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2-sphere affine conjugate assume attracting fixed points branched cover chapter closed curve component of S2 connected Julia sets corollary counted without multiplicities critical fixed points critical value critically finite curve Levy cycle definition degenerate Levy cycle degree d polynomial denote disk component distinct critical fixed external rays landing Fatou components Fatou set Figure filled Julia set fixed critical points formal mating forward orbit Hence holomorphic homeomorphic homotopic Hubbard tree hyperbolic hyperbolic component immediate attraction basin inflection point inflection value internal ray isotopic Jordan curve Jordan domain lemma locally connected Mandelbrot set Newton's method Note open disk parabubble ray points of F preimage prove puzzle partition quadratic polynomials quadratic rational map rational map ray with angle Remark repelling fixed point Ri(t Riemann sphere Riemann-Hurwitz formula satisfying GA1 simply connected subset theorem 3.2 Thurston equator Thurston equivalent Thurston obstruction topological Newton map Yoccoz polynomial