Classification of Fatou components

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In mathematics, Fatou components are components of the Fatou set.

Rational case[edit]

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
  4. is a Herman ring.

A Siegel disk is 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. A Herman ring is 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.

Examples[edit]

Attracting periodic point[edit]

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[edit]

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.

Transcendental case[edit]

Baker domain[edit]

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] Example function :[5]

Wandering domain[edit]

Finally, transcendental maps also may have wandering domains: these are Fatou components that are not eventually periodic.

See also[edit]

References[edit]

  • Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993.
  • Alan F. Beardon Iteration of Rational Functions, Springer 1991.
  1. ^ wikibooks : parabolic Julia sets
  2. ^ Milnor, John W. (1990), Dynamics in one complex variable, arXiv:math/9201272Freely accessible 
  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