Synopses & Reviews
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.
Synopsis
In 1920, Perre 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 book provides a rigorous proof of the Real Fatou Conjecture--that in spite of the apparently elementary nature of a problem, its solution requires advanced tools of complex analysis.
Synopsis
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.
Description
Includes bibliographical references (p. 143-146) and index.
Table of Contents
1 | Review of Concepts | 3 |
1.1 | Theory of Quadratic Polynomials | 3 |
1.2 | Dense Hyperbolicity | 6 |
1.3 | Steps of the Proof of Dense Hyperbolicity | 12 |
2 | Quasiconformal Gluing | 25 |
2.1 | Extendibility and Distortion | 26 |
2.2 | Saturated Maps | 30 |
2.3 | Gluing of Saturated Maps | 35 |
3 | Polynomial-Like Property | 45 |
3.1 | Domains in the Complex Plane | 45 |
3.2 | Cutting Times | 47 |
4 | Linear Growth of Moduli | 67 |
4.1 | Box Maps and Separation Symbols | 67 |
4.2 | Conformal Roughness | 87 |
4.3 | Growth of the Separation Index | 100 |
5 | Quasiconformal Techniques | 109 |
5.1 | Initial Inducing | 109 |
5.2 | Quasiconformal Pull-back | 120 |
5.3 | Gluing Quasiconformal Maps | 129 |
5.4 | Regularity of Saturated Maps | 133 |
5.5 | Straightening Theorem | 139 |
| Bibliography | 143 |
| Index | 147 |