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

2000
ISBN: 978-3-540-67434-4
Verlag: Springer Berlin Heidelberg

Buch, Englisch, Band 1732, 208 Seiten, Format (B × H): 155 mm x 233 mm, Gewicht: 338 g

Reihe: Lecture Notes in Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Keller, Karsten

Fachgebiete

• Mathematik | Informatik Mathematik Topologie Mengentheoretische Topologie
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis

Inhaltsverzeichnis

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.