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Rational case

If f is a rational function

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

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

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

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

1. U {\displaystyle U} contains an attracting periodic point
2. U {\displaystyle U} is parabolic¹
3. U {\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. U {\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.

Attracting periodic point

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

Herman ring

The map

f ( z ) = e 2 π i t z 2 ( z − 4 ) / ( 1 − 4 z ) {\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

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)"³⁴ one example of such a function is:⁵ f ( z ) = z − 1 + ( 1 − 2 z ) e z {\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 Domains⁶
• Basins of attraction

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

References

1. wikibooks : parabolic Julia sets https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/parabolic ↩

2. Milnor, John W. (1990), Dynamics in one complex variable, arXiv:math/9201272, Bibcode:1992math......1272M /wiki/John_Milnor ↩

3. An Introduction to Holomorphic Dynamics (with particular focus on transcendental functions)by L. Rempe http://pcwww.liv.ac.uk/~lrempe/Talks/liverpool_seminar_2006.pdf ↩

4. Siegel Discs in Complex Dynamics by Tarakanta Nayak http://www.ncnsd.org/proceedings/proceeding05/paper/185.pdf ↩

5. A transcendental family with Baker domains by Aimo Hinkkanen, Hartje Kriete and Bernd Krauskopf http://www.math.uiuc.edu/~aimo/anim.html ↩

6. JULIA AND JOHN REVISITED by NICOLAE MIHALACHE https://arxiv.org/abs/0803.3889 ↩