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Dynamics in One Complex Variable. (AM-160), 3rd Edition by John Milnor

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§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 : SS of an arbitrary Riemann surface, the Fatou set of f is the union of all open sets US 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, ...

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