Real Fatou Conjecture.

GRACZYK, Jacek and SWIATEK, Grzegorz.

ISBN 10: 0691002576 / ISBN 13: 9780691002576

Editore: Princeton University Press,, Princeton:, 1998

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First printing. Fine in orange cloth. No dust jacket.

Codice articolo 49132

Dati bibliografici

• Titolo: Real Fatou Conjecture.
• Editore: Princeton University Press,, Princeton:
• Data di pubblicazione: 1998
• Lingua: Inglese
• ISBN 10: 0691002576
• ISBN 13: 9780691002576
• Rilegatura: Hardcover
• Edizione: prima edizione
• Condizione: Fine
• Cataloghi dei venditori:
  • General Nonfiction IV

Descrizione articolo

Riassunto

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.

Contenuti

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