In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou.

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 parabolicwikibooks : parabolic Julia sets is a Siegel disc: 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. is a Herman ring: 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.

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.

More than one type of component If degree d is greater than 2 then there is more than one critical point and then can be more than one type of component

Transcendental case 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)"An Introduction to Holomorphic Dynamics (with particular focus on transcendental functions)by L. RempeSiegel Discs in Complex Dynamics by Tarakanta Nayak one example of such a function is:A transcendental family with Baker domains by Aimo Hinkkanen, Hartje Kriete and Bernd Krauskopf

Wandering domain Transcendental maps may have wandering domains: these are Fatou components that are not eventually periodic.

See also No-wandering-domain theorem Montel's theorem John DomainsJULIA AND JOHN REVISITED by NICOLAE MIHALACHE Basins of attraction

References

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