Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
ATZ-Fachkongress

Chassis.tech plus 2024


4 Kongresse in einer Veranstaltung

Treffen Sie in München die Fachleute von Chassis, Lenkung, Bremsen sowie Reifen/Räder.
Erweiterte Suche
Anmelden
nach oben

1994 | Buch

Schröder, Cayley and Newton’s Method Kapitel lesen Erstes Kapitel lesen

A History of Complex Dynamics

From Schröder to Fatou and Julia

verfasst von: Professor Daniel S. Alexander

Verlag: Vieweg+Teubner Verlag

Buchreihe : Aspects of Mathematics

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

In late 1917 Pierre Fatou and Gaston Julia each announced several results regarding the iteration ofrational functions of a single complex variable in the Comptes rendus of the French Academy of Sciences. These brief notes were the tip of an iceberg. In 1918 Julia published a long and fascinating treatise on the subject, which was followed in 1919 by an equally remarkable study, the first instalIment of a three­ part memoir by Fatou. Together these works form the bedrock of the contemporary study of complex dynamics. This book had its genesis in a question put to me by Paul Blanchard. Why did Fatou and Julia decide to study iteration? As it turns out there is a very simple answer. In 1915 the French Academy of Sciences announced that it would award its 1918 Grand Prix des Sciences mathematiques for the study of iteration. However, like many simple answers, this one doesn't get at the whole truth, and, in fact, leaves us with another equally interesting question. Why did the Academy offer such a prize? This study attempts to answer that last question, and the answer I found was not the obvious one that came to mind, namely, that the Academy's interest in iteration was prompted by Henri Poincare's use of iteration in his studies of celestial mechanics.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Schröder, Cayley and Newton’s Method
Abstract
The body of work on the iteration of complex analytic functions which culminated in the major studies of Fatou and Julia has its origins in two detailed examinations of Newton’s method. The first was a remarkable paper by the German mathematician Ernst Schröder (1841–1902), published in two parts in 1870 and 1871, and the second, written by the British mathematician Arthur Cayley (1821–1895), appeared in 1879.
Daniel S. Alexander
Chapter 2. The Next Wave: Korkine and Farkas
Abstract
Schröder developed several notions which are central to the study of complex dynamics. Despite his failure to rigorously establish his fixed point theorem, it is a fundamental result, and his belief that iteration of an arbitrary function ∅(z) could be reduced to the solution of the so-called Abel and Schröder functional equations was prophetic. Not only was the next phase in the development of complex dynamics ushered in by an interest in the solution of the Schröder and Abel equations, but the study and solution of functional equations is fundamental in many contemporary studies of iteration.
Daniel S. Alexander
Chapter 3. Gabriel Koenigs
Abstract
Gabriel Koenigs was the dominant figure in the nineteenth century study of the iteration of complex functions. Drawing on the papers of Schröder, Korkine and Farkas, Koenigs turned the study of the iteration of complex functions into a coherent and rigorously established body of work. His influence on the study of iteration continued throughout the 1890’s. Not only did two of his students, Leopold Leau and Auguste Grévy (1865–1930) each examine a special case which he did not treat, but his work also stimulated papers by Paul Appell (1855–1930), Ernest Lémeray (1860-?) and others. In this chapter I treat Koenigs’ own work. In the next two, I discuss the responses to his work.
Daniel S. Alexander
Chapter 4. Iteration in the 1890’s: Grévy
Abstract
Although Koenigs contributed no new works to the study of iteration of complex functions during the 1890’s, he nonetheless remained its central figure. The infusion of new mathematical ideas—in particular Montels theory of normal families—which would prove so useful to the studies of Fatou and Julia did not occur until after the turn of the century. Consequently, the developments of the nineties consisted largely of either the application of Koenigs’ theory into other branches of mathematics or in the extension of Koenigs’ local study into two special cases he did not examine, namely, the case where the derivative at the fixed point x = 0 is 0 or 1 in modulus.
Daniel S. Alexander
Chapter 5. Iteration in the 1890’s: Leau
Abstract
