## Dynamics on the Riemann Sphere: A Bodil Branner FestschriftDynamics on the Riemann Sphere presents a collection of original research articles by leading experts in the area of holomorphic dynamics. These papers arose from the symposium Dynamics in the Complex Plane, held on the occasion of the 60th birthday of Bodil Branner. Topics covered range from Lattès maps to cubic polynomials over rational maps with Sierpinsky Carpets and Gaskets as Julia sets, as well as rational and entire transcendental maps with Herman rings. Contributors include Artur Avila (Paris VI, France), Arnault Chéritat (Toulouse, France), Robert L. Devaney (Boston, USA), Adrien Douady (Orsay, France), Nuria Fagella (Barcelona, Spain), Christian Henriksen (Lyngby, Denmark), Wolf Jung (Aachen, Germany), Tomoki Kawahira (Kyoto, Japan), Tan Lei (Cergy Pontoise, France), Mikhail Lyubich (Stony Brook, USA), Carsten Lunde Petersen (Roskilde, Denmark), John Milnor (Stony Brook, USA), Pascale Roesch (Lille, France). |

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### Contents

Introduction | 3 |

BrannerHubbard motions and attracting dynamics | 45 |

Examples of Feigenbaum Julia sets with small Hausdorff dimension | 71 |

Parabolic explosion and the size of Siegel disks in the quadratic family | 89 |

On capture zones for the family z z2 + hz2 | 121 |

Homeomorphisms of the Mandelbrot Set | 139 |

Arnold Disks and the Moduli of Herman Rings of | 161 |

Stretching rays and their accumulations following Pia Willumsen | 183 |

Conjectures about the BrannerHubbard motion of Cantor sets in C | 209 |

### Common terms and phrases

affine map analytic angles annulus attracting dynamics base point Beltrami form BH-motion Bottcher coordinate boundary bounded Bq(oo Branner Branner-Hubbard Brjuno Cantor set Chebyshev combinatorial compact conformal radius conjugacy conjugate connected component connectedness locus construction contains converges corresponding critical orbit critical points critical values cubic polynomials cycle cylinder defined denote disjoint domain Douady dynamical plane external ray Fatou Figure filled Julia set follows function Fx(z Hence Herman ring holomorphic motion homeomorphism Hubbard hyperbolic component immediate basin injective integer invariant iterates J(Fx Lattes maps Lemma linear linearizable Mandelbrot set Math metric Misiurewicz points Mobius modulus multiplier neighborhood orbifold parameter space periodic point perturbation points of Q postcritical points preimage proof of Theorem properties Proposition prove quadratic polynomials quasi-conformal rational maps repelling Riemann sphere rotation number satisfies Section Siegel disk Sierpinski curve subset Theorem 1.1 topological torus unique unit circle Yoccoz