title: Classification of Fatou components
text: # Classification of Fatou components
  
  In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou.
  
  ## Rational case
  
  If f is a rational function
  
      $f={\frac {P(z)}{Q(z)}}$
  
  defined in the extended complex plane, and if it is a nonlinear function (degree > 1)
  
      $d(f)=\max(\deg(P),\,\deg(Q))\geq 2,$
  
  then for a periodic component $U$ of the Fatou set, exactly one of the following holds:
  
  * $U$ contains an attracting periodic point
  * $U$ is parabolic
  * $U$ 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.
  * $U$ 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 $f:z\mapsto z-{\frac {g}{g'}}(z)$ with $g:z\mapsto z^{3}-1$ 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 $f(z)=z-(z^{3}-1)/3z^{2}$ 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.
  
  * 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 ( perieod 2) and the exterior (period 1)
  
  ### Herman ring
  
  The map
  
      $f(z)=e^{2\pi it}z^{2}(z-4)/(1-4z)$
  
  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:
  
  $f(z)=z-1+(1-2z)e^{z}$
  
  ### Wandering domain
  
  Transcendental maps may have wandering domains: these are Fatou components that are not eventually periodic.
  
  