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A K Peters/CRC Press
663 pages
Complex Dynamics: Families and Friends features contributions by many of the leading mathematicians in the field, such as Mikhail Lyubich, John Milnor, Mitsuhiro Shishikura, and William Thurston. Some of the chapters, including an introduction by Thurston to the general subject of complex dynamics, are classic manuscripts that were never published before but have influenced the field for more than two decades. Other chapters contain fresh, original work and bring readers to the current frontier of research. The title reflects the fruitful interplay between diverse mathematical fields bound together by the common theme of complex dynamics, including hyperbolic geometry, number theory, group theory, combinatorics, general dynamics, and many more. At the same time, the title alludes to the spirit of mathematical friendship among the researchers in this area. This book is a tribute to John Hubbard, one of the most inspiring pioneers in the field of complex dynamics.
The current volume is a collection of seventeen research articles, all of a very high standard. … give insight into the human side of Hubbard. The quality of the editing is superb, making this volume an absolute pleasure to read.
—Sebastian van Strien, Nieuw Archief voor Wiskunde, June 2012
In 2005 John Hamal Hubbard celebrated his sixtieth birthday. His families and friends organised a conference in his honour for him, for themselves, and for the mathematical community, especially those with an interest in complex dynamics in any shape or form. … [this volume] serves as a memento for the conference and much more. … For nearly thirty years, the preprint produced by Bill Thurston has been repeatedly photocopied (and, more recently, scanned). Here, it has been faithfully reproduced with discreet, but occasionally significant, well-judged editing by Dierk Schleicher and Nikita Selinger. Schleicher has also provided a crucial missing section: what he reckons would have been Thurston’s II.7. This spells out the way in which the quadratic laminations determine the topological dynamics on the corresponding Julia set in the cases when the Julia set is locally connected. … Thus, the first article of the book is the first full print version of a monograph that has been in existence since the early days … The illustrations are not only beautiful but also an important reminder of the role that pictures play in research in this field, and in the work of Hubbard and his students and co-workers in particular.
—Mary Rees, The Mathematical Intelligencer, 2011
Foreword
Jean-Christophe Yoccoz
Introduction
Dierk Schleicher
Polynomial Dynamics from Combinatorics to Topology
On the Geometry and Dynamics of Iterated Rational Maps
The Basic Theory
Polynomials and the Riemann Mapping Near Infinity
William P. Thurston
Appendix: Laminations, Julia Sets, and the Mandelbrot Set
Dierk Schleicher
Wandering Gaps for Weakly Hyperbolic Polynomials
Alexander Blokh and Lex Oversteegen
Combinatorics of Polynomial Iterations
Volodymyr Nekrashevych
The Unicritical Branner-Hubbard Conjecture
Tan Lei and Yin Yongcheng
A Priori Bounds for Some Infinitely Renormalizable Quadratics, III: Molecules
Jeremy Kahn and Mikhail Lyubich
Beyond Polynomials: Rational and Transcendental Dynamics
The Connectivity of the Julia Set and Fixed Points
Mitsuhiro Shishikura
The Rabbit and Other Julia Sets Wrapped in Sierpinski Carpets
Paul Blanchard, Robert L. Devaney, Antonio Garijo, Sebastian M. Marotta, and Elizabeth D. Russell
The Teichmuller Space of an Entire Function
Nuria Fagella and Christian Henriksen
Two Complex Dimensions
Cubic Polynomial Maps with Periodic Critical Orbit, Part I
John Milnor
Analytic Coordinates Recording Cubic Dynamics
Carsten Lunde Petersen and Tan Lei
Cubic Polynomials: A Measurable View of Parameter Space
Romain Dujardin
Bifurcation Measure and Postcritically Finite Rational Maps
Xavier Buff and Adam Epstein
Real Dynamics of a Family of Plane Birational Maps: Trapping Regions and Entropy Zero
Eric Bedford and Jeffrey Diller
Making New Friends
The Hunt for Julia Sets with Positive Measure
Arnaud Cheritat
On Thurston’s Pullback Map
Xavier Buff, Adam Epstein, Sarah Koch, and Kevin Pilgrim
On the Boundary Behavior of Thurston’s Pullback Map
Nikita Selinger
Computing Arithmetic Invariants for Hyperbolic Reflection Groups
Omar Antolın-Camarena, Gregory R. Maloney, and Roland Roeder
Contributors
