Classification of Fatou components

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

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 )

$\max(\deg(P),\,\deg(Q))\geq 2,$

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
4. $U$ is a Herman ring.

One can prove that case 3 only occurs when f(z) is analytically conjugate to a Euclidean rotation of the unit disc onto itself, and case 4 only occurs when f(z) is analytically conjugate to a Euclidean rotation of some annulus onto itself.

Examples

• 

  Julia set with attracting cycle

• 

  Parabolic Julia set

• 

  Julia set with Siegel disc

• 

  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^{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.

Transcendental case

In case of transcendental functions there is also Baker domain: "domains on which the iterates tend to an essential singularity (not possible for polynomials and rational functions)" Example function :

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

See also

• No wandering domain theorem

External links

• Cremer Julia sets

References

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