Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

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Overview

Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday by Araceli Bonifant, Misha Lyubich, Scott Sutherland

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing. This collection will be useful to students and researchers for decades to come.

The contributors are Marco Abate, Marco Arizzi, Alexander Blokh, Thierry Bousch, Xavier Buff, Serge Cantat, Tao Chen, Robert Devaney, Alexandre Dezotti, Tien-Cuong Dinh, Romain Dujardin, Hugo García-Compeán, William Goldman, Rotislav Grigorchuk, John Hubbard, Yunping Jiang, Linda Keen, Jan Kiwi, Genadi Levin, Daniel Meyer, John Milnor, Carlos Moreira, Vincente Muñoz, Viet-Anh Nguyên, Lex Oversteegen, Ricardo Pérez-Marco, Ross Ptacek, Jasmin Raissy, Pascale Roesch, Roberto Santos-Silva, Dierk Schleicher, Nessim Sibony, Daniel Smania, Tan Lei, William Thurston, Vladlen Timorin, Sebastian van Strien, and Alberto Verjovsky.

Product Details

ISBN-13: 9780691159294
Publisher: Princeton University Press
Publication date: 03/16/2014
Series: Princeton Mathematical Series
Pages: 824
Product dimensions: 7.00(w) x 10.10(h) x 2.30(d)

About the Author

Araceli Bonifant is associate professor of mathematics at the University of Rhode Island. Misha Lyubich is director of the Institute for Mathematical Sciences and professor of mathematics at Stony Brook University. Scott Sutherland is associate professor of mathematics at Stony Brook University.

Table of Contents

Preface xi

Introduction 1

1 Bibliography 8

Part I One Complex Variable 13

Arithmetic of Unicritical Polynomial Maps J. Milnor 15

1 Introduction 15

2 Periodic orbits 15

3 Proofs 18

4 Postcritically finite maps 20

5 Bibliography 23

Les racines des composantes hyperboliques de M sont des quarts d'entiers algébriques T. Bousch 25

Dynamical cores of topological polynomials A. Blokh L. Oversteegen R. Ptacek V. Timorin 27

1 Introduction and the main result 27

2 Preliminaries 29

3 Dynamical core 35

4 Bibliography 47

The quadratic dynatomic curves are smooth and irreducible X. Buff Tan Lei 49

1 Introduction 49

2 Dynatomic polynomials 50

3 Smoothness of the dynatomic curves 53

4 Irreducibility of the dynatomic curves 61

5 Bibliography 71

Multicorns are not path connected J. H. Hubbard D. Schleicher 73

1 Introduction 73

2 Antiholomorphic and parabolic dynamics 77

3 Bifurcation along arcs and the fixed-point index 80

4 Parabolic perturbations 85

5 Parabolic trees and combinatorics 89

6 Non-pathwise connectivity 95

7 Further results 98

8 Bibliography 101

Leading monomials of escape regions J. Kiwi 103

1 Introduction 103

2 Statement of the results 104

3 Puiseux series dynamics statement 106

4 One-parameter families 108

5 Bibliography 118

Limiting behavior of Julia sets of singularly perturbed rational maps R. L. Devaney 121

1 Introduction 121

2 Elementary mapping properties 123

3 Julia sets converging to the unit disk 126

4 The case n > 2 128

5 Other c-values 131

6 Bibliography 133

On (non-)local connectivity of some Julia sets A. Dezotti P. Roesch 135

1 Local connectivity 136

2 Rational maps 141

3 Douady-Sullivan criterion 143

4 The case of infinitely satellite-renormalizable quadratic polynomials 150

5 Bibliography 161

Perturbations of weakly expanding critical orbits G. Levin 163

1 Introduction 163

2 Polynomials 167

3 Rational functions 176

4 Part (a) of Theorem 3.6 182

5 Part (b) of Theorem 3.6 189

6 Bibliography 195

Unmating of rational maps: Sufficient criteria and examples D. Meyer 197

1 Introduction 197

2 Moore's theorem 199

3 Mating of polynomials 201

4 Equators and hyperbolic rational maps 204

5 An example 209

6 A sufficient criterion for mating 211

7 Connections 216

8 Critical portraits 220

9 Unmating the map 221

10 Examples of unmatings 224

11 A mating not arising from a pseudo-equator 228

12 Open questions 232

13 Bibliography 232

A framework toward understanding the characterization of holomorphic dynamics Y. Jiang 235

1 Characterization 235

2 Obstruction 237

3 Review 237

4 Geometry 242

5 Geometrization 248

6 Appendix on Transcendental Functions by Tao Chen, Yunping Jiang, and Linda Keen 242

7 Bibliography 254

Part II One Real Variable 259

Metric stability for random walks (with applications in renormalization theory) C. G. Moreira D. Smania 261

1 Introduction 261

2 Expanding Markov maps, random walks, and their perturbations 265

3 Statements of results 269

4 Preliminaries 272

5 Stability of transience 276

6 Stability of recurrence 288

7 Stability of the multifractal spectrum 296

8 Applications to one-dimensional renormalization theory 312

9 Bibliography 321

Milnor's conjecture on monotonicity of topological entropy: Results and questions S. Van Strien 323

1 Motivation 323

2 Milnor's monotonicity of entropy conjecture 324

3 Idea of the proof 326

4 Open problems 331

5 Bibliography 334

Entropy in dimension one W. P. Thurston 339

1 Introduction 339

2 Special case: Pisot numbers 346

3 Constructing interval maps: First steps 351

4 Second step: Constructing a map for λN 355

5 Powers and roots: Completion of proof of Theorem 1.3 355

6 Maps of asterisks 358

7 Entropy in bounded degree 359

8 Traintracks 365

9 Splitting hairs 367

10 Dynamic extensions 369

11 Bipositive matrices 370

12 Tracks, doubletracks, zipping and a sketch of the proof of Theorem 1.11 374

13 Supplementary notes (mostly by John Milnor) 376

14 Bibliography 383

Part III Several Complex Variables 385

On Écalle-Hakim's theorems in holomorphic dynamics M. Arizzi J. Raissy 387

1 Introduction 387

2 Notation 389

3 Preliminaries 389

4 Characteristic directions 390

5 Changes of coordinates 397

6 Existence of parabolic curves 407

7 Existence of attracting domains 421

8 Parabolic manifolds 423

9 Fatou coordinates 439

10 Fatou-Bieberbach domains 444

11 Bibliography 447

Index theorems for meromorphic self-maps of the projective space M. Abate 451

1 Introduction 451

2 The proof 453

3 Bibliography 460

Dynamics of automorphisms of compact complex surfaces S. Cantat 463

1 Introduction 463

2 Hodge theory and automorphisms 466

3 Groups of automorphisms 476

4 Periodic curves, periodic points, and topological entropy 478

5 Invariant currents 485

6 Entire curves, stable manifolds, and laminarity 489

7 Fatou and Julia sets 498

8 The measure of maximal entropy and periodic points 502

9 Complements 505

10 Appendix: Classification of surfaces 507

11 Bibliography 509

Bifurcation currents and equidistribution in parameter space R. Dujardin 515

1 Prologue: normal families, currents and equidistribution 518

2 Bifurcation currents for families of rational mappings on P1 524

3 Higher bifurcation currents and the bifurcation measure 534

4 Bifurcation currents for families of Möbius subgroups 547

5 Further settings, final remarks 557

6 Bibliography 559

Part IV Laminations and Foliations 567

Entropy for hyperbolic Riemann surface laminations I T.-C. Dinh V.-A. Nguyên N. Sibony 569

1 Introduction 569

2 Poincaré metric on laminations 571

3 Hyperbolic entropy for foliations 578

4 Entropy of harmonic measures 585

5 Bibliography 591

Entropy for hyperbolic Riemann surface laminations II T.-C. Dinh V.-A. Nguyên N. Sibony 593

1 Introduction 593

2 Local models for singular points 595

3 Poincaré metric on leaves 604

4 Finiteness of entropy: the strategy 611

5 Adapted transversals and their coverings 614

6 Finiteness of entropy: end of the proof 619

7 Bibliography 621

Intersection theory for ergodic solenoids V. Muñoz R. Pérez-Marco 623

1 Introduction 623

2 Measured solenoids and generalized currents 624

3 Homotopy of solenoids 627

4 Intersection theory of solenoids 630

5 Almost everywhere transversality 638

6 Intersection of analytic solenoids 641

7 Bibliography 644

Invariants of four-manifolds with flows via cohomological field theory H. García-Compeán R. Santos-Silva A. Verjovsky 645

1 Introduction 645

2 Asymptotic cycles and currents 647

3 Overview of cohomological quantum field theory: Donaldson-Witten invariants 650

4 Donaldson-Witten invariants for flows 653

5 Donaldson-Witten invariants for Kähler manifolds with flows 659

6 Survey on Seiberg-Witten invariants 667

7 Seiberg-Witten invariants for flows 669

8 A physical interpretation 672

9 Final remarks 673

10 Bibliography 674

Color Plates C-1

Part V Geometry and Algebra 677

Two papers which changed my life: Milnor's seminal work on flat manifolds and bundles W. Goldman 679

1 Introduction 679

2 Gauss-Bonnet beginnings 679

3 The Milnor-Wood inequality 682

4 Maximal representations 684

5 Complete affine manifolds 687

6 Margulis spacetimes 691

7 Bibliography 699

Milnor's problem on the growth of groups and its consequences R. Grigorchuk 705

1 Introduction 705

2 Acknowledgments 709

3 Preliminary facts 709

4 The problem and the conjecture of Milnor 713

5 Relations between group growth and Riemannian geometry 715

6 Results about group growth obtained before 1981 717

7 Growth and amenability 719

8 Polynomial growth 722

9 Intermediate growth: The construction 726

10 The gap conjecture 734

11 Intermediate growth: The upper and lower bounds 738

12 Asymptotic invariants of probabilistic and analytic nature and corresponding gap-type conjectures 743

13 Inverse orbit growth and examples with explicit growth 750

14 Miscellaneous 753

15 Bibliography 758

Contributors 775

Index 779

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