The Real Fatou Conjecture

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.

< TR> 1Review of Concepts3< TR> 1. 1Theory of Quadratic Polynomials3< TR> 1. 2Dense Hyperbolicity6< TR> 1. 3Steps of the Proof of Dense Hyperbolicity12< TR> 2Quasiconformal Gluing25< TR> 2. 1Extendibility and Distortion26< TR> 2. 2Saturated Maps30< TR> 2. 3Gluing of Saturated Maps35< TR> 3Polynomial-Like Property45< TR> 3. 1Domains in the Complex Plane45< TR> 3. 2Cutting Times47< TR> 4Linear Growth of Moduli67< TR> 4. 1Box Maps and Separation Symbols67< TR> 4. 2Conformal Roughness87< TR> 4. 3Growth of the Separation Index100< TR> 5Quasiconformal Techniques109< TR> 5. 1Initial Inducing109< TR> 5. 2Quasiconformal Pull-back120< TR> 5. 3Gluing Quasiconformal Maps129< TR> 5. 4Regularity of Saturated Maps133< TR> 5. 5Straightening Theorem139< TR> Bibliography143< TR> Index147