The most troublesome behavior involving fixed points occurs when the derivative at the fixed point, often called the multiplier of the fixed point, has modulus one. Consequently, Koenigs made no headway with this case, and it was not until the mid-1890’s that any progress was made.
Daniel S. Alexander
Chapter 6. The Flower Theorem of Fatou and Julia
Abstract
Julia studied the case where ∅′(0) = 1 in his lengthy Mémoire sur l’iteration des fonctions rationnelles published in 1918, which was his principal work on the theory of iteration. Fatou discussed this case in the even longer Sur les èquations fonctionnelles which was published in three parts in 1919 and 1920. Each of these works represents a fresh and innovative approach to the study of iteration. Although the work of Fatou and Julia will be discussed at length in Chapter 11, I will discuss their contributions to the ∅′(0) = 1 case in the present chapter, somewhat out of chronological order. However, before discussing their respective approaches to the ∅′(0) = 1 case, it will be worthwhile to say a few words regarding the scope of their studies of iteration.
Daniel S. Alexander
Chapter 7. Fatou’s 1906 Note
Abstract
Previous to Pierre Fatou’s note “Sur les solutions uniformes de certaines équations fonctionnelles” which appeared in the Comptes rendus of the French Academy of Sciences in 1906, studies of iteration focused on a given analytic function ∅(z) in the vicinity of an attracting fixed point.1 Although much was known about the behavior of ∅(z) under iteration near a fixed point, little was known about the global behavior of such functions, that is, the behavior of the iterates of an arbitrary point in the extended complex plane.
Daniel S. Alexander
Chapter 8. Montel’s Theory of Normal Families
Abstract
The key to understanding the behavior under iteration of an arbitrary point in the complex plane lies in understanding the set of points whose orbits do not converge to an attracting or neutral orbit. Fatou’s note [1906a] described this set, often denoted J, in detail for a class of complex rational functions possessing a unique attracting fixed point. Although his technique of examining the intersection of the preimages under ∅ n (z) of the complement of a neighborhood of an attracting fixed point led to his discovery that when ∅(z) has a unique attracting fixed point the set J can be a totally disconnected perfect set, this technique did not reveal enough about J when ∅(z) has more than one attracting orbit.
Daniel S. Alexander
Chapter 9. The Contest
Abstract
In 1915 the French Academy of Sciences announced that it would award its 1918 Grand Prix des Sciences mathématiques—and 3000 francs—for the study of iteration. This announcement evidently motivated both Julia’s Mémoire sur l’iteration des fonctions rationnelles, referred to as [1918], and Fatou’s Sur les équations fonctionnelles, published in three parts as [1919], [1920a] and [1920b], as well as a third effort by Samuel Lattès.
Daniel S. Alexander
Chapter 10. Lattès and Ritt
Abstract
Joseph Fels Ritt was born six months after Julia in 1893. He received his doctorate from Columbia University in 1917 for his work on differential operators, written under the supervision of Edward Kasner (1878–1955). Following World War I, Ritt taught at Columbia until his death in 1951. Although he wrote several articles on iteration in the late teens and early twenties, his chief interests involved the study of differential equations, and both he and his students made many important contributions to the field of algebraic differential equations.
Daniel S. Alexander
Chapter 11. Fatou and Julia
Abstract
Although the mathematical content of the respective approaches of Fatou and Julia to the iteration of rational functions is quite similar, there are considerable differences in both style and emphasis. Julia on the whole argued more precisely than Fatou. He presented his results in a more organized fashion, and he did a better job of utilizing important theorems from the theory of complex functions. Their works differ on another more subtle level which has as much to do with aesthetics as with mathematics. Fatou wrote in a gently meandering style, reminiscent of a certain nineteenth century style of mathematics, while Julia’s paper is closer to the axiomatic style which predominates in contemporary mathematics.
Daniel S. Alexander
Backmatter
Metadaten
Titel
A History of Complex Dynamics
verfasst von
Professor Daniel S. Alexander
Copyright-Jahr
1994
Verlag
Vieweg+Teubner Verlag
Electronic ISBN
978-3-663-09197-4
Print ISBN
978-3-663-09199-8
DOI
https://doi.org/10.1007/978-3-663-09197-4