Classification of Fatou components

Rational case

If f is a rational function :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:

1. U contains an attracting periodic point
2. U is parabolic
3. 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.
4. 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. 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+Az+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+az+ 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)" one example of such a function is: f(z) = z - 1 + (1 - 2z)e^z

Wandering domain

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

References

Bibliography

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

References

1. [[:wikibooks:Fractals/Iterations in the complex plane/parabolic. wikibooks : parabolic Julia sets]]
2. Milnor, John W.. (1990). "Dynamics in one complex variable".
3. [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]
4. [http://www.ncnsd.org/proceedings/proceeding05/paper/185.pdf Siegel Discs in Complex Dynamics by Tarakanta Nayak]
5. [http://www.math.uiuc.edu/~aimo/anim.html A transcendental family with Baker domains by Aimo Hinkkanen, Hartje Kriete and Bernd Krauskopf ]
6. [https://arxiv.org/abs/0803.3889 JULIA AND JOHN REVISITED by NICOLAE MIHALACHE]