Classification of Fatou components

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

Rational case

: $f = \frac{P(z)}{Q(z)}$

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

: $d(f) = \max(\deg(P),\, \deg(Q))\geq 2,$

then for a periodic component $U$ of the Fatou set, exactly one of the following holds:

$U$ contains an attracting periodic point
$U$ is parabolicwikibooks : parabolic Julia sets
$U$ 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.
$U$ 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.

File:Julia-set_N_z3-1.png|Julia set (white) and Fatou set (dark red/green/blue) for $f: z\mapsto z-\frac{g}{g'}(z)$ with $g: z \mapsto z^3-1$ 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

=Attracting periodic point=

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

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)

=Herman ring=

The map

: $f(z) = e^{2 \pi i t} z^2(z - 4)/(1 - 4z)$

and t = 0.6151732... will produce a Herman ring.{{citation|first=John W.|last=Milnor|authorlink=John Milnor|title=Dynamics in one complex variable|year=1990|arxiv=math/9201272|bibcode=1992math......1272M}} 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

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

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)"[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][http://www.ncnsd.org/proceedings/proceeding05/paper/185.pdf Siegel Discs in Complex Dynamics by Tarakanta Nayak] one example of such a function is:[http://www.math.uiuc.edu/~aimo/anim.html A transcendental family with Baker domains by Aimo Hinkkanen, Hartje Kriete and Bernd Krauskopf ]

$f(z) = z - 1 + (1 - 2z)e^z$