.

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