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Classification of Fatou components

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In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou.

Rational case

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If f is a rational function

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

then for a periodic component of the Fatou set, exactly one of the following holds:

  1. contains an attracting periodic point
  2. is parabolic[1]
  3. 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. 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

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The components of the map contain the attracting points that are the solutions to . This is because the map is the one to use for finding solutions to the equation by Newton–Raphson formula. The solutions must naturally be attracting fixed points.

Herman ring

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The map

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

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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

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Baker domain

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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]

Wandering domain

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Transcendental maps may have wandering domains: these are Fatou components that are not eventually periodic.

See also

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References

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  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