Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot Set

by Karsten Keller

Overview

This book is mainly devoted to the combinatorics of quadratic holomorphic dynamics. The conceptual kernel is a self-contained abstract counterpart of connected quadratic Julia sets which is built on Thurston's concept of a quadratic invariant lamination and on symbolic descriptions of the angle-doubling map. The theory obtained is illustrated in the complex plane. It is used to give rigorous proofs of some well-known and some partially new statements on the structure of the Mandelbrot set. The text is intended for graduate students and researchers. Some elementary knowledge in topology and in functions of one complex variable is assumed.

Product Details

ISBN-13:	9783540674344
Publisher:	Springer Berlin Heidelberg
Publication date:	06/08/2000
Series:	Lecture Notes in Mathematics , #1732
Edition description:	2000
Pages:	208
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

Table of Contents

1. Introduction: Quadratic iteration and Julia equivalences. The Mandelbrot set.- 2. Abstract Julia sets: Symbolic dynamics of the angle-doubling map. Invariant laminations. Julia equivalences.- 3. The Abstract Mandelbrot set: The Abstract Mandelbrot set - an atlas of Abstract Julia sets. The ordered Abstract Mandelbrot set. Renormalization. Correspondence and Translation Principles.- 4. Abstract and concrete theory: Quadratic iteration. Miscellaneous. Appendix: Invariant and completely invariant factors. Simple statements. Shift-invariant factors. Further interesting examples.