# Classification of Fatou components

In mathematics, Fatou components are components of the Fatou set.

## Rational case

If f is a rational function

$f={\frac {P(z)}{Q(z)}}$ defined in the extended complex plane, and if it is a nonlinear function (degree > 1)

$\max(\deg(P),\,\deg(Q))\geq 2,$ then for a periodic component $U$ of the Fatou set, exactly one of the following holds:

1. $U$ contains an attracting periodic point
2. $U$ is parabolic
3. $U$ is a Siegel disc
4. $U$ is a Herman ring.

A Siegel disk is 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. A Herman ring is 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.

### Examples

#### Attracting periodic point

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.

#### 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. It is shown by Shishikura that the degree of such map must be at least 3, as in this example.

## 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 :

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

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