Buch, Englisch, 288 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 418 g
Reihe: Mathematical Notes
Buch, Englisch, 288 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 418 g
Reihe: Mathematical Notes
ISBN: 978-0-691-14735-2
Verlag: Princeton University Press
This book provides a detailed exposition of William Thurston's work on surface homeomorphisms, available here for the first time in English. Based on material of Thurston presented at a seminar in Orsay from 1976 to 1977, it covers topics such as the space of measured foliations on a surface, the Thurston compactification of Teichmüller space, the Nielsen-Thurston classification of surface homeomorphisms, and dynamical properties of pseudo-Anosov diffeomorphisms. Thurston never published the complete proofs, so this text is the only resource for many aspects of the theory.Thurston was awarded the prestigious Fields Medal in 1982 as well as many other prizes and honors, and is widely regarded to be one of the major mathematical figures of our time. Today, his important and influential work on surface homeomorphisms is enjoying continued interest in areas ranging from the Poincaré conjecture to topological dynamics and low-dimensional topology.Conveying the extraordinary richness of Thurston's mathematical insight, this elegant and faithful translation from the original French will be an invaluable resource for the next generation of researchers and students.
Fachgebiete
Weitere Infos & Material
Preface ix
Foreword to the First Edition ix
Foreword to the Second Edition x
Translators? Notes xi
Acknowledgments xii
Abstract xiii
Chapter 1 An Overview of Thurston?s Theorems on Surfaces 1
Valentin Po?naru
1.1 Introduction 1
1.2 The Space of Simple Closed Curves 2
1.3 Measured Foliations 3
1.4 Teichm?ller Space 5
1.5 Pseudo-Anosov Diffeomorphisms 6
1.6 The Case of the Torus 8
Chapter 2 Some Reminders about the Theory of Surface Diffeomorphisms 14
Valentin Po?naru
2.1 The Space of Homotopy Equivalences of a Surface 14
2.2 The Braid Groups 15
2.3 Diffeomorphisms of the Pair of Pants 19
Chapter 3 Review of Hyperbolic Geometry in Dimension 2 25
Valentin Po?naru
3.1 A Little Hyperbolic Geometry 25
3.2 The Teichm?ller Space of the Pair of Pants 27
3.3 Generalities on the Geometric Intersection of Simple Closed Curves 35
3.4 Systems of Simple Closed Curves and Hyperbolic Isometries 42
V4 The Space of Simple Closed Curves in a Surface 44
Valentin Po?naru
4.1 The Weak Topology on the Space of Simple Closed Curves 44
4.2 The Space of Multicurves 46
4.3 An Explicit Parametrization of the Space of Multicurves 47
A Pair of Pants Decompositions of a Surface 53
Albert Fathi
Chapter 5 Measured Foliations 56
Albert Fathi and Fran?ois Laudenbach
5.1 Measured Foliations and the Euler-Poincar? Formula 56
5.2 Poincar? Recurrence and the Stability Lemma 59
5.3 Measured Foliations and Simple Closed Curves 62
5.4 Curves as Measured Foliations 71
B Spines of Surfaces 74
Valentin Po?naru
Chapter 6 The Classification of Measured Foliations 77
Albert Fathi
6.1 Foliations of the Annulus 78
6.2 Foliations of the Pair of Pants 79
6.3 The Pants Seam 84
6.4 The Normal Form of a Foliation 87
6.5 Classification of Measured Foliations 92
6.6 Enlarged Curves as Functionals 97
6.7 Minimality of the Action of the Mapping Class Group on PMF 98
6.8 Complementary Measured Foliations 100
C Explicit Formulas for Measured Foliations 101
Albert Fathi
Chapter 7 Teichm?ller Space 107
Adrien Douady; notes by Fran?ois Laudenbach
Chapter 8 The Thurston Compactification of Teichm?ller Space 118
Albert Fathi and Fran?ois Laudenbach
8.1 Preliminaries 118
8.2 The Fundamental Lemma 121
8.3 The Manifold T 125
D Estimates of Hyperbolic Distances 128
Albert Fathi
D.1 The Hyperbolic Distance from i to a Point z0 128
D.2 Relations between the Sides of a Right Hyperbolic Hexagon 129
D.3 Bounding Distances in Pairs of Pants 131
Chapter 9 The Classification of Surface Diffeomorphisms 135
Valentin Po?naru
9.1 Preliminaries 135
9.2 Rational Foliations (the Reducible Case) 136
9.3 Arational Measured Foliations 137
9.4 Arational Foliations with ? = 1 (the Finite Order Case) 140
9.5 Arational Foliations with ? 6= 1 (the Pseudo-Anosov Case) 141
9.6 Some Properties of Pseudo-Anosov Diffeomorphisms 150
Chapter 10 Some Dynamics of Pseudo-Anosov Diffeomorphisms 154
Albert Fathi and Michael Shub
10.1 Topological Entropy 154
10.2 The Fundamental Group and Entropy 157
10.3 Subshifts of Finite Type 162
10.4 The Entropy of Pseudo-Anosov Diffeomorphisms 165
10.5 Constructing Markov Partitions for Pseudo-Anosov Diffeomorphisms
171
10.6 Pseudo-Anosov Diffeomorphisms are Bernoulli 173
Chapter 11 Thurston?s Theory for Surfaces with Boundary 177
Fran?ois Laudenbach
11.1 The Spaces of Curves and Measured Foliations 177
11.2 Teichm?ller Space and Its Compactification 179
11.3 A Sketch of the Classification of Diffeomorphisms 180
11.4 Thurston?s Classification and Nielsen?s Theorem 184
11.5 The Spectral Theorem 188
Chapter 12 Uniqueness Theorems for Pseudo-Anosov Diffeomorphisms 191
Albert Fathi and Valentin Po?naru
12.1 Statement of Results 191
12.2 The Perron-Frobenius Theorem and Markov Partitio
