Classification of Fatou components
In mathematics, Fatou components are components of the Fatou set.
Contents
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.
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.
Examples
-
Fatou componenets 3.png
Julia set with attracting cycle
-
Parabolic Julia set for internal angle 1 over 15.png
Parabolic Julia set
-
Quadratic Golden Mean Siegel Disc Average Velocity - Gray.png
Julia set with Siegel disc
-
Herman Standard.png
Julia set with Herman ring
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
Baker doimain
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)"[3][4] Example function :[5]
Wandering domain
Finally, transcendental maps also may have wandering domains: these are Fatou components that are not eventually periodic.
See also
References
- Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993.
- Alan F. Beardon Iteration of Rational Functions, Springer 1991.
- ↑ wikibooks : parabolic Julia sets
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 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