### §5. Dynamics on Hyperbolic Surfaces

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 *f*^{onj}|_{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, ...