Combinations of Complex Dynamical Systems, Issue 1827

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Springer Science & Business Media, Oct 8, 2003 - Mathematics - 118 pages
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This 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
References
111
Index
117
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