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Teichmüller Theory and Applications to Geometry, Topology, and Dynamics

Volume 2: Surface Homeomorphisms and Rational Functions

by John H. Hubbard

262 pages, hardcover, smythe-sewn binding, 108 color illustrations.  $63.
April 2016, ISBN 978-1-943863-00-6

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See also
Volume 1: Teichmüller Theory

For a preview of Volumes 2 and 3, see

 Volume 3: Manifolds that Fiber over the Circle

Volume 4: Hyberbolization of Haken Manifolds

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 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, 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 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

From 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 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.

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  pseudo-Anosov 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 Perron-Frobenius 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 pseudo-Anosov 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 1-dimensional complex dynamics and statements about 3-dimensional 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 (Perron-Frobenius, 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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book cover


successive blowups of Mandelbrot set



108 color illustrations

cartoon of Mandelbrot set


repelling fixed point