Combinations of Complex Dynamical Systems

Kevin M. Pilgrim (auth.)9783540201731, 3-540-20173-4

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.

Table of contents :
1 Introduction….Pages 1-35
2 Preliminaries….Pages 37-48
3 Combinations….Pages 49-57
4 Uniqueness of combinations….Pages 59-68
5 Decomposition….Pages 69-77
6 Uniqueness of decompositions….Pages 79-81
7 Counting classes of annulus maps….Pages 83-88
8 Applications to mapping class groups….Pages 89-94
9 Examples….Pages 95-103
10 Canonical Decomposition Theorem….Pages 105-109
References….Pages 111-116