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 Garcia-Compean, William Goldman, Rotislav Grigorchuk, John Hubbard, Yunping Jiang, Linda Keen, Jan Kiwi, Genadi Levin, Daniel Meyer, John Milnor, Carlos Moreira, Vincente Munoz, Viet-Anh Nguyen, Lex Oversteegen, Ricardo Perez-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.
Author(s): Araceli Bonifant, Misha Lyubich, Scott Sutherland
Series: Princeton Mathematical Series, 51
Publisher: Princeton University Press
Year: 2014
Language: English
Pages: 800
City: Princeton
Cover
Title
Copyright
Contents
Preface
Introduction
1. Bibliography
Part I. One Complex Variable
Arithmetic of Unicritical Polynomial Maps
1. Introduction
2. Periodic orbits
3. Proofs
4. Postcritically finite maps
5. Bibliography
Les racines des composantes hyperboliques de M sont des quarts d’entiers algébriques
Dynamical cores of topological polynomials
1. Introduction and the main result
2. Preliminaries
3. Dynamical core
4. Bibliography
The quadratic dynatomic curves are smooth and irreducible
1. Introduction
2. Dynatomic polynomials
3. Smoothness of the dynatomic curves
4. Irreducibility of the dynatomic curves
5. Bibliography
Multicorns are not path connected
1. Introduction
2. Antiholomorphic and parabolic dynamics
3. Bifurcation along arcs and the fixed-point index
4. Parabolic perturbations
5. Parabolic trees and combinatorics
6. Non-pathwise connectivity
7. Further results
8. Bibliography
Leading monomials of escape regions
1. Introduction
2. Statement of the results
3. Puiseux series dynamics statement
4. One-parameter families
5. Bibliography
Limiting behavior of Julia sets of singularly perturbed rational maps
1. Introduction
2. Elementary mapping properties
3. Julia sets converging to the unit disk
4. The case n > 2
5. Other c-values
6. Bibliography
On (non-)local connectivity of some Julia sets
1. Local connectivity
2. Rational maps
3. Douady-Sullivan criterion
4. The case of infinitely satellite-renormalizable quadratic polynomials: a model
5. Bibliography
Perturbations of weakly expanding critical orbits
1. Introduction
2. Polynomials
3. Rational functions
4. Part (a) of Theorem 3.6
5. Part (b) of Theorem 3.6
6. Bibliography
Unmating of rational maps: Sufficient criteria and examples
1. Introduction
2. Moore’s theorem
3. Mating of polynomials
4. Equators and hyperbolic rational maps
5. An example
6. A sufficient criterion for mating
7. Connections
8. Critical portraits
9. Unmating the map
10. Examples of unmatings
11. A mating not arising from a pseudo-equator
12. Open questions
13. Bibliography
A framework toward understanding the characterization of holomorphic dynamics
1. Characterization
2. Obstruction
3. Review
4. Geometry
5. Geometrization
6. Appendix on Transcendental Functions by Tao Chen, Yunping Jiang, and Linda Keen
7. Bibliography
Part II. One Real Variable
Metric stability for random walks (with applications in renormalization theory)
1. Introduction
2. Expanding Markov maps, random walks, and their perturbations
3. Statements of results
4. Preliminaries
5. Stability of transience
6. Stability of recurrence
7. Stability of the multifractal spectrum
8. Applications to one-dimensional renormalization theory
9. Bibliography
Milnor’s conjecture on monotonicity of topological entropy: Results and questions
1. Motivation
2. Milnor’s monotonicity of entropy conjecture
3. Idea of the proof
4. Open problems
5. Bibliography
Entropy in dimension one
1. Introduction
2. Special case: Pisot numbers
3. Constructing interval maps: First steps
4. Second step: Constructing a map for λN
5. Powers and roots: Completion of proof of Theorem 1.3
6. Maps of asterisks
7. Entropy in bounded degree
8. Traintracks
9. Splitting hairs
10. Dynamic extensions
11. Bipositive matrices
12. Tracks, doubletracks, zipping and a sketch of the proof of Theorem 1.11
13. Supplementary notes (mostly by John Milnor)
14. Bibliography
Part III. Several Complex Variables
On Écalle-Hakim’s theorems in holomorphic dynamics
1. Introduction
2. Notation
3. Preliminaries
4. Characteristic directions
5. Changes of coordinates
6. Existence of parabolic curves
7. Existence of attracting domains
8. Parabolic manifolds
9. Fatou coordinates
10. Fatou-Bieberbach domains
11. Bibliography
Index theorems for meromorphic self-maps of the projective space
1. Introduction
2. The proof
3. Bibliography
Dynamics of automorphisms of compact complex surfaces
1. Introduction
2. Hodge theory and automorphisms
3. Groups of automorphisms
4. Periodic curves, periodic points, and topological entropy
5. Invariant currents
6. Entire curves, stable manifolds, and laminarity
7. Fatou and Julia sets
8. The measure of maximal entropy and periodic points
9. Complements
10. Appendix: Classification of surfaces
11. Bibliography
Bifurcation currents and equidistribution in parameter space
1. Prologue: normal families, currents and equidistribution
2. Bifurcation currents for families of rational mappings on P1
3. Higher bifurcation currents and the bifurcation measure
4. Bifurcation currents for families of Möbius subgroups
5. Further settings, final remarks
6. Bibliography
Part IV. Laminations and Foliations
Entropy for hyperbolic Riemann surface laminations I
1. Introduction
2. Poincaré metric on laminations
3. Hyperbolic entropy for foliations
4. Entropy of harmonic measures
5. Bibliography
Entropy for hyperbolic Riemann surface laminations II
1. Introduction
2. Local models for singular points
3. Poincaré metric on leaves
4. Finiteness of entropy: the strategy
5. Adapted transversals and their coverings
6. Finiteness of entropy: end of the proof
7. Bibliography
Intersection theory for ergodic solenoids
1. Introduction
2. Measured solenoids and generalized currents
3. Homotopy of solenoids
4. Intersection theory of solenoids
5. Almost everywhere transversality
6. Intersection of analytic solenoids
7. Bibliography
Invariants of four-manifolds with flows via cohomological field theory
1. Introduction
2. Asymptotic cycles and currents
3. Overview of cohomological quantum field theory: Donaldson-Witten invariants
4. Donaldson-Witten invariants for flows
5. Donaldson-Witten invariants for Kähler manifolds with flows
6. Survey on Seiberg-Witten invariants
7. Seiberg-Witten invariants for flows
8. A physical interpretation
9. Final remarks
10. Bibliography
Color Plates
Part V. Geometry and Algebra
Two papers which changed my life: Milnor’s seminal work on flat manifolds and bundles
1. Introduction
2. Gauss-Bonnet beginnings
3. The Milnor-Wood inequality
4. Maximal representations
5. Complete affine manifolds
6. Margulis spacetimes
7. Bibliography
Milnor’s problem on the growth of groups and its consequences
1. Introduction
2. Acknowledgments
3. Preliminary facts
4. The problem and the conjecture of Milnor
5. Relations between group growth and Riemannian geometry
6. Results about group growth obtained before 1981
7. Growth and amenability
8. Polynomial growth
9. Intermediate growth: The construction
10. The gap conjecture
11. Intermediate growth: The upper and lower bounds
12. Asymptotic invariants of probabilistic and analytic nature and corresponding gap-type conjectures
13. Inverse orbit growth and examples with explicit growth
14. Miscellaneous
15. Bibliography
Contributors
Index
