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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
-
U contains an attracting periodic point -
U is parabolic -
U is a Siegel disc -
U 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.
Attracting periodic point
The components of the map
Herman ring
The map
and t = 0.6151732... will produce a Herman ring. It is shown by Shishikura that the degree of such map must be at least 3, as in this example.
Baker domain
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 :
Wandering domain
Finally, transcendental maps also may have wandering domains: these are Fatou components that are not eventually periodic.