Teichmüller
Theory and Applications to Geometry, Topology, and Dynamics
Volume 2: Surface Homeomorphisms and Rational Functions
by John
H. Hubbard
262 pages, hardcover, smythesewn binding, 108 color illustrations. $63.
April 2016, ISBN 9781943863006 Add to shopping cart
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See also Volume 1: Teichmüller Theory
Scroll down for links to sample pages.
Table of Contents
Chapter 8 The classification of homeomorphisms of surfaces
8.1 The classification theorem
8.2 Periodic and reducible homeomorphisms
8.3 PseudoAnosov 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, Böttcher 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 mapps 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 s_f
10.7 The derivative and coderivative of s_f
10.8 The necessity of the eigenvalue criterion
10.9 Convergence in moduli spaces implies convergence in Teichmüller space
10.10 Asymptotic geometry of Riemann surfaces
10.11 Sufficiency of the eigenvalue criterion
Appendix C1 The PerronFrobenius 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 dictionaryFrom the introduction by the author:
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 3manifolds that fiber over the circle
 The hyperbolization theorem for Haken 3manifolds
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.
In this second volume, in Chapters 8 and 10, I prove the first
two of these theorems. Both proofs use most of the contents of Volume 1.
Chapters 8 and 10 also contain an extensive collection of examples. In
Chapter 8 we exhibit some constructions of pseudoAnosov
mappings; Chapter 10 describes spiders and uses them to analyze the
structure of the Mandelbrot set.
This requires the basics of dynamics in one complex variable; Chapter 9 contains the necessary material.
There are nine appendices. The first proves the PerronFrobenius
theorem; the next three treat some technical issues concerning
the topology of surfaces and their homeomorphisms: the Alexander
trick, homotopies and isotopies for surfaces, the algebraic
definition of the mapping class group, and a list of generators for the
mapping class group.
Appendix C5 outlines a recent result of Ursula Hamenstadt saying that
``most'' stretch factors of pseudoAnosov homeomorphisms are totally
real.
Appendix C6 gives a bare bones treatment of irrationally indifferent
fixed points to complete the cases treated in Chapter 9. It isn't
really relevant to this book, but it is of such importance
to complex dynamics that it seems a good idea to include it.
Appendices C7 and C8 are relevant to Chapter 10. Appendix
C7 gives some examples of Thurston pullback maps. Appendix C8completes
the main theorem by an analysis of postcritically finite branched maps
with nonhyperbolic orbifolds.
Appendix C9 gives an introduction to Sullivan's dictionary, which
pairs statements about 1dimensional complex dynamics and
statements about 3dimensional hyperbolic geometry. Several topics are
relevant to this volume, but the appendix is really about the
relations of these topics to those that will be treated in Volumes 3
and 4.
Sample pages
index (in
pdf)
Chapter 8 (first 4 pages, pdf)
Chapter 9 (first 4 pages, pdf)
Chapter 10 (first 4 pages, pdf)
Mandelbrot set, 2 figures
Appendix C1 (PerronFrobenius, first page, pdf)
Appendix C2 (Alexander trick, first page, pdf)
Appendix C3 (Homotopy implies isotopy, first page, pdf)
Appendix C4 (The mapping class group and outer automorphisms, first page, pdf)
Appendix C5 (Totally real stretch factors, first page, pdf)
Appendix C6 (Linearizing at irrationally indifferent fixed points, first page, pdf)
Appendix C7 (Thurston pullback maps, first page, pdf)
Appendix C8 (Thurston maps with nonhyperbolic orbifolds, first page, pdf)
Appendix C9 (Sullivan's dictionary, first page, pdf)
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