a history of complex dynamics from schroder to fatou and julia.by daniel s. alexander

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Page 1: A History of Complex Dynamics from Schroder to Fatou and Julia.by Daniel S. Alexander

A History of Complex Dynamics from Schroder to Fatou and Julia. by Daniel S. AlexanderReview by: R. B. BurckelSIAM Review, Vol. 36, No. 4 (Dec., 1994), pp. 663-664Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2132728 .

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Page 2: A History of Complex Dynamics from Schroder to Fatou and Julia.by Daniel S. Alexander

SIAM REVIEW ? 1994 Society for Industrial and Applied Mathematics Vol. 36. No. 4, pp. 663-686. Decenmber 1994 009

BOOK REVIEWS

EDITED BY C. W. GROETSCH AND K. R. MEYER

A History of Complex Dynamics from Schroder to Fatou and Julia. By Daniiel S. Alexander, Aspects of Mathematics E24. Vieweg & Sohn, Wiesbaden/Braunschweig, 1994. vii + 165 pp. $42.00. ISBN 3-528-06520-6.

If f maps a subset D of (C into itself, the iterates f:= f, f,,+1 := f o f, make sense for all n E N. If for some a E D the sequence (f, (a)) converges to b E C, then b E D. If in fact b E D, and if f is continuous, then b is a fixed point of f: b = limf,,+1(a) = lim,, f(f,,(a)) =

f (b). On the hand, if b is a fixed point of f and f is holomorphic in a neighborhood of b with If'(b)l < 1, then it is not hard to show that in a neighborhood of b, (f,) converges to the constant value b, which is consequently called an attracting fixed point. The number f '(b) is called the multiplier of f at b. One thus has a symphony involving iteration, fixed points, multipliers, and basins of attraction (or repulsion).

Notice that b is a zero of the function F if and only if it is a fixed point of the function f (x) := x - F(x). Since iteration can produce fixed points, it can find roots of equations. One of the earliest investigations of iteration was in con- nection with the Newton-Raphson root-finding method, and this is where the author's book be- gins (third quarter of the 19th century). It is not a complete history of complex dynamics (mean- ing, roughly, the study of iteration) but starts at a reasonable point where the Moldau is still but a trickle and traces it to the broad river it became at the hands of G. Julia and P. Fatou in 1920. The torrential white-water of contemporary re- search is not discussed. The initial protagonists are A. Cayley and E. Schroder (known to all con- temporary mathematicians as half of the impor- tant Schroder-Bernstein theorem of set theory), working on Newton's method. It was soon real- ized that if 0 is a bijection of D, then studying

Publishers are invited to send books for review to Professors C. W Groetsch and K. R. Meyer, Department of Mathematics, University of Cinicinnati, Cincinniiati, nOH 4522 1-002.S

the sequence (f,,) is the same as studying the se- quence (F,,), where F := 4' o f o 4; this func- tion, whose nth iterate is simply 0-1 o f,, o 0, is said to be conjugate to f. Iteration of functions F of the form F(x) := x + X or F(x) := Xx (X E (C constant) is trivial. Consequently, if f is conjugate to such an F, its iteration theory is easy. This explains why the fiunctional equationis 4,o F = f o4and FoV/ = irf of(f given, F one of the above two functions) were studied con- comitantly with iteration. Even finding solutions 4 and il, which are not necessarily bijective is use- ful. With F(x) := x + X we have Abel's equia- tionI and with F(x) := Xx we have Schroder's equation. Some early studies even started with these F and looked for nontrivial f conjugate to them. (Example: F(x) := 2x, ?(z) := tanz, f (z) := 2z/(1 - Z2).)

About half the book concerns developments up to 1906, involving A. Korkine, J. Farkas, G. Koenigs, A. Grevy, L. Leau, and E. Lemeray. The second half is mainly the work of Montel, Julia, and Fatou, culminating in the 1918 prize competi- tion. Paul Montel's vital role was in supplying the concept of and fundamental theorems about nor- mal families of holomorphic functions, i.e., sets of holomorphic functions which are precompact in the topology of local uniform convergence. In 1915 the French Academy of Sciences offered a 3000-franc prize for a work on iteration. Long and important memoirs were produced by Pierre Fatou and Gaston Julia. Fatou's memoirs were not entered in the competition, which was won by the war hero Julia. These papers, which dealt mainly with rational functions, are universally re- garded as the germ of the modern theory, com- plete treatments of which the interested reader can find in the following monographs: A. Bear- don, Iteration of Rationcal Functions: Complex Analytical Dyniamical Systems, Springer-Verlag, 1991; L. Carleson and T.W. Gamelin, Complex Dynamics, Springer-Verlag, 1993; N. Steinmetz, RationalIterationi: ComplexAnalyticDynamnical Systems, deGruyter Verlag, 1993; and the survey article W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29 (1993), pp. 151-188.

