Classification of Fatou components

In mathematics, if f = P(z) / Q(z) is a rational function defined in the extended complex plane, and if

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

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

Attracting periodic point

Interior of Julia set contains periodic Fatou components marked with different colors

The components of the map f(z) = z − (z³ − 1) / 3z² that contains the attracting points that are the solutions to z³ = 1. This is because the map is the one to use for finding solutions to the equation z³ = 1 by Newton-Raphson formula. The solutions must naturally be attracting fixed points.

Herman ring

image of Julia set of a rational function which possess a Herman ring

The map

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

and t = 0.6151732... will produce a Herman ring.^[1] It is shown by Shishikura that the degree of such map must be at least 3, as in this example.

References

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

1. ^ Milnor, John W. (1990), Dynamics in one complex variable, arXiv:math/9201272 

See also

• No wandering domain theorem