McMullen Complex Dynamics and Renormalization

1. Auflage 2016
ISBN: 978-1-4008-8255-7
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 214 Seiten

Reihe: Annals of Mathematics Studies

Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c. As discovered by Feigenbaum, such a mapping exhibits a repetition of form at infinitely many scales. Drawing on universal estimates in hyperbolic geometry, this work gives an analysis of the limiting forms that can occur and develops a rigidity criterion for the polynomial f. This criterion supports general conjectures about the behavior of rational maps and the structure of the Mandelbrot set.

The course of the main argument entails many facets of modern complex dynamics. Included are foundational results in geometric function theory, quasiconformal mappings, and hyperbolic geometry. Most of the tools are discussed in the setting of general polynomials and rational maps.

Autoren/Hrsg.

McMullen, Curtis T.

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Nicht-Euklidische Geometrie

Inhaltsverzeichnis

- Background in conformal geometry

- Dynamics of rational maps

- Holomorphic motions and the Mandelbrot set

- Compactness in holomorphic dynamics

- Polynomials and external rays

- Renormalization

- Puzzles and infinite renormalization

- Robustness

- Limits of renormalization

- Real quadratic polynomials

Über Autor(innen)

Curtis T. McMullen is Professor of Mathematics at the University of California, Berkeley.