The author does a nice job of organizing this piece of history, detailing the contributions and

663

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Page 3: A History of Complex Dynamics from Schroder to Fatou and Julia.by Daniel S. Alexander

664 BOOK REVIEWS

limitations of each worker and of his influence on the others. He also supplies evidence that (contrary to folklore) the remarkable work of H. Poincare in celestial mechanics (precursor of er- godic theory) was not the only factor influencing the Academy's choice of topic for the prize. His occasional summaries and repetitions of defini- tions facilitate browsing. Important ideas (like the later so-called Fatou set and its comple- ment the Julia set) and theorems from the two big papers are analyzed. We see, for example, the pathological curves that so horrified Hermite coming up "naturally" as the boundary of these two sets; and we see the beginnings of the cur- rently popular fractal theory. (Mercifully we're spared more pictures of fractals - although it's a shame that, the principal cast of characters being as small as it is, no photos of them are included.) The significance of the nascent set theory and topology of the early 20th century in Julia's and Fatou's work is also pointed out.

The book's main weakness lies in the techni- cal mathematical discussions. Some are garbled, use confusing or contradictory notation (such as "0- (x) is the total inverse of 0(z) '" to mean that 0-l (A) := {z ?(z) E A), or "lim,, Ef, ") and contain actual mathematical errors. For ex- ample, the author's version of Dirichlet's Princi- ple is not that a function minimizing the Dirichlet integral is harmonic, but that it has Dirichlet inte- gral 0 (which, of course, entails its constant). Not a little trouble is caused by randomly reverting to the old-fashioned habit of writing "the func- tion f (z)," usually intending "the function f," in spite of the appearance nearby of f evaluated at some definite point z. The distinction between uniform convergence in a domain (rare) and the dominant mode, local uniform convergence, is ignored; which also results in some statements being, strictly speaking, false. For a historical work, the degree of attention to orthography in the bibliography is disappointing. ("Volterra" is con- sistently misspelled and a wrong volume number is attached to the prize paper of Julia.) At least one of the author's historical observations is not unimpeachable: The assertion that J.F. Ritt did not use normal families (p. 122) is belied by his paper in Ann. Math., (2) 22 (1920-21), pp. 157- 160 (not in the author's bibliography). The book is attractively produced but, for its length (165 pages), perhaps a bit expensive ($42).

R. B. BURCKEL Kansas State University

Nonlinear Ordinary Differential Equations. By R. Grimishaw, Blackwell Scientific Publica- tions, Oxford. 328 pp. $59.95, cloth. ISBN 0-632-02708-8.

This book is an introduction to the geometrical theory of ordinary differential equations and to the theory of nonlinear oscillations. The author has written this book as a text for students of mathematics, engineering, and the physical sci- ences. In the setting of an American university, the exposition is at a level appropriate to students in their fourth year of undergraduate study or first two years of graduate work.

The book is organized into eleven chapters. In Chapter 1 the author formulates the notion of an initial value problem in R". For the solutions of such a problem, he then develops the standard results concerning existence, uniqueness, contin- uation, and continuity with respect to data.

Chapter 2 is an introduction to the theory of lin- ear systems in R". The topics treated include: the notion of a principal matrix for a homogeneous linear system, the variation of constants formula for a nonhomogeneous linear system; and, the role played by eigenvalues and eigenvectors for a homogeneous linear system with constant co- efficients.

Chapter 3 is an exposition of Floquet's the- ory for homogeneous linear systems in R" with periodic coefficients. In this chapter the author develops the notions of characteristic multiplier and characteristic exponent. He also renders an extensive treatment of Hill's and Mathieu's equa- tions.

Chapter 4 concerns the theory of stability. Herein the author sets forth the concepts of sta- bility and asymptotic stability as formulated by Liapunov and the concepts of orbital stability and orbital asymptotic stability as formulated by Poincare. The two principal topics of this chap- ter are linearized stability about an equilibrium point and Liapunov's Second Method. In con- nection with the latter, the author shows how, in the context of Hamiltonian mechanics, the ex- tremal properties of a potential energy function determine the stability properties of a given equi- librium point.

Chapters 5 and 6 deal with autonomous sys- tems in the plane-their phase portraits and their periodic orbits. As one might expect, these two chapters have a highly geometrical flavor. With regard to phase portraits, the author sets forth Poincare's classification of equilibria-nodes,

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