View source for Classification of Fatou components

In [[mathematics]], Fatou components are [[connected component (analysis)|component]]s of the [[Fatou set]].  They were named after [[Pierre Fatou]].

==Rational case==
If f is a [[rational function]]
:<math>f = \frac{P(z)}{Q(z)}</math>

defined in the [[extended complex plane]], and if it is a nonlinear function (degree > 1)

: <math> d(f) = \max(\deg(P),\, \deg(Q))\geq 2,</math>

then for a periodic [[connected component (analysis)|component]] <math>U</math> of the [[Fatou set]], exactly one of the following holds:

# <math>U</math> contains an '''[[attracting periodic point]]'''
# <math>U</math> is '''parabolic'''<ref>[[:wikibooks:Fractals/Iterations in the complex plane/parabolic|wikibooks : parabolic Julia sets]]</ref>
# <math>U</math> 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.
# <math>U</math> is a '''[[Herman ring]]''': a double connected Fatou component (an [[Annulus (mathematics)|annulus]]) on which ''f''(''z'') is analytically conjugate to a Euclidean rotation of a round annulus, again by an irrational rotation angle.

<gallery>
File:Julia-set_N_z3-1.png|Julia set (white) and Fatou set (dark red/green/blue) for <math>f: z\mapsto z-\frac{g}{g'}(z)</math> with  <math>g: z \mapsto z^3-1</math> in the complex plane.

Cauliflower Julia set DLD field lines.png|Julia set with parabolic cycle
Quadratic Golden Mean Siegel Disc Average Velocity - Gray.png|Julia set with [[Siegel disc]] (elliptic case)
Herman Standard.png|Julia set with [[Herman ring]]

</gallery>

===Attracting periodic point===
The components of the map <math>f(z) = z - (z^3-1)/3z^2</math>  contain the attracting points that are the solutions to <math>z^3=1</math>. This is because the map is the one to use for finding solutions to the equation <math>z^3=1</math> by [[Newton–Raphson]] formula. The solutions must naturally be attracting fixed points.

<gallery>
Julia-Set z2+c 0 0.png|Dynamic plane consist of Fatou 2 superattracting period 1 basins, each has only one component.
Basilica_Julia_set_-_DLD.png|Level curves and rays in superattractive case
Basilica Julia set, level curves of escape and attraction time.png|Julia set with superattracting cycles (hyperbolic) in the interior (period 2) and the exterior (period 1)
</gallery>

===Herman ring===
The map
:<math>f(z) = e^{2 \pi i t} z^2(z - 4)/(1 - 4z)</math>
and t = 0.6151732... will produce a Herman ring.<ref>{{citation|first=John W.|last=Milnor|authorlink=John Milnor|title=Dynamics in one complex variable|year=1990|arxiv=math/9201272|bibcode=1992math......1272M}}</ref> It is shown by [[Mitsuhiro Shishikura|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
<gallery>
Herman+Parabolic.png|Herman+Parabolic
Cubic set z^3+A*z+c with two cycles of length 3 and 105.png|Period 3 and 105
Julia set z+0.5z2-0.5z3.png|attracting and parabolic
Geometrically finite Julia set.png|period 1 and period 1
Julia set f(z)=1 over az5+z3+bz.png|period 4 and 4 (2 attracting basins)
Julia set for f(z)=1 over (z3+a*z+ b) with a = 2.099609375 and b = 0.349609375.png|two period 2 basins

</gallery>

==Transcendental case==
===Baker domain===
In case of [[transcendental functions]] there is another type of periodic Fatou components, called '''Baker domain''': these are "[[Domain (mathematical analysis)|domains]] on which the iterates tend to an [[essential singularity]] (not possible for polynomials and rational functions)"<ref>[http://pcwww.liv.ac.uk/~lrempe/Talks/liverpool_seminar_2006.pdf An Introduction to Holomorphic Dynamics  (with particular focus on transcendental functions)by L. Rempe]</ref><ref>[http://www.ncnsd.org/proceedings/proceeding05/paper/185.pdf Siegel Discs in Complex Dynamics by Tarakanta Nayak]</ref> one example of such a function is:<ref>[http://www.math.uiuc.edu/~aimo/anim.html A transcendental family with Baker domains by Aimo Hinkkanen, Hartje Kriete and Bernd Krauskopf ]</ref>
<math display="block">f(z) = z - 1 + (1 - 2z)e^z</math>

===Wandering domain===
Transcendental maps may have [[Wandering set|wandering domains]]: these are Fatou components that are not eventually periodic.

==See also==
* [[No-wandering-domain theorem]]
* [[Montel's theorem]]
* [[Fritz John|John]] Domains<ref>[https://arxiv.org/abs/0803.3889 JULIA AND JOHN REVISITED by NICOLAE MIHALACHE]</ref>
* [[Attractor#Basins of attraction|Basins of attraction]]

==References==
* [[Lennart Carleson]] and Theodore W. Gamelin, ''Complex Dynamics'', Springer 1993.
* [[Alan F. Beardon]] ''Iteration of Rational Functions'', Springer 1991.
<references/>

[[Category:Fractals]]
[[Category:Limit sets]]
[[Category:Theorems in complex analysis]]
[[Category:Complex dynamics]]
[[Category:Theorems in dynamical systems]]
[[Category:Mathematical classification systems]]