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
then for a periodic component of the Fatou set, exactly one of the following holds:
  1. contains an attracting periodic point
  2. is parabolic
  3. is a Siegel disc: a simply connected Fatou component on which f is analytically conjugate to a Euclidean rotation of the unit disc onto itself by an irrational rotation angle.
  4. is a Herman ring: a double connected Fatou component on which f is analytically conjugate to a Euclidean rotation of a round annulus, again by an irrational rotation angle.

    Attracting periodic point

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.

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 than 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 " one example of such a function is:

Wandering domain

Transcendental maps may have wandering domains: these are Fatou components that are not eventually periodic.