Progress in Holomorphic Dynamics: 1st Edition (Hardback) book cover

Progress in Holomorphic Dynamics

1st Edition

By Hartje Kriete

Chapman and Hall/CRC

200 pages

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Hardback: 9780582323889
pub: 1998-05-20

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In the last few decades, complex dynamical systems have received widespread public attention and emerged as one of the most active fields of mathematical research. Starting where other monographs in the subject end, Progress in Holomorphic Dynamics advances the theoretical aspects and recent results in complex dynamical systems, with particular emphasis on Siegel discs.

Organized into four parts, the papers in this volume grew out of three workshops: two hosted by the Georg-August-Universität Göttingen and one at the "Mathematisches Forschungsinstitut Oberwolfach." Part I addresses linearization. The authors review Yoccoz's proof that the Brjuno condition is the optimal condition for linearizability of indifferent fixed points and offer a treatment of Perez-Marco's refinement of Yoccoz's work. Part II discusses the conditions necessary for the boundary of a Siegel disc to contain a critical point, builds upon Herman's work, and offers a survey of the state-of-the-art regarding the boundaries of Siegel discs.

Part III deals with the topology of Julia sets with Siegel discs and contains a remarkable highlight: C.L. Petersen establishes the existence of Siegel discs of quadratic polynomials with a locally connected boundary. Keller, taking a different approach, explains the relations between locally connected "real Julia sets" with Siegel discs and the abstract concepts of kneading sequences and itineraries.

Part IV closes the volume with four papers that review the different directions of present research in iteration theory. It includes discussions on the relations between commuting rational functions and their Julia sets, interactions between the iteration of polynomials and the iteration theory of entire transcendental functions, a deep analysis of the topology of the limbs of the Mandelbrot set, and an overview of complex dynamics in higher dimensions.

Table of Contents


Part I

On the Brjuno Condition, Part I, S. Petersen

On the Brjuno Condition, Part II, P. Bonfert

Linearization of Structurally Stable Polynomials, L. Geyer

Part II

Herman's Proof of the Existence of Critical Points on the Boundary of Singular Domains, H. Kriete

Recent Results on the Boundaries of Siegel Discs, J.T. Rogers Jr.

Part III

Puzzles and Siegel Discs, C.L. Petersen

Julia Equivalences and Abstract Siegel Discs, K. Keller

Part IV

Sharing a Julia Set: The Polynomial Case, P. Atela

Approximating Transcendental Julia Sets, B. Krauskopf and H. Kriete

Surgery on the Limbs of the Mandelbrot Set, N. Fagella

Julia Sets in Cn, S.M. Heinemann

About the Series

Chapman & Hall/CRC Research Notes in Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations
MATHEMATICS / Functional Analysis