Combinations of Complex Dynamical Systems Book

1	Introduction	1
1.1	Motivation from dynamics - a brief sketch	1
1.2	Thurston's Characterization and Rigidity Theorem. Standard definitions	2
1.3	Examples	6
1.4	Summary of this work	12
1.5	Survey of previous results	14
1.6	Analogy with three-manifolds	24
1.7	Connections	27
1.8	Discussion of combinatorial subtleties	29
1.9	Tameness assumptions	33
2	Preliminaries	37
2.1	Mapping trees	39
2.2	Map of spheres over a mapping tree	44
2.3	Map of annuli over a mapping tree	46
3	Combinations	49
3.1	Topological gluing data	49
3.2	Critical gluing data	50
3.3	Construction of combination	52
3.4	Summary: statement of Combination Theorem	53
3.5	Properties of combinations	53
4	Uniqueness of combinations	59
4.1	Structure data and amalgamating data	59
4.2	Combinatorial equivalence of sphere and annulus maps	60
4.3	Statement of Uniqueness of Combinations Theorem	61
4.4	Proof of Uniqueness of Combinations Theorem	62
5	Decomposition	69
5.1	Statement of Decomposition Theorem	69
5.2	Standard form with respect to a multicurve	71
5.3	Maps in standard forms are amalgams	71
5.4	Proof of Decomposition Theorem	76
6	Uniqueness of decompositions	79
6.1	Statement of Uniqueness of Decompositions Theorem	79
6.2	Proof of Uniqueness of Decomposition Theorem	79
7	Counting classes of annulus maps	83
7.1	Statement of Number of Classes of Annulus Maps Theorem	83
7.2	Proof of Number of Classes of Annulus Maps Theorem	84
8	Applications to mapping class groups	89
8.1	The Twist Theorem	89
8.2	Proof of Twist Theorem	90
8.3	When Thurston obstructions intersect	92
9	Examples	95
9.1	Background from complex dynamics	95
9.2	Matings	96
9.3	Generalized matings	98
9.4	Integral Lattes examples	101
10	Canonical Decomposition Theorem	105
10.1	Cycles of a map of spheres, and their orbifolds	105
10.2	Statement of Canonical Decomposition Theorem	107
10.3	Proof of Canonical Decomposition Theorem	108
	References	111
	Index	117