**Fields Institute Communications**

Volume: 53;
2008;
395 pp;
Hardcover

MSC: Primary 20; 37;

Print ISBN: 978-0-8218-4275-1

Product Code: FIC/53

List Price: $130.00

Individual Member Price: $104.00

**Electronic ISBN: 978-1-4704-3087-0
Product Code: FIC/53.E**

List Price: $130.00

Individual Member Price: $104.00

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# Holomorphic Dynamics and Renormalization: A Volume in Honour of John Milnor’s 75th Birthday

Share this page *Edited by *
*Mikhail Lyubich; Michael Yampolsky*

A co-publication of the AMS and Fields Institute

The papers collected in this volume reflect
some of the directions of research in two closely related fields:
Complex Dynamics and Renormalization in Dynamical Systems.

While dynamics of polynomial mappings, particularly quadratics, has
by now reached a mature state of development, much less is known about
non-polynomial rational maps. The reader will be introduced into this
fascinating world and a related area of transcendental dynamics by the
papers in this volume. A graduate student will find an area rich with
open problems and beautiful computer simulations.

A survey by V. Nekrashevych introduces the reader to iterated
monodromy groups of rational mappings, a recently developed subject
that links geometric group theory to combinatorics of rational
maps. In this new language, many questions related to Thurston's
theory of branched coverings of the sphere can be answered
explicitly.

Renormalization theory occupies a central place in modern
Complex Dynamics. The progress in understanding the structure of the
Mandelbrot set, polynomial Julia sets, and Feigenbaum-type
universalities stems from renormalization techniques. Renormalization
of circle maps and rotation domains, such as Siegel disks, can be
understood in the context of the classical KAM theory. Corresponding
phenomena in higher dimensions, such as universal scaling in
area-preserving maps in 2D, on the boundary of KAM, pose a challenging
problem. A survey by H. Koch and several other papers in the volume
will introduce the reader to this direction of study.

Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

#### Readership

Graduate students and research mathematicians interested in complex dynamical systems and renormalization in dynamics.

# Table of Contents

## Holomorphic Dynamics and Renormalization: A Volume in Honour of John Milnor's 75th Birthday

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Schwarzian derivatives and cylinder maps 110
- Holomorphic dynamics 2332
- Symbolic dynamics and self-similar groups 2534
- Are there critical points on the boundaries of mother hedgehogs? 7584
- Finiteness for degenerate polynomials 8998
- Cantor webs in the parameter and dynamical planes of rational maps 105114
- Simple proofs of uniformization theorems 125134
- The Yoccoz combinatorial analytic invariant 145154
- Bifurcation loci of exponential maps and quadratic polynomials: Local connectivity, triviality of fibers, and density of hyperbolicity 177186
- Rational and transcendental Newton maps 197206
- Newton’s method as a dynamical system: Efficient root finding of polynomials and the Riemann 𝜁 function 213222
- The external boundary of 𝑀₂ 225234
- Renormalization 267276
- Renormalization of vector fields 269278
- Renormalization of arbitrary weak noises for one-dimensional critical dynamical systems: Summary of results and numerical explorations 331340
- KAM for the nonlinear Schrödinger equation—A short presentation 361370
- Siegel disks and renormalization fixed points 377386
- Back Cover Back Cover1408