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: The components of the map
The solutions must naturally be attracting fixed points.
The map and t = 0.6151732... will produce a Herman ring.
[2] 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 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)"[3][4] one example of such a function is:[5]
Transcendental maps may have wandering domains: these are Fatou components that are not eventually periodic.