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

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Appendix F. No Wandering Fatou Components

This appendix will outline a proof of the following. (Compare §16.)

Theorem F.l (Sullivan Nonwandering Theorem). Every Fatou component of a rational map is eventually periodic.

The intuitive idea of the proof is the following. Let U be any Fatou component for f, that is, any connected component of image\J(f). Suppose that we try to change the conformal structure on U. If f is to preserve this new conformal structure, then we must also change the conformal structure everywhere throughout the grand orbit of U in a compatible manner. If f(U) = U, then the condition that f preserves this structure imposes very strong ...

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