Poincaré–Birkhoff theorem

In symplectic topology and dynamical systems, 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 of an annulus that rotates the two boundaries in opposite directions has at least two fixed points.

For the result on the universal enveloping algebra of a Lie algebra, see Poincaré–Birkhoff–Witt theorem.
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