Classification of Fatou components

If f is a rational function

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

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

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

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

1. ${\displaystyle U}$ contains an attracting periodic point
2. ${\displaystyle U}$ is parabolic[1]
3. ${\displaystyle U}$ 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. ${\displaystyle U}$ 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 ${\displaystyle f:z\mapsto z-{\frac {g}{g'}}(z)}$ with ${\displaystyle g:z\mapsto z^{3}-1}$ in the complex plane.
• 
  Julia set with superattracting cycles (hyperbolic) in the interior and the exterior
• 
  Level curves and rays in superattractive case
• 
  Julia set with parabolic cycle
• 
  Julia set with Siegel disc (elliptic case)
• 
  Julia set with Herman ring

• No wandering domain theorem
• Montel's theorem
• John Domains[6]

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