# 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[1]
3. $U$ is a Siegel disc
4. $U$ is a Herman ring.

One can prove that case 3 only occurs when f(z) is analytically conjugate to a Euclidean rotation of the unit disc onto itself, and case 4 only occurs when f(z) is analytically conjugate to a Euclidean rotation of some annulus onto itself.

### 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 i t} 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.

## Transcendental case

In case of transcendental functions there is also Baker domain: "domains on which the iterates tend to an essential singularity (not possible for polynomials and rational functions)"[3][4] Example function :[5]

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