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

[math]\displaystyle{ f = \frac{P(z)}{Q(z)} }[/math]

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

[math]\displaystyle{ d(f) = \max(\deg(P),\, \deg(Q))\geq 2, }[/math]

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

1. [math]\displaystyle{ U }[/math] contains an attracting periodic point
2. [math]\displaystyle{ U }[/math] is parabolic
3. [math]\displaystyle{ U }[/math] 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. [math]\displaystyle{ U }[/math] 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.

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  Julia set (white) and Fatou set (dark red/green/blue) for [math]\displaystyle{ f: z\mapsto z-\frac{g}{g'}(z) }[/math] with [math]\displaystyle{ g: z \mapsto z^3-1 }[/math] in the complex plane.

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  Julia set with parabolic cycle

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  Julia set with Siegel disc (elliptic case)

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  Julia set with Herman ring

Attracting periodic point

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

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  Dynamic plane consist of Fatou 2 superattracting period 1 basins, each has only one component.

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  Level curves and rays in superattractive case

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  Julia set with superattracting cycles (hyperbolic) in the interior ( perieod 2) and the exterior (period 1)

Herman ring

The map

[math]\displaystyle{ f(z) = e^{2 \pi i t} z^2(z - 4)/(1 - 4z) }[/math]

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

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  Herman+Parabolic

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  Period 3 and 105

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  attracting and parabolic

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  period 1 and period 1

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  period 4 and 4 (2 attracting basins)

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  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: [math]\displaystyle{ f(z) = z - 1 + (1 - 2z)e^z }[/math]

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

References

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