Classification of Fatou components

From formulasearchengine
Revision as of 11:49, 15 June 2014 by en>Adam majewski (→‎Transcendental case: link int)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


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


Rational case

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.

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

Attracting periodic point

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

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

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]

See also

External links

References

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