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. The program is motivated by the dictionary between the theories of iterated rational maps and Kleinian groups.
Author(s): Kevin M. Pilgrim
Series: Lecture Notes in Mathematics
Edition: 1
Publisher: Springer Berlin Heidelberg
Year: 2008
Language: English
Pages: 120
Title......Page 1
Preface......Page 5
Contents......Page 6
1.1 Motivation from dynamics–a brief sketch......Page 9
1.2 Thurston’s Characterization and Rigidity Theorem. Standard definitions......Page 10
1.3.2 An obstructed mating......Page 14
1.3.3 An obstructed expanding Thurston map......Page 16
1.3.4 A subdivision rule......Page 19
1.4 Summary of this work......Page 20
1.5.1 Enumeration......Page 22
1.5.2 Combinations and decompositions......Page 25
1.5.3 Parameter space......Page 28
1.5.4 Combinations via quasiconformal surgery......Page 30
1.5.5 From p.f. to geometrically finite and beyond......Page 31
1.6 Analogy with three-manifolds......Page 32
1.7.1 Geometric Galois theory......Page 35
1.7.2 Gromov hyperbolic spaces and interesting groups......Page 36
1.8 Discussion of combinatorial subtleties......Page 37
1.8.1 Overview of decomposition and combination......Page 38
1.8.2 Embellishments. Technically convenient assumption.......Page 39
1.8.3 Invariant multicurves for embellished map of spheres. Thurston linear map.......Page 40
1.9 Tameness assumptions......Page 41
2 Preliminaries.pdf......Page 44
2.1 Mapping trees......Page 46
2.2 Map of spheres over a mapping tree......Page 51
2.3 Map of annuli over a mapping tree......Page 53
3.1 Topological gluing data......Page 56
3.2 Critical gluing data......Page 57
3.3 Construction of combination......Page 59
3.5 Properties of combinations......Page 60
4 Uniqueness of combinations.pdf......Page 65
5.1 Statement of Decomposition Theorem......Page 75
5.3 Maps in standard forms are amalgams......Page 77
5.4 Proof of Decomposition Theorem......Page 82
6.2 Proof of Uniqueness of Decomposition Theorem......Page 84
7.1 Statement of Number of Classes of Annulus Maps Theorem......Page 87
7.2.1 Homeomorphism of annuli. Index.......Page 88
7.2.2 Characterization of combinatorial equivalence by group action.......Page 89
7.2.4 Computations and conclusion of proof......Page 90
8.1 The Twist Theorem......Page 93
8.2.1 Combinatorial automorphisms of annulus maps......Page 94
8.2.2 Conclusion of proof of Twist Theorem......Page 95
8.3.1 Statement of Intersecting Obstructions Theorem......Page 96
8.3.2 Maps with intersecting obstructions have large mapping class groups......Page 97
9.1 Background from complex dynamics......Page 99
9.2 Matings......Page 100
9.3 Generalized matings......Page 102
9.4 Integral Lattès examples......Page 105
10.1 Cycles of a map of spheres, and their orbifolds......Page 108
10.2 Statement of Canonical Decomposition Theorem......Page 110
10.3.1 Characterization of rational cycles with hyperbolic orbifold......Page 111
10.3.2 Conclusion of proof......Page 112
References......Page 113
Index......Page 119
