Complex Dynamics: Families and Friends, 1st Edition (Hardback) book cover

Complex Dynamics

Families and Friends, 1st Edition

Edited by Dierk Schleicher

A K Peters/CRC Press

663 pages

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pub: 2009-11-03
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Description

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.

Reviews

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

Table of Contents

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

About the Editor

Dierk Schleicher is a professor of mathematics at Jacobs University in Bremen, Germany.

Subject Categories

BISAC Subject Codes/Headings:
MAT037000
MATHEMATICS / Functional Analysis