Iteration of Rational Functions: Complex Analytic Dynamical Systems

analytic map argument assume attracting fixed point Beltrami coefficient bounded Cantor set closed curve closure compact subset completely invariant completes the proof complex sphere component F component of F conjugacy conjugate constant construct contains covering map D₁ deduce define deg(R denote disjoint equicontinuous Euclidean example EXERCISE F₁ Fatou set finite critical point finite number follows forward orbit given hence Herman ring homeomorphism hyperbolic metric indifferent fixed point infinitely connected iterates Jordan curve Julia set lies map of degree Möbius map N₁ neighbourhood non-constant normal obtain open set origin periodic points petals polynomial positive integer positive number proof of Theorem prove Theorem quasiconformal rational function rational map rationally indifferent cycle repelling fixed point result Riemann-Hurwitz satisfies sequence Siegel disc simply connected simply connected domain Suppose topological union unit disc W₁ wandering domain z₁ zero