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

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

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

\max(\deg(P),\, \deg(Q))\geq 2,

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

  1. U contains an attracting periodic point
  2. U is parabolic[1]
  3. U is a Siegel disc
  4. U is a Herman ring.

One can prove that case 3 only occurs when f(z) is analytically conjugate to a Euclidean rotation of the unit disc onto itself, and case 4 only occurs when f(z) is analytically conjugate to a Euclidean rotation of some annulus onto itself.

Examples[edit]

Attracting periodic point[edit]

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

Herman ring[edit]

The map

f(z) = e^{2 \pi i t} 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.

Transcendental case[edit]

In case of transcendental functions there is also Baker domain: "domains on which the iterates tend to an essential singularity (not possible for polynomials and rational functions)"[3][4] Example function :[5]

f(z) = z - 1 + (1 - 2z)e^z

See also[edit]

External links[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/9201272 
  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