## Combinations of Complex Dynamical Systems, Issue 1827This work is a research-level monograph whose goal is to develop a general combination, decomposition, and structure theory for branched coverings of the two-sphere to itself, regarded as the combinatorial and topological objects which arise in the classification of certain holomorphic dynamical systems on the Riemann sphere. It is intended for researchers interested in the classification of those complex one-dimensional dynamical systems which are in some loose sense tame. The program is motivated by the dictionary between the theories of iterated rational maps and Kleinian groups. |

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### Contents

Introduction | 1 |

12 Thurstons Characterization and Rigidity Theorem Standard definitions | 2 |

13 Examples | 6 |

133 An obstructed expanding Thurston map | 8 |

134 A subdivision rule | 11 |

14 Summary of this work | 12 |

15 Survey of previous results | 14 |

Theorem 16 Quadratic Mating Theorem | 17 |

43 Statement of Uniqueness of Combinations Theorem | 61 |

44 Proof of Uniqueness of Combinations Theorem | 62 |

443 Reduction to simple form | 63 |

444 Conclusion of proof of Uniqueness Theorem | 67 |

Decomposition | 69 |

52 Standard form with respect to a multicurve | 71 |

54 Proof of Decomposition Theorem | 76 |

Uniqueness of decompositions | 79 |

153 Parameter space | 20 |

154 Combinations via quasiconformal surgery | 22 |

155 From pf to geometrically finite and beyond | 23 |

16 Analogy with threemanifolds | 24 |

17 Connections | 27 |

172 Gromov hyperbolic spaces and interesting groups | 28 |

173 Cannons conjecture | 29 |

181 Overview of decomposition and combination | 30 |

182 Embellishments Technically convenient assumption | 31 |

183 Invariant multicurves for embellished map of spheres Thurston linear map | 32 |

19 Tameness assumptions | 33 |

Preliminaries | 37 |

21 Mapping trees | 39 |

22 Map of spheres over a mapping tree | 44 |

23 Map of annuli over a mapping tree | 46 |

Combinations | 49 |

32 Critical gluing data | 50 |

33 Construction of combination | 52 |

statement of Combination Theorem | 53 |

Uniqueness of combinations | 59 |

42 Combinatorial equivalence of sphere and annulus maps | 60 |

Counting classes of annulus maps | 83 |

72 Proof of Number of Classes of Annulus Maps Theorem | 84 |

722 Characterization of combinatorial equivalence by group action | 85 |

723 Reduction to abelian groups | 86 |

Applications to mapping class groups | 89 |

82 Proof of Twist Theorem | 90 |

822 Conclusion of proof of Twist Theorem | 91 |

83 When Thurston obstructions intersect | 92 |

832 Maps with intersecting obstructions have large mapping class groups | 93 |

Examples | 95 |

92 Matings | 96 |

93 Generalized matings | 98 |

94 Integral Lattes examples | 101 |

Canonical Decomposition Theorem | 105 |

102 Statement of Canonical Decomposition Theorem | 107 |

103 Proof of Canonical Decomposition Theorem | 108 |

1032 Conclusion of proof | 109 |

111 | |

117 | |

### Common terms and phrases

amalgamating data annular edge annulus maps basepoint bijection boundary component boundary values branched covering Canonical Decomposition classes of annulus combination procedure Combinations Theorem combinatorial automorphisms combinatorial class combinatorially equivalent conjugacy conjugate connected components covering map critical gluing critical points cycle defined Dehn twists denote disjoint dynamics embellished branched covering embellished map finite rational maps fixed follows form with respect geometrically finite given gluing data Hence homeomorphism homotopic hyperbolic components invariant multicurve irreducible isotopic Julia set lamination Lemma Let F Mandelbrot set map F map of spheres mapping class group mapping scheme mapping tree mating matrix missing annulus missing disk maps multicurve F nonperipheral orbifold orientations pair parabolic postcritical set preimages Proposition quasiconformal quotient quotient space rational functions rational map So(x sphere maps standard form structure data subset Thurston linear map Thurston map Thurston's characterization topological gluing torus Twist Theorem vertices

### References to this book

Applications of Group Theory to Combinatorics Jack Koolen,Jin Ho Kwak,Ming-Yao Xu Limited preview - 2008 |