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 )
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.
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 also Baker domain: "domains on which the iterates tend to an essential singularity (not possible for polynomials and rational functions)" Example function :
- 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:math/9201272
- 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