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math04hw5 - z 2 shows that the convergence need not be...

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Homework 5: Due May 17th, Monday, 2010 1. (1) If a holomorphic map f : D D (where D is the unit disk) fixes the origin and is not a rotation, prove that the successive images f n ( z ) (here f n means the successive compositions f f ... by n times) converge to zero for all z in the open disk D . (2) Prove that this convergence is uniform on compact subsets of D . (The example f ( z ) =
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Unformatted text preview: z 2 shows that the convergence need not be uniform on all of D .) 2. If f : b C → b C is rational of degree d = 1, then show that the Julia set J ( f ) is either vacuous or consists of a single repelling or parabolic fixed point. 3. Prove that the family { F n } is not normal at any repelling periodic point for F . 1...
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