# Classification of Fatou components

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

## 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 )

${\displaystyle \max(\deg(P),\,\deg(Q))\geq 2,}$

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

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 ${\displaystyle f(z)=z-(z^{3}-1)/3z^{2}}$ 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.[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]