The Classification of Fatou Components of rational maps

There are four families for a connected Fatou Component $U$ (this is a component of the complement to the julia set obtained from some map $z \to P(z)/Q(z)$). The families are

• $U$ contains an attracting periodic point. (This is somehow very generic).
• $U$ contains a parabolic, (has an indifferent attracting point on the boundary).
• $U$ is a Siegel disk.
• $U$ is a Herman ring.

The two latter cases are somewhat special, and do not appear generically.