$U$ 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.

$U$ 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.

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.

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

Dynamic plane consist of Fatou 2 superattracting period 1 basins, each has only one component.

Level curves and rays in superattractive case

Julia set with superattracting cycles (hyperbolic) in the interior ( perieod 2) and the exterior (period 1)

Herman ring

The map

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

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.

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

Herman+Parabolic

Period 3 and 105

attracting and parabolic

period 1 and period 1

period 4 and 4 (2 attracting basins)

two period 2 basins

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)"^{
[3]}^{
[4]} one example of such a function is:^{
[5]}

$f(z)=z-1+(1-2z)e^{z}$

Wandering domain

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