Daniel Alexander

Iteration in the 1890’s: Leau

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.

Montel’s Theory of Normal Families

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.

Schröder, Cayley and Newton’s Method

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 Schroder (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.

The Next Wave: Korkine and Farkas

Schroder 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 Schroder 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 Schroder and Abel equations, but the study and solution of functional equations is fundamental in many contemporary studies of iteration.

The Flower Theorem of Fatou and Julia

Julia studied the case where ∅′(0) = 1 in his lengthy Memoire 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 equations 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.

Lattès and Ritt

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.

Fatou and Julia

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.

Iteration in the 1890’s: Grévy

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.