For the rational curve of an unperturbed system with rotation number r/s under a map T (for which every point is a fixed point of T^s), only an even number of fixed points 2k s (k = 1, 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.