BOOK REVIEWS
A First Course in Discrete Dynamical Systems
by Richard A. Holmgren, Second Edition, Universitext,
Springer-Verlag, New York, 1996, 223pp, US$29.95, ISBN 0-387-94780-9
This book is an inviting introduction to one-dimensional discrete dynamical systems. The main theorems of the subject are carefully developed and supported by numerous concrete examples and illustrations. Holmgren's style of exposition is lucid and personable. His book has been tested in the classroom and is suitable for good second or upper-year mathematics or science students who have at least two courses in calculus with some analysis. This book fills two kinds of needs. First, it is definitely a text of choice for a single or two-semester terminal course in discrete dynamical systems at the undergraduate level. As well, it is a solid preparatory text for those wishing to pursue dynamics at a higher level such as that found in Devaney's book, An Introduction to Chaotic Dynamical Systems, or the definitive works, Iterated Maps on the Interval as a Dynamical System, by Collet and Eckmann, and, One-Dimensional Dynamics, by de Melo and van Strien.
In simple terms a one-dimensional discrete dynamical system is a continuous real-valued function of a real variable which is composed with itself over and over again. The main goal in the subject is to understand the asymptotic behavior of the set of points obtained by successive compositions of the function at a given point. Such iterative systems are very easy to motivate and illustrate via examples. They appear in diverse settings such as models of biological, chemical, and economic phenomena, fractal geometry, and numerical solutions of differential equations, just to name a few. Given this, and the wide availability of inexpensive and powerful graphing computers, one can quickly access complex mathematical phenomena which lie at the heart of dynamics.
In Holmgren's book the subject of discrete dynamical systems is developed in fifteen chapters. There are 186 exercises, 56 illustrations, and appendices containing a half-dozen Mathematica programs for graphing. Beyond the introduction, the second and third chapters give a solid review of elementary real analysis and topology. This background material could well be assigned reading for students who have taken a respectable course in analysis where the writing of rigorous proofs was emphasized. However, this reviewer agrees with the author's remarks in the preface that most students will probably benefit from a careful revision of these topics right at the start of a course. The fourth chapter marks the beginning of the subject of one-dimensional dynamics proper. Here the basic concepts of fixed point, periodic point and periodicity, orbit, and stable sets are introduced and illustrated. A remarkable result in dynamical systems, and in all of analysis, is Sarkovskii's theorem. This beautiful theorem describes a certain ordering on the natural numbers which completely characterizes the periodicity of prime periods of a continuous function. The fifth chapter contains a statement and full proof of the so-called "period three" special case of Sarkovskii's theorem. This two-page proof is detailed and is a nice example of some rather intricate analysis which is accessible to undergraduate students. Although the general statement of Sarkovskii's theorem is not proved in Holmgren's book, he does encourage further study by suggesting two excellent references: Huang's proof in the 1992 issue of the Mathematics Magazine and a proof in Devaney's text.
Differentiation enters the subject matter in the sixth chapter. This chapter is particularly appealing because it contains many applications of theorems in classical analysis to fundamental results in dynamics. For example, the mean value theorem is used to prove the result on the uniqueness of fixed points, and the size of the derivative in terms of absolute value is used to motivate the concepts of hyperbolic, attracting, and repelling Families of functions which are indexed by a single parameter are introduced in chapter seven. The goal in studying such families is to understand how dynamics are affected by changing the parameter. This idea leads naturally into the notion of bifurcations. In chapter seven several types of bifurcations are described and motivated by examples and illustrations.
The next four chapters are devoted to exploring four core topics in dynamical systems via the logistic function. This function is the parameterized family h(r;x) = rx(1-x) where r > 0 is the parameter and x is the variable. The approach Holmgren takes by using the logistic equation as the recurring example is concrete. He first uses this example in chapter eight to introduce Cantor sets and chaos. In studying the dynamics of the logistic function on a Cantor set, the concepts of chaos and topological transitivity appear in a straightforward way. Devaney's definition of chaos is adopted. There are thirteen exercises in chapter eight and one of these is devoted to exploring periodicity, transitivity, and chaos via the so-called tent map.
Two real functions are said to be topologically conjugate if they commute with a homeomorphism. Topological conjugacy is introduced in chapter nine and is developed by way of commutative diagrams. In fact, it is this part of the book that may well be an undergraduate student's first encounter with commutative diagrams, particularly in analysis. In chapter ten the period-doubling cascade is studied also via the logistic map. There are several bifurcation and iteration diagrams which encourage the reader to explore dynamics further with a computer. The Feigenbaum constant is discussed both in the text and in the exercises Basic facts of metric spaces are reviewed in the beginning of chapter eleven. A very important metric space in dynamics is the symbol space consisting of sequences of 0's and 1's. In chapter eleven one sees how the dynamics of the logistic map are encoded via topological conjugacy in the symbolic dynamics of the 0-1-sequence space.
Chapters twelve and thirteen are devoted to two important applications of one-dimensional dynamical systems: Newton's method for approximating zeros of a function and numerical solutions to differential equations. Chapter twelve on Newton's method will be of considerable interest to students who have had some exposure to numerical analysis. The treatment of Newton's method is quite thorough and Holmgren is careful in showing how the theory of dynamics developed in previous chapters is put to good use. One of the appendices contains Mