# Holomorphic Dynamics

In Chapter 3 , we explain the dynamics of entire functions , considered as holomorphic maps of the complex plane into itself , and give several examples of wandering domains and so - called Baker domains , which never appear in the case ...

Author: S. Morosawa

Publisher: Cambridge University Press

ISBN: 9780521662581

Category: Mathematics

Page: 338

View: 643

This book, first published in 2000, is a comprehensive introduction to holomorphic dynamics, that is the dynamics induced by the iteration of various analytic maps in complex number spaces. This has been the focus of much attention in recent years, with, for example, the discovery of the Mandelbrot set, and work on chaotic behaviour of quadratic maps. The treatment is mathematically unified, emphasizing the substantial role played by classical complex analysis in understanding holomorphic dynamics as well as giving an up-to-date coverage of the modern theory. The authors cover entire functions, Kleinian groups and polynomial automorphisms of several complex variables such as complex Henon maps, as well as the case of rational functions. The book will be welcomed by graduate students and professionals in pure mathematics and science who seek a reasonably self-contained introduction to this exciting area.

# Holomorphic Dynamics and Renormalization

A Volume in Honour of John Milnor's 75th Birthday Mikhail Lyubich, Michael Yampolsky. Holomorphic Dynamics Volume 53 , 2008 Symbolic Dynamics and Self - similar Holomorphic Dynamics.

Author: Mikhail Lyubich

Publisher: American Mathematical Soc.

ISBN: 9780821871560

Category: Mathematics

Page: 395

View: 348

Schwarzian derivatives and cylinder maps by A. Bonifant and J. Milnor Holomorphic dynamics: Symbolic dynamics and self-similar groups by V. Nekrashevych Are there critical points on the boundaries of mother hedgehogs? by D. K. Childers Finiteness for degenerate polynomials by L. DeMarco Cantor webs in the parameter and dynamical planes of rational maps by R. L. Devaney Simple proofs of uniformization theorems by A. A. Glutsyuk The Yoccoz combinatorial analytic invariant by C. L. Petersen and P. Roesch Bifurcation loci of exponential maps and quadratic polynomials: Local connectivity, triviality of fibers, and density of hyperbolicity by L. Rempe and D. Schleicher Rational and transcendental Newton maps by J. Ruckert Newton's method as a dynamical system: Efficient root finding of polynomials and the Riemann $\zeta$ function by D. Schleicher The external boundary of $M_2$ by V. Timorin Renormalization: Renormalization of vector fields by H. Koch Renormalization of arbitrary weak noises for one-dimensional critical dynamical systems: Summary of results and numerical explorations by O. Diaz-Espinosa and R. de la Llave KAM for the nonlinear Schrodinger equation--A short presentation by H. L. Eliasson and S. B. Kuksin Siegel disks and renormalization fixed points by M. Yampolsky

# Progress in Holomorphic Dynamics

T Pitman Research Notes in Mathematics Series 387 Hartje Kriete ( Editor ) Progress in holomorphic dynamics 5 LONGMAN GMA Pitman Research Notes in Mathematics Series Main Editors H. Brezis. Front Cover.

Author: Hartje Kriete

Publisher: CRC Press

ISBN: 9780582323889

Category: Mathematics

Page: 200

View: 587

In the last few decades, complex dynamical systems have received widespread public attention and emerged as one of the most active fields of mathematical research. Starting where other monographs in the subject end, Progress in Holomorphic Dynamics advances the theoretical aspects and recent results in complex dynamical systems, with particular emphasis on Siegel discs. Organized into four parts, the papers in this volume grew out of three workshops: two hosted by the Georg-August-Universität Göttingen and one at the "Mathematisches Forschungsinstitut Oberwolfach." Part I addresses linearization. The authors review Yoccoz's proof that the Brjuno condition is the optimal condition for linearizability of indifferent fixed points and offer a treatment of Perez-Marco's refinement of Yoccoz's work. Part II discusses the conditions necessary for the boundary of a Siegel disc to contain a critical point, builds upon Herman's work, and offers a survey of the state-of-the-art regarding the boundaries of Siegel discs. Part III deals with the topology of Julia sets with Siegel discs and contains a remarkable highlight: C.L. Petersen establishes the existence of Siegel discs of quadratic polynomials with a locally connected boundary. Keller, taking a different approach, explains the relations between locally connected "real Julia sets" with Siegel discs and the abstract concepts of kneading sequences and itineraries. Part IV closes the volume with four papers that review the different directions of present research in iteration theory. It includes discussions on the relations between commuting rational functions and their Julia sets, interactions between the iteration of polynomials and the iteration theory of entire transcendental functions, a deep analysis of the topology of the limbs of the Mandelbrot set, and an overview of complex dynamics in higher dimensions.

# Holomorphic Dynamics

1344. J. Král, J. Lukeš, J. Netuka, J. Vesely (Eds.), Potential Theory – Surveys and Problems. Proceedings, 1987. VIII, 271 pages. 1988. Vol. 1345; X. Gomez-Mont, J. Seade, A. Verjovski (Eds.), Holomorphic Dynamics. Proceedings, 1986.

Author: Xavier Gomez-Mont

Publisher: Springer

ISBN: 354045957X

Category: Mathematics

Page: 328

View: 156

The objective of the meeting was to have together leading specialists in the field of Holomorphic Dynamical Systems in order to present their current reseach in the field. The scope was to cover iteration theory of holomorphic mappings (i.e. rational maps), holomorphic differential equations and foliations. Many of the conferences and articles included in the volume contain open problems of current interest. The volume contains only research articles.

# Quasiconformal Surgery in Holomorphic Dynamics

Sullivan's first crucial work was merely the beginning of the uses to which quasiconformal surgery could be put in holomorphic dynamics. Quasiconformal surgery has been at the heart of many important advances and is now part of the ...

Author: Bodil Branner

Publisher: Cambridge University Press

ISBN: 1107042917

Category: Mathematics

Page: 428

View: 175

A comprehensive introduction to quasiconformal surgery in holomorphic dynamics. Contains a wide variety of applications and illustrations.

# Dynamics in Several Complex Variables

Holomorphic Dynamics in Cn In the previous sections we have mainly been concerned with complex dynamics on complex projective space P2 . The maps we have studied can always be lifted to homogeneous maps on 03. In the remaining lectures ...

Author: John Erik Fornæss

Publisher: American Mathematical Soc.

ISBN: 9780821889312

Category: Mathematics

Page: 59

View: 482

This is part of the CBMS lecture series, held in Albany, New York in June 1994 aimed to introduce the audience to the literature on complex dynamics in higher dimension. These notes provide an easy to read introduction into the field. This monograph then points readers towards technically more advanced literature.

# Combinatorics and Holomorphic Dynamics

glum gh hr ) COMBINATORICS AND HOLOMORPHIC DYNAMICS : CAPTURES , MATINGS , NEWTON'S METHOD Jiaqi Luo , Ph.D. Cornell University 1995 This thesis studies three inter - related topics of one dimensional holomorphic dynamics : captures ...

Author: Jiaqi Luo

Publisher:

ISBN:

Category:

Page: 232

View: 866

# Introduction to the Modern Theory of Dynamical Systems

This sets holomorphic dynamics apart from general differentiable dynamics ( where different locally defined maps can be easily glued together ) and to a lesser extent Hamiltonian dynamics ( where there are no local restrictions either ...

Author: Anatole Katok

Publisher: Cambridge University Press

ISBN: 9780521575577

Category: Mathematics

Page: 802

View: 750

A self-contained comprehensive introduction to the mathematical theory of dynamical systems for students and researchers in mathematics, science and engineering.

# Frontiers in Complex Dynamics

Holomorphic dynamics is one of the earliest branches of dynamical systems which is not part of classical mechanics. As a prominent field in its own right, it was founded in the classical work of Fatou and Julia (see [Fa1, Fa2] and [J]) ...

Author: Araceli Bonifant

Publisher: Princeton University Press

ISBN: 1400851319

Category: Mathematics

Page: 800

View: 305

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.

# Complex Dynamics

In the preface to the book The Mandelbrot Set , Theme and Variations ( edited by Tan Lei ) , John describes his first brush with holomorphic dynamics in the academic year 1976–77 in Orsay : “ I was no computer whiz : at the time I was a ...

Author: Dierk Schleicher

Publisher: CRC Press

ISBN: 1439865426

Category: Mathematics

Page: 663

View: 711

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