In iteration theory of rational functions, it is well known that any Fatou component is mapped onto another in an n-to-1 manner, and that periodic components are simply, doubly or infinitely connected.

For meromorphic functions, the situation is much more complicated. Using Ahlfors' theory of covering surfaces, we prove that Fatou components are mapped ‘nearly’ onto others, and that periodic components are again simply, doubly or infinitely connected. Instead of considering meromorphic functions with only one essential singularity, we allow countable sets of singularities and partly even sets of logarithmic capacity zero.

It remains open whether doubly connected periodic components of meromorphic functions with only one singularity are necessarily Herman rings (as holds for rational functions). However, there is a function with two singularities and a doubly connected periodic component which is not an Herman ring.