# Classification of Fatou components

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

## Rational case

If f is a rational function

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

then for a periodic component of the Fatou set, exactly one of the following holds:

- contains an
**attracting periodic point** - is
**parabolic**^{[1]} - is a
**Siegel disc** - 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 contain the attracting points that are the solutions to . This is because the map is the one to use for finding solutions to the equation by Newton-Raphson formula. The solutions must naturally be attracting fixed points.

#### Herman ring

The map

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]}

## See also

## External links

## References

- Lennart Carleson and Theodore W. Gamelin,
*Complex Dynamics*, Springer 1993. - Alan F. Beardon
*Iteration of Rational Functions*, Springer 1991.

- ↑ wikibooks : parabolic Julia sets
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
- ↑ An Introduction to Holomorphic Dynamics (with particular focus on transcendental functions)by L. Rempe
- ↑ Siegel Discs in Complex Dynamics by Tarakanta Nayak
- ↑ A transcendental family with Baker domains by Aimo Hinkkanen , Hartje Kriete and Bernd Krauskopf