Classification of Fatou components

In mathematics, Fatou components are components of the Fatou set.

Rational case

If f is a rational function

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

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

${\displaystyle \max(\deg(P),\,\deg(Q))\geq 2,}$

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

1. ${\displaystyle U}$ contains an attracting periodic point
2. ${\displaystyle U}$ is parabolic[1]
3. ${\displaystyle U}$ is a Siegel disc
4. ${\displaystyle U}$ is a Herman ring.

A Siegel disk is 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. A Herman ring is 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.

Examples

Attracting periodic point

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

Herman ring

The map

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

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.

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] Example function :[5]

${\displaystyle f(z)=z-1+(1-2z)e^{z}}$

Wandering domain

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