The Real Fatou Conjecture

by Graczyk, Jacek; Swiatek, Grzegorz

• ISBN13: 9780691002583
• ISBN10: 0691002584
• Format: Paperback
• Copyright: 1998-10-05
• Publisher: Princeton Univ Pr

Summary

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.

Table of Contents

Review of Concepts	p. 3
Theory of Quadratic Polynomials	p. 3
Dense Hyperbolicity	p. 6
Steps of the Proof of Dense Hyperbolicity	p. 12
Quasiconformal Gluing	p. 25
Extendibility and Distortion	p. 26
Saturated Maps	p. 30
Gluing of Saturated Maps	p. 35
Polynomial-Like Property	p. 45
Domains in the Complex Plane	p. 45
Cutting Times	p. 47
Linear Growth of Moduli	p. 67
Box Maps and Separation Symbols	p. 67
Conformal Roughness	p. 87
Growth of the Separation Index	p. 100
Quasiconformal Techniques	p. 109
Initial Inducing	p. 109
Quasiconformal Pull-back	p. 120
Gluing Quasiconformal Maps	p. 129
Regularity of Saturated Maps	p. 133
Straightening Theorem	p. 139
Bibliography	p. 143
Index	p. 147
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