Classification of Fatou components

Rational case

If f is a rational function

${\displaystyle f=\frac{P(z)}{Q(z)}}$

defined in the extended complex plane, and if it is a nonlinear function (degree > 1)

${\displaystyle d(f)=max(\mathrm{deg}(P),\mathrm{deg}(Q))\ge 2,}$

then for a periodic component ${\displaystyle U}$ of the Fatou set, exactly one of the following holds:

1. ${\displaystyle U}$ contains an attracting periodic point
2. ${\displaystyle U}$ is parabolic
3. ${\displaystyle U}$ 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.
4. ${\displaystyle U}$ 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.

Attracting periodic point

The components of the map ${\displaystyle f(z)=z-(z^3-1)/3z^2}$ contain the attracting points that are the solutions to ${\displaystyle z^3=1}$. This is because the map is the one to use for finding solutions to the equation ${\displaystyle z^3=1}$ by Newton–Raphson formula. The solutions must naturally be attracting fixed points.

Herman ring

The map

${\displaystyle f(z)=e^{2\pi it}{z}^2(z-4)/(1-4z)}$

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