---
title: "Classification of Fatou components"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Classification_of_Fatou_components"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T09:08:18.797856+00:00"
instance: "kb-cron"
---

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:

  
    
      
        U
      
    
    {\displaystyle U}
  
 contains an attracting periodic point

  
    
      
        U
      
    
    {\displaystyle U}
  
 is parabolic

  
    
      
        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.

  
    
      
        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.


=== 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.


=== 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


== 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 ==
No-wandering-domain theorem
Montel's theorem
John Domains
Basins of attraction


== References ==


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