- Paperback: 310 pages
- Publisher: Princeton University Press; 3rd edition edition (2 Jan 2006)
- Language English
- ISBN-10: 0691124884
- ISBN-13: 978-0691124889
- Product Dimensions: 25.1 x 17.8 x 2 cm
- Amazon Bestsellers Rank: 750,308 in Books (See Top 100 in Books)
Dynamics in One Complex Variable.
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Review
John Milnor's book provides a solid foundation and the kind of bird's eye view that perhaps only a mathematician of his caliber can offer. -- William J. Satzer, MAA Reviews
Product Description
This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Latts map has been made more inclusive, and the calle-Voronin theory of parabolic points is described. The rsidu itratif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated.
Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field.
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Front Cover | Copyright | Table of Contents | Excerpt | Index
The first three sections will present an overview of some background material. Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index
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3 reviews
5 of 5 people found the following review helpful
Great Book
1 Feb 2005
Format:Paperback
This is a wonderful book by a master of the subject.
It is packed with penetrating insights, illuminating
some of the deepest results on dynamics. Its great
strength is the bird's-eye view it gives.
It is a great place to start the study of complex dynamics.
It is not self-contained, so to get the detail on essential
ingredients such as uniformization and solving the Beltrami
equation one has to go elsewhere. I recommend students
to read it alongside Carleson and Gamelin, who take pains
to supply full details.
It is packed with penetrating insights, illuminating
some of the deepest results on dynamics. Its great
strength is the bird's-eye view it gives.
It is a great place to start the study of complex dynamics.
It is not self-contained, so to get the detail on essential
ingredients such as uniformization and solving the Beltrami
equation one has to go elsewhere. I recommend students
to read it alongside Carleson and Gamelin, who take pains
to supply full details.
Excellent introduction to a marvelous subject
6 Feb 2012
Format:Paperback
This is a great book for, say, a beginning graduate student or really anyone else with a solid undergraduate math background who wants to understand what complex dynamics is all about. The exposition is amazingly crisp and clear. Milnor has a knack for efficiently driving towards the central issue in a proof. Some of the material is quite complicated, but you get the sense that you are learning it from a master who has spent a long time grappling with the concepts and has found the cleanest possible presentation. There are lots of helpful diagrams throughout the text. The subject is known for beautifully complex fractal images, some of which are rendered in the book (albeit in fairly low quality black and white). There are places where a little more detail might have made it easier to read, but the independent thinking that is needed to fill in the gaps is ultimately rewarding. For some of the more advanced material (eg Herman rings), only sketches of proofs are given, which is somewhat annoying, but I have a feeling that going through all the details of some of the more difficult proofs would not really be worth the time for someone new to the field. There are many interesting exercises at the end of almost every section, none of them trivial, and some quite difficult. The core of the book concerns the Julia and Fatou sets and the behavior of rational maps around periodic points. Discussion of the famous Mandelbrot set is relegated to an appendix.
As for prerequisites, you definitely need to know complex analysis. You also should know some topology, in particular, about covering spaces. Milnor writes in the preface that the reader should be familiar with 2-D differential geometry. I didn't really find that necessary, but it might help in understanding the context of the frequently used hyperbolic metric.
As for prerequisites, you definitely need to know complex analysis. You also should know some topology, in particular, about covering spaces. Milnor writes in the preface that the reader should be familiar with 2-D differential geometry. I didn't really find that necessary, but it might help in understanding the context of the frequently used hyperbolic metric.
2 of 5 people found the following review helpful
Very good if..
28 May 2009
Format:Paperback
you have a heavy package in
differentiel geometry
complex analysis
topological algebra
riemann geometry.
If not, forget it.
differentiel geometry
complex analysis
topological algebra
riemann geometry.
If not, forget it.
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