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

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

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

$d(f)=\max (\mathrm{deg}(P),\mathrm{deg}(Q))\ge 2,$

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.

• Julia set (white) and Fatou set (dark red/green/blue) for f:z↦z−gg′(z) with g:z↦z3−1 in the complex plane.
  Julia set (white) and Fatou set (dark red/green/blue) for $f:z\mapsto z-\frac{g}{g^{\prime }}(z)$ with $g:z\mapsto z^3-1$ in the complex plane.
• Julia set with parabolic cycle
  Julia set with parabolic cycle
• Julia set with Siegel disc (elliptic case)
  Julia set with Siegel disc (elliptic case)
• Julia set with Herman ring
  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.
  Dynamic plane consist of Fatou 2 superattracting period 1 basins, each has only one component.
• Level curves and rays in superattractive case
  Level curves and rays in superattractive case
• Julia set with superattracting cycles (hyperbolic) in the interior (period 2) and the exterior (period 1)
  Julia set with superattracting cycles (hyperbolic) in the interior (period 2) and the exterior (period 1)

Herman ring

The map

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

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
  Herman+Parabolic
• Period 3 and 105
  Period 3 and 105
• attracting and parabolic
  attracting and parabolic
• period 1 and period 1
  period 1 and period 1
• period 4 and 4 (2 attracting basins)
  period 4 and 4 (2 attracting basins)
• two period 2 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)" 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 Domains
• Basins of attraction

Bibliography

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