Classification Of Fatou Components

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

Rational case

If f is a rational function
:$f = \frac$

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
# $U$ 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 $f: z\mapsto z-\frac(z)$ with $g: z \mapsto z^3-1$ in the complex plane.
Basilica Julia set, level curves of escape and attraction time.png, Julia set with superattracting cycles (hyperbolic) in the interior and the exterior
Basilica_Julia_set_-_DLD.png, Level curves and rays in superattractive case
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.

Herman ring

The map
:$f(z) = e^ z^2(z - 4)/(1 - 4z)$
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+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)" 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.

See also

* No-wandering-domain theorem
* Montel's theorem
* John DomainsJULIA AND JOHN REVISITED by NICOLAE MIHALACHE
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References

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

Fractals
Limit sets
Theorems in complex analysis
Complex dynamics
Theorems in dynamical systems
Mathematical classification systems