This section will begin the discussion of dynamics on Riemann surfaces other than the Riemann sphere. It turns out that the possibilities for dynamics on a hyperbolic surface are rather limited. Let us first restate the definition in greater generality, allowing Riemann surfaces which may be noncompact.
Definition. For a holomorphic map f : S → S of an arbitrary Riemann surface, the Fatou set of f is the union of all open sets U ⊂ S such that every sequence of iterates fonj|U either
(1) contains a locally uniformly convergent subsequence, or
(2) contains a subsequence which diverges locally uniformly from S, so that the images of a compact subset of U eventually leave any compact subset of S.
(If S is compact, ...