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Paperback

Published: 11th November 1998
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RRP $89.99
$81.45

In 1920, Pierre Fatou expressed the conjecture that--except for special cases--all critical points of a rational map of the Riemann sphere tend to periodic orbits under iteration. This conjecture remains the main open problem in the dynamics of iterated maps. For the logistic family x- ax(1-x), it can be interpreted to mean that for a dense set of parameters "a," an attracting periodic orbit exists. The same question appears naturally in science, where the logistic family is used to construct models in physics, ecology, and economics.

In this book, Jacek Graczyk and Grzegorz Swiatek provide a rigorous proof of the Real Fatou Conjecture. In spite of the apparently elementary nature of the problem, its solution requires advanced tools of complex analysis. The authors have written a self-contained and complete version of the argument, accessible to someone with no knowledge of complex dynamics and only basic familiarity with interval maps. The book will thus be useful to specialists in real dynamics as well as to graduate students.

Review of Conceptsp. 3
Theory of Quadratic Polynomialsp. 3
Dense Hyperbolicityp. 6
Steps of the Proof of Dense Hyperbolicityp. 12
Quasiconformal Gluingp. 25
Extendibility and Distortionp. 26
Saturated Mapsp. 30
Gluing of Saturated Mapsp. 35
Polynomial-Like Propertyp. 45
Domains in the Complex Planep. 45
Cutting Timesp. 47
Linear Growth of Modulip. 67
Box Maps and Separation Symbolsp. 67
Conformal Roughnessp. 87
Growth of the Separation Indexp. 100
Quasiconformal Techniquesp. 109
Initial Inducingp. 109
Quasiconformal Pull-backp. 120
Gluing Quasiconformal Mapsp. 129
Regularity of Saturated Mapsp. 133
Straightening Theoremp. 139
Bibliographyp. 143
Indexp. 147
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691002583
ISBN-10: 0691002584
Series: Annals of Mathematics Studies (Paperback)
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 158
Published: 11th November 1998
Dimensions (cm): 15.2 x 23.3  x 1.277
Weight (kg): 0.26