Classification of Fatou components

In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou. If f is a rational function defined in the extended complex plane, and if it is a nonlinear function (degree > 1) then for a periodic component of the Fatou set, exactly one of the following holds: contains an attracting periodic point is parabolic is a Siegel disc: a simply connected Fatou component on which f(z) is analytically conjugate to a Euclidean rotation of the unit disc onto itself by an irrational rotation angle. is a Herman ring: a double connected Fatou component (an annulus) on which f(z) is analytically conjugate to a Euclidean rotation of a round annulus, again by an irrational rotation angle. File:Julia-set_N_z3-1.png|Julia set (white) and Fatou set (dark red/green/blue) for f: z\mapsto z-\frac{g}{g'}(z) with g: z \mapsto z^3-1 in the complex plane. Cauliflower Julia set DLD field lines.png|Julia set with parabolic cycle Quadratic Golden Mean Siegel Disc Average Velocity - Gray.png|Julia set with [[Siegel disc]] (elliptic case) Herman Standard.png|Julia set with [[Herman ring]] The components of the map contain the attracting points that are the solutions to . This is because the map is the one to use for finding solutions to the equation by Newton–Raphson formula. The solutions must naturally be attracting fixed points. Julia-Set z2+c 0 0.png|Dynamic plane consist of Fatou 2 superattracting period 1 basins, each has only one component. Basilica_Julia_set_-_DLD.png|Level curves and rays in superattractive case Basilica Julia set, level curves of escape and attraction time.png|Julia set with superattracting cycles (hyperbolic) in the interior ( perieod 2) and the exterior (period 1) The map and t = 0.6151732... will produce a Herman ring. It is shown by Shishikura that the degree of such map must be at least 3, as in this example. If degree d is greater than 2 then there is more than one critical point and then can be more than one type of component Herman+Parabolic.