Complex Dynamics : Families and Friends

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. I. Polynomial Dynamics From Combinatorics To Topology -- On The Geometry And Dynamics Of Iterated Rational Maps / William P. Thurston, 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 -- Ii. 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 Sierpiński Carpets / Paul Blanchard [and Others] -- The Teichmüller Space Of An Entire Function / Núria Fagella And Christian Henriksen -- Iii. 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 -- Iv. Making New Friends -- The Hunt For Julia Sets With Positive Measure / Arnaud Chéritat -- On Thurston's Pullback Map / Xavier Buff [and Others] -- On The Boundary Behavior Of Thurston's Pullback Map / Nikita Selinger -- Computing Arithmetic Invariants For Hyperbolic Reflection Groups / Omar Anatolín-camarena [and Others]. Edited By Dierk Schleicher. Includes Bibliographical References. Contents......Page 8 Foreword......Page 10 Introduction......Page 12 I. Polynomial Dynamics from Combinatorics to Topology......Page 22 1. On the Geometry and Dynamics of Iterated Rational Maps......Page 24 I. The Basic Theory......Page 34 II. Polynomials and the Riemann Mapping Near Infinity......Page 80 Appendix: Laminations, Julia Sets, and the Mandelbrot Set......Page 132 2. Wandering Gaps for Weakly Hyperbolic Polynomials......Page 152 3. Combinatorics of Polynomial Iterations......Page 182 4. The Unicritical Branner-Hubbard Conjecture......Page 228 5. A Priori Bounds for Some Infinitely Renormalizable Quadratics, III. Molecules......Page 242 II. Beyond Polynomials: Rational and Transcendental Dynamics......Page 268 6. The Connectivity of the Julia Set and Fixed Points......Page 270 7. The Rabbit and Other Julia Sets Wrapped in Sierpiński Carpets......Page 290 8. The Teichmüller Space of an Entire Function......Page 310 III. Two Complex Dimensions......Page 344 9. Cubic Polynomial Maps with Periodic Critical Orbit, Part I......Page 346 10. Analytic Coordinates Recording Cubic Dynamics......Page 426 11. Cubic Polynomials: A Measurable View of Parameter Space......Page 464 12. Bifurcation Measure and Postcritically Finite Rational Maps......Page 504 13. Real Dynamics of a Family of Plane Birational Maps: Trapping Regions and Entropy Zero......Page 526 IV. Making New Friends......Page 550 14. The Hunt for Julia Sets with Positive Measure......Page 552 15. On Thurston's Pullback Map......Page 574 16. On the Boundary Behavior of Thurston’s Pullback Map......Page 598 17. Computing Arithmetic Invariants for Hyperbolic Reflection Groups......Page 610 Contributors......Page 646