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

Rational case

If f is a rational function

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

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

d ( f ) = max ( deg ⁡ ( P ) , deg ⁡ ( Q ) ) ≥ 2 , {\displaystyle d(f)=\max(\deg(P),\,\deg(Q))\geq 2,}

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

  1. U {\displaystyle U} contains an attracting periodic point
  2. U {\displaystyle U} is parabolic
  3. U {\displaystyle U} 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. U {\displaystyle U} 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 f : z ↦ z − g g ′ ( z ) {\displaystyle f:z\mapsto z-{\frac {g}{g'}}(z)} with g : z ↦ z 3 − 1 {\displaystyle g:z\mapsto z^{3}-1} in the complex plane.
  • Julia set with parabolic cycle
  • Julia set with Siegel disc (elliptic case)
  • Julia set with Herman ring

Attracting periodic point

The components of the map f ( z ) = z − ( z 3 − 1 ) / 3 z 2 {\displaystyle f(z)=z-(z^{3}-1)/3z^{2}} contain the attracting points that are the solutions to z 3 = 1 {\displaystyle z^{3}=1}. This is because the map is the one to use for finding solutions to the equation z 3 = 1 {\displaystyle z^{3}=1} 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.
  • Level curves and rays in superattractive case
  • Julia set with superattracting cycles (hyperbolic) in the interior (period 2) and the exterior (period 1)

Herman ring

The map

f ( z ) = e 2 π i t z 2 ( z − 4 ) / ( 1 − 4 z ) {\displaystyle f(z)=e^{2\pi it}z^{2}(z-4)/(1-4z)}

and t = 0.6151732... will produce a Herman ring. 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
  • Period 3 and 105
  • attracting and parabolic
  • period 1 and period 1
  • period 4 and 4 (2 attracting 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)" one example of such a function is: f ( z ) = z − 1 + ( 1 − 2 z ) e z {\displaystyle f(z)=z-1+(1-2z)e^{z}}

Wandering domain

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

See also

Bibliography