Synopses & Reviews
A discussion of the properties of conformal mappings in the complex plane, closely related to the study of fractals and chaos. Indeed, the book ends in a detailed study of the famous Mandelbrot set, which describes very general properties of such mappings. Focusing on the analytic side of this contemporary subject, the text was developed from a course taught over several semesters and aims to help students and instructors to familiarize themselves with complex dynamics. Topics covered include: conformal and quasi-conformal mappings, fixed points and conjugations, basic rational iteration, classification of periodic components, critical points and expanding maps, some applications of conformal mappings, the local geometry of the Fatou set, and quadratic polynomials and the Mandelbrot set.
Synopsis
Complex dynamics is today very much a focus of interest. Though several fine expository articles were available, by P. Blanchard and by M. Yu. Lyubich in particular, until recently there was no single source where students could find the material with proofs. For anyone in our position, gathering and organizing the material required a great deal of work going through preprints and papers and in some cases even finding a proof. We hope that the results of our efforts will be of help to others who plan to learn about complex dynamics and perhaps even lecture. Meanwhile books in the field a. re beginning to appear. The Stony Brook course notes of J. Milnor were particularly welcome and useful. Still we hope that our special emphasis on the analytic side will satisfy a need. This book is a revised and expanded version of notes based on lectures of the first author at UCLA over several \Vinter Quarters, particularly 1986 and 1990. We owe Chris Bishop a great deal of gratitude for supervising the production of course notes, adding new material, and making computer pictures. We have used his computer pictures, and we will also refer to the attractive color graphics in the popular treatment of H. -O. Peitgen and P. Richter. We have benefited from discussions with a number of colleagues, and from suggestions of students in both our courses.
Synopsis
Discusses the properties of conformal mappings in the complex plane, a contemporary subject that is closely connected to the study of fractals and chaos. The text was developed out of a course taught over several semesters, with a focus on helping students and instructors to familiarize themselves with complex dynamics. 28 illustrations.
Synopsis
This book discusses the properties of conformal mappings in the complex plane, a subject closely connected to the study of fractals and chaos. Focusing on analytic aspects of this contemporary topic, it includes a detailed study of the famous Mandelbrot set.