SIAM REVIEWVol. 36, No. 4, pp. 663-686, December 1994

() 1994 Society for Industrial and Applied Mathematics0O9

BOOK REVIEWS

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

A History of Complex Dynamics fromSchr6der to Fatou and Julia. By Daniel S.Alexander, Aspects ofMathematics 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,,+! := fof,, make sense for alln 6 N. If for some a 6 D the sequence (f,,(a))converges tob 6 C, then b 6 D. If in factb 6 D, and if f is continuous, then b is a fixedpoint of f b lim f,,+(a) lim,, f(f,,(a))f(b). On the hand, if b is a fixed point of fand f is holomorphic in a neighborhood of bwith If’(b)l < 1, then it is not hard to showthat in a neighborhood of b, (f,,) converges to theconstant value b, which is consequently called an

attracting fixed point. The number f’ (b) is calledthe multiplier of f at b. One thus has a symphonyinvolving iteration, fixed points, multipliers, andbasins of attraction (or repulsion).

Notice that b is a zero of the function F ifand only if it is a fixed point of the function

f(x) := x F(x). Since iteration can producefixed points, it can find roots of equations. One ofthe earliest investigations of iteration was in con-nection with the Newton-Raphson root-findingmethod, and this is where the author’s book be-

gins (third quarter of the 19th century). It is nota 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 becameat the hands of G. Julia and P. Fatou in 1920.The torrential white-water of contemporary re-

search is not discussed. The initial protagonistsare A. Cayley and E. Schr6der (known to all con-

temporary mathematicians as half of the impor-tant Schr6der-Bernstein theorem of set theory),working on Newton’s method. It was soon real-ized that if 4 is a bijection of D, then studying

Publishers are invited to send booksfor review to

Professors C. W. Groetsch and K. R. Meyer, Departmentof Mathematics, University of Cincinnati, Cincinnati,OH 45221-0025.

the sequence (f,,) is the same as studying the se-quence (F,,), where F := p- o f p; this func-tion, whose nth iterate is simply b- c f,, 4, issaid to be conjugate to f. Iteration of functionsF of the form F(x) := x + or F(x) := ,x() 6 ( constant) is trivial. Consequently, if fis conjugate to such an F, its iteration theory iseasy. This explains why thefunctional equations4) F f o 4) and F p f (f given, Fone of the above two functions) were studied con-comitantly with iteration. Even finding solutions

4 and p which are not necessarily bijective is use-ful. With F(x) := x + ) we have Abel’s equa-tion and with F(x) := ,kx we have Schr6der’sequation. Some early studies even started withthese F and looked for nontrivial f conjugate to

them. (Example: F(x) := 2x, 4(z) := tanz,

f(z) := 2z/(1 -z2).)About half the book concerns developments

up to 1906, involving A. Korkine, J. Farkas, G.Koenigs, A. Gravy, L. Leau, and E. L6meray. Thesecond 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 theconcept of and fundamental theorems about nor-

real families of holomorphic functions, i.e., setsof holomorphic functions which are precompactin the topology of local uniform convergence. In1915 the French Academy of Sciences offered a3000-franc prize for a work on iteration. Longand important memoirs were produced by PierreFatou and Gaston Julia. Fatou’s memoirs werenot entered in the competition, which was wonby the war hero Julia. These papers, which dealtmainly with rational functions, are universally re-garded as the germ of the modern theory, com-

plete treatments of which the interested readercan find in the following monographs: A. Bear-don, Beration of Rational Functions: ComplexAnalytical Dynamical Systems, Springer-Verlag,1991; L. Carleson and T.W. Gamelin, ComplexDynamics, Springer-Verlag, 1993; N. Steinmetz,Rational Iteration: ComplexAnalytic DynamicalSystems, deGruyter Verlag, 1993; and the surveyarticle W. Bergweiler, Iteration of meromorphicfunctions, Bull. Amen Math. Soc., 29 (1993), pp.151-188.The author does a nice job of organizing this

piece of history, detailing the contributions and

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664 BOOK REVIEWS

limitations of each worker and of his influenceon the others. He also supplies evidence that(contrary to folklore) the remarkable work of H.Poincar6 in celestial mechanics (precursor of er-

godic theory) was not the only factor influencingthe Academy’s choice of topic for the prize. Hisoccasional summaries and repetitions of defini-tions facilitate browsing. Important ideas (likethe 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 Hermitecoming up "naturally" as the boundary of thesetwo sets; and we see the beginnings of the cur-

rently popular fractal theory. (Mercifully we’respared more pictures of fractals although it’s ashame that, the principal cast of characters beingas small as it is, no photos of them are included.)The significance of the nascent set theory andtopology of the early 20th century in Julia’s andFatou’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

"4-1 (x) is the total inverse of 4(z) ’" to mean

that b-1(A):= {z 4)(z) A}, or "lim,, Zf,,")and contain actual mathematical errors. For ex-ample, the author’s version of Dirichlet’s Princi-

ple is not that a function minimizing the Dirichletintegral is harmonic, but that it has Dirichlet inte-gral 0 (which, of course, entails its constant). Nota little trouble is caused by randomly revertingto 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 betweenuniform convergence in a domain (rare) and thedominant mode, local uniform convergence, is

ignored; which also results in some statements

being, strictly speaking, false. For a historicalwork, the degree of attention to orthography in the

bibliography is disappointing. ("Volterra" is con-

sistently misspelled and a wrong volume numberis attached to the prize paper of Julia.) At leastone of the author’s historical observations is not

unimpeachable: The assertion that J.F. Ritt didnot 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 bookis attractively produced but, for its length (165pages), perhaps a bit expensive ($42).

R. B. BURCKELKansas State University

Nonlinear Ordinary Differential Equations.By R. Grimshaw, Blackwell Scientific Publica-tions, Oxford. 328 pp. $59.95, cloth. ISBN0-632-02708-8.

This book is an introduction to the geometricaltheory of ordinary differential equations and to

the theory of nonlinear oscillations. The authorhas written this book as a text for students ofmathematics, engineering, and the physical sci-ences. In the setting of an American university,the exposition is at a level appropriate to studentsin their fourth year of undergraduate study or firsttwo years of graduate work.

The book is organized into eleven chapters. InChapter the author formulates the notion of aninitial value problem in R". For the solutionsof such a problem, he then develops the standardresults 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: thenotion of a principal matrix for a homogeneouslinear system: the variation of constants formulafor a nonhomogeneous linear system; and, therole played by eigenvalues and eigenvectors fora 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 authordevelops the notions of characteristic multiplierand 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 byLiapunov and the concepts of orbital stabilityand orbital asymptotic stability as formulated byPoincar6. The two principal topics of this chap-ter are linearized stability about an equilibriumpoint and Liapunov’s Second Method. In con-

nection with the latter, the author shows how, inthe 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. Withregard to phase portraits, the author sets forthPoincar6’s classification of equilibrianodes,

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