Classification of Fatou components
If f is a rational function
defined in the extended complex plane, and if it is a nonlinear function ( degree > 1 )
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.
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.
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 :
Finally, transcendental maps also may have wandering domains: these are Fatou components that are not eventually periodic.
- Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993.
- Alan F. Beardon Iteration of Rational Functions, Springer 1991.
- wikibooks : parabolic Julia sets
- Milnor, John W. (1990), Dynamics in one complex variable, arXiv:
- 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