Fixed Point Theorems And Their Applications

by Ioannis Farmakis, Martin Moskowitz

Overview

This is the only book that deals comprehensively with fixed point theorems throughout mathematics. Their importance is due, as the book demonstrates, to their wide applicability. Beyond the first chapter, each of the other seven can be read independently of the others so the reader has much flexibility to follow his/her own interests. The book is written for graduate students and professional mathematicians and could be of interest to physicists, economists and engineers.

Product Details

ISBN-13:	9789814458917
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	09/18/2013
Pages:	248
Product dimensions:	5.90(w) x 9.10(h) x 0.80(d)

Table of Contents

Preface and Acknowledgments ix

Introduction 1

1 Early Fixed Point Theorems 3

1.1 The Picard-Banach Theorem 3

1.2 Vector Fields on Spheres 5

1.3 Proof of the Brouwer Theorem and Corollaries 9

1.3.1 A Counter Example 12

1.3.2 Applications of the Brouwer Theorem 15

1.3.3 The Perron-Frobenius Theorem 16

1.3.4 Google; A Billion Dollar Fixed Point Theorem 21

1.4 Fixed Point Theorems for Groups of Affine Maps of Rⁿ 24

1.4.1 Affine Maps and Actions 25

1.4.2 Affine Actions of Non Compact Groups 30

2 Fixed Point Theorems in Analysis 35

2.1 The Schaüder-Tychonoff Theorem 36

2.1.1 Proof of the Schaüder-Tychonoff Theorem 38

2.2 Applications of the Schaüder-Tychonoff Theorem 43

2.3 The Theorems of Hahn, Kakutani and Markov-Kakutahi 46

2.4 Amenable Groups 51

2.4.1 Amenable Groups 52

2.4.2 Structure of Connected Amenable Lie Groups 54

3 The Lefschetz Fixed Point Theorem 57

3.1 The Lefschetz Theorem for Compact Polyhedra 58

3.1.1 Projective Spaces 62

3.2 The Lefschetz Theorem for a Compact Manifold 64

3.2.1 Preliminaries from Differential Topology 64

3.2.2 Transversality 67

3.3 Proof of the Lefschetz Theorem 76

3.4 Some Applications 81

3.4.1 Maximal Tori in Compact Lie Groups 83

3.4.2 The Poincaré-Hopf's Index Theorem 87

3.5 The Atiyah-Bott Fixed Point Theorem 94

3.5.1 The Case of the de Rham Complex 104

4 Fixed Point Theorems in Geometry 109

4.1 Some Generalities on Riemannian Manifolds 110

4.2 Hadamard Manifolds and Cartan's Theorem 124

4.3 Fixed Point Theorems for Compact Manifolds 135

5 Fixed Points of Volume Preserving Maps 143

5.1 The Poincaré Recurrence Theorem 143

5.2 Symplectic Geometry and its Fixed Point Theorems 146

5.2.1 Introduction to Symplectic Geometry 146

5.2.2 Fixed Points of Symplectomorphisms 153

5.2.3 Arnold's Conjecture 154

5.3 Poincaré's Last Geometric Theorem 155

5.4 Automorphisms of Lie Algebras 163

5.5 Hyperbolic Automorphisms of a Manifold 167

5.5.1 The Case of a Torus 169

5.5.2 Anosov Diffeomorphisms 173

5.5.3 Nilmanifold Examples of Anosov Diffeomorphisms 177

5.6 The Lefschetz Zeta Function 179

6 Borel's Fixed Point Theorem in Algebraic Groups 187

6.1 Complete Varieties and Borel's Theorem 187

6.2 The Projective and Grassmann Spaces 190

6.3 Projective Varieties 193

6.4 Consequences of Borel's Fixed Point Theorem 197

6.5 Two Conjugacy Theorems for Real Linear Lie Groups 200

7 Miscellaneous Fixed Point Theorems 203

7.1 Applications to Number Theory 203

7.1.1 The Little Fermat Theorem 203

7.1.2 Fermat's Two Squares Theorem 205

7.2 Fixed Points in Group Theory 207

7.3 A Fixed Point Theorem in Complex Analysis 209

8 A Fixed Point Theorem in Set Theory 211

Afterword 217

Bibliography 219

Index 229