Classification of Fatou components

Remarks

In the first two cases $z_0$ is a periodic point of $f$, and the superattracting case $g'(z_0) = 0$ is included in the first case. In the last two cases $g$ restricted to $U$ is conformally conjugate to an irrational rotation of a disc or of a round annulus. The proof uses the theory of normal families: limits of subsequences of $g^n$ on $U$ are either constant, giving the first two cases, or not, giving the last two. For $f(z) = z^2$, the unit disc is an attracting component with $z_0 = 0$. For $f(z) = z + z^2$, the fixed point $0$ has multiplier $1$ and lies on the boundary of a parabolic component. By Sullivan's theorem every Fatou component is eventually periodic, so these four types describe the long-run behaviour on the whole Fatou set.

Going deeper

For a student

The theorem is the starting point for describing the Fatou set of any rational map. Attracting and parabolic cycles each attract at least one critical point. So a map of degree $d$ has at most $2d - 2$ such cycles, the number of its critical points counted with multiplicity. Siegel discs and Herman rings are rotation domains; their boundaries lie in the closure of the critical orbits. Polynomials have no Herman rings, so for them only three cases occur.