Classification of Fatou components

Source: Wikipedia, the free encyclopedia.

In

Fatou set. They were named after Pierre Fatou
.

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: 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.
  • Julia set (white) and Fatou set (dark red/green/blue) for '"`UNIQ--postMath-00000008-QINU`"' with '"`UNIQ--postMath-00000009-QINU`"' in the complex plane.
    Julia set (white) and Fatou set (dark red/green/blue) for with in the complex plane.
  • Julia set with parabolic cycle
    Julia set with parabolic cycle
  • Julia set with Siegel disc (elliptic case)
    Julia set with Siegel disc (elliptic case)
  • Julia set with Herman ring
    Julia set with Herman ring

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.

  • Dynamic plane consist of Fatou 2 superattracting period 1 basins, each has only one component.
    Dynamic plane consist of Fatou 2 superattracting period 1 basins, each has only one component.
  • Level curves and rays in superattractive case
    Level curves and rays in superattractive case
  • Julia set with superattracting cycles (hyperbolic) in the interior (period 2) and the exterior (period 1)
    Julia set with superattracting cycles (hyperbolic) in the interior (period 2) and the exterior (period 1)

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.

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

  • Herman+Parabolic
    Herman+Parabolic
  • Period 3 and 105
    Period 3 and 105
  • attracting and parabolic
    attracting and parabolic
  • period 1 and period 1
    period 1 and period 1
  • period 4 and 4 (2 attracting basins)
    period 4 and 4 (2 attracting basins)
  • two period 2 basins
    two period 2 basins

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)"[3][4] one example of such a function is:[5]

Wandering domain

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

See also

References

Bibliography

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