Classification of Fatou components

In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou.
Rational case.
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 formula_3 of the Fatou set, exactly one of the following holds:
Attracting periodic point.
The components of the map formula_8 contain the attracting points that are the solutions to formula_9. This is because the map is the one to use for finding solutions to the equation formula_9 by Newton–Raphson formula. The solutions must naturally be attracting fixed points.
Herman ring.
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.
More than one type of component.
If degree d is greater then 2 then there is more than one critical point and then can be more than one type of component
Transcendental case.
Baker domain.
In case of transcendental functions there is another type of periodic Fatou components, called Baker domain: these are "domains on which the iterates tend to an essential singularity (not possible for polynomials and rational functions)" Example function :
formula_12
Wandering domain.
Transcendental maps may have wandering domains: these are Fatou components that are not eventually periodic.