Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 2: Surface Homeomorphisms and Rational Functions, タイヒミュラー理論と幾何学、トポロジー、および 動力学への応用, 第2巻:表面同相写像と有理関数, 9781943863006, 978-1-943863-00-6
学術書籍 理工学 Teichmüller Theory: タイヒミュラー理論
Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 2
数量
Desciption
This volume is the second of four volumes devoted to Teichmüller theory and its applications to geometry, topology, and dynamics. The first volume gave an introduction to Teichmüller theory. Volumes 2 through 4 prove four theorems by William Thurston:
- The classification of homeomorphisms of surfaces
- The topological characterization of rational maps
- The hyperbolization theorem for 3-manifolds that fiber over the circle
- The hyperbolization theorem for Haken 3-manifolds
These theorems are of extraordinary beauty in themselves, and the methods Thurston used to prove them were so novel and displayed such amazing geometric insight that to this day they have barely entered the accepted methods of mathematicians in the field.
The results sound more or less unrelated, but they are linked by a common thread: each one goes from topology to geometry.
Each says that either a topological problem has a natural geometry, or there is an understandable obstruction.
The proofs are closely related: you use the topology to set up an analytic mapping from a Teichmüller space to itself; the geometry arises from a fixed point of this mapping. Thurston proceeds to show that if there is no fixed point, then some system of simple closed curves is an obstruction to finding a solution.These theorems have been quite difficult to approach, in part because Thurston never published complete proofs of any of them.
Contents:
Chapter 8 The classification of homeomorphisms of surfaces
8.1 The classification theorem
8.2 Periodic and reducible homeomorphisms
8.3 Pseudo-Anosov homeomorphisms
8.4 Proof of the classification theorem
8.5 The structure in the reducible case
Chapter 9 Dynamics of polynomials
9.1 Julia sets
9.2 Fixed points
9.3 Green's functions, Bottcher coordinates
9.4 Extending f_0 to S^1
9.5 External rays at rational angles land
Chapter 10 Rational functions
10.1 Introduction
10.1 Thurston mappings
10.2 Thurston maps associated to spiders
10.3 Thurston obstructions for spider maps and Levy cycles
10.4 Julia sets of quadratic polynomials with superattracting cycles
10.5 Parameter spaces for quadratic polynomials
10.6 The Thurston pullback mapping
10.7 The derivative and coderivative of Thurston pullback mapping
10.8 The necessity of the eigenvalue criterion
10.9 Convergence in moduli spaces implies convergence in Teichmuller space
10.10 Asymptotic geometry of Riemann surfaces
10.11 Sufficiency of the eigenvalue criterion
Appendix C1 The Perron-Frobenius theorem
Appendix C2 The Alexander trick
Appendix C3 Homotopy implies isotopy
Appendix C4 The mapping class group and outer automorphisms
Appendix C5 Totally real stretch factors
Appendix C6 Irrationally indifferent fixed points
Appendix C7 Examples of Thurston pullback maps
Appendix C8 Branched maps with nonhyperbolic orbifolds
Appendix C9 The Sullivan dictionary
Bibliography
Index
