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 parabolic[1]
$U$ 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.

$Julia set (white) and Fatou set (dark red/green/blue) for f : z ↦ z − g g ′ ( z ) {\displaystyle f:z\mapsto z-{\frac {g}{g'}}(z)} with g : z ↦ z 3 − 1 {\displaystyle g:z\mapsto z^{3}-1} in the complex plane.$
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.
Julia set with parabolic cycle
Julia set with Siegel disc (elliptic case)
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.

Dynamic plane consist of Fatou 2 superattracting period 1 basins, each has only one component.
Level curves and rays in superattractive case
Julia set with superattracting cycles (hyperbolic) in the interior ( perieod 2) and the exterior (period 1)

Herman ring

The map

f(z)=e^{2\pi it}z^{2}(z-4)/(1-4z)

and t = 0.6151732... will produce a Herman ring.[2] 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
Period 3 and 105
attracting and parabolic
period 1 and period 1
period 4 and 4 (2 attracting basins)
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)"[3][4] one example of such a function is:[5]

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

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

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[1] wikibooks : parabolic Julia sets

[2] Milnor, John W. (1990), Dynamics in one complex variable, arXiv:math/9201272, Bibcode:1992math......1272M

[3] An Introduction to Holomorphic Dynamics (with particular focus on transcendental functions)by L. Rempe

[4] Siegel Discs in Complex Dynamics by Tarakanta Nayak

[5] A transcendental family with Baker domains by Aimo Hinkkanen , Hartje Kriete and Bernd Krauskopf

[6] JULIA AND JOHN REVISITED by NICOLAE MIHALACHE