The Real Fatou Conjecture. (am-144) (ebook)

Argumento de The Real Fatou Conjecture. (am-144) (ebook)

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.01 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