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

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

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

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

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 : 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. ${\displaystyle U}$ 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 ${\displaystyle f:z\mapsto z-{\frac {g}{g'}}(z)}$ with ${\displaystyle g:z\mapsto z^{3}-1}$ in the complex plane.
• 
  Julia set with parabolic cycle
• 
  Julia set with Siegel disc (elliptic case)
• 
  Julia set with Herman ring

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.

• 
  Dynamic plane consist of Fatou 2 superattracting period 1 basins, each has only one component.
• 
  Level curves and rays in superattractive case
• 
  Julia set with superattracting cycles (hyperbolic) in the interior (period 2) and the exterior (period 1)

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.

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
• 
  Period 3 and 105
• 
  attracting and parabolic
• 
  period 1 and period 1
• 
  period 4 and 4 (2 attracting 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)" ^{[3] [4]} one example of such a function is: ^[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.

See also

• No-wandering-domain theorem
• Montel's theorem
• John Domains ^[6]
• Basins of attraction

References

1. wikibooks : parabolic Julia sets
2. Milnor, John W. (1990), Dynamics in one complex variable, arXiv: math/9201272 , Bibcode:1992math......1272M
3. An Introduction to Holomorphic Dynamics (with particular focus on transcendental functions)by L. Rempe
4. Siegel Discs in Complex Dynamics by Tarakanta Nayak
5. A transcendental family with Baker domains by Aimo Hinkkanen, Hartje Kriete and Bernd Krauskopf
6. JULIA AND JOHN REVISITED by NICOLAE MIHALACHE

Bibliography

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