Jacek Graczyk, Grzegorz Swiatek

The Real Fatou Conjecture

• Broschiertes Buch

Produktbeschreibung

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.

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Produktdetails

• Verlag: Princeton University Press
• Seitenzahl: 158
• Erscheinungstermin: 25. Oktober 1998
• Englisch
• Abmessung: 234mm x 156mm x 9mm
• Gewicht: 251g
• ISBN-13: 9780691002583
• ISBN-10: 0691002584
• Artikelnr.: 21079245

Autorenporträt

Jacek Graczyk is Assistant Professor of Mathematics at Michigan State University. Grzegorz Swiatek is Associate Professor of Mathematics at Pennsylvania State University.

Inhaltsangabe

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