Poincaré-Birkhoff Fixed Point Theorem
For the rational curve of an unperturbed system with Rotation Number 
under a map 
(for which every point is aFixed Point of 
), only an even number of Fixed Points 
(
, 2, ...) will remainunder perturbation. These Fixed Points are alternately stable (Elliptic) and unstable (Hyperbolic). Around eachelliptic fixed point there is a simultaneous application of the Poincaré-Birkhoff fixed point theorem and the KAMTheorem, which leads to a self-similar structure on all scales.
The original formulation was: Given a Conformal One-to-One transformation from an Annulus to itselfthat advances points on the outer edge positively and on the inner edge negatively, then there are at least two fixedpoints.
It was conjectured by Poincaré from a consideration of the three-body problem 
incelestial mechanics and proved by Birkhoff.
- Critical Line
- Critical Point
- Critical Strip
- Crook
- Crookedness
- Cross
- Cross-Cap
- Cross-Correlation
- Cross-Correlation Coefficient
- Cross-Correlation Theorem
- Cross Curve
- Crossed Ladders Problem
- Crossed Trough
- Cross Fractal
- Crossing Number (Graph)
- Crossing Number (Link)
- Cross of Lorraine
- Cross Polytope
- Cross Product
- Cross-Ratio
- Cross-Section
- Crout's Method
- Crowd
- Crown
- Crucial Point