For the rational curve of an unperturbed system with rotation number 
under a map 
(for which every point is a fixed
point of 
), only an even number of fixed
points 
(
, 2, ...) will remain under perturbation. These fixed
points are alternately stable (elliptic
and unstable (hyperbolic). Around each elliptic
fixed point there is a simultaneous application of the Poincaré-Birkhoff fixed
point theorem and the Kolmogorov-Arnold-Moser
theorem, 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 itself that advances points on the outer edge positively and on the inner edge negatively, then there are at least two fixed points.
It was conjectured by Poincaré from a consideration of the three-body problem in celestial mechanics and proved by Birkhoff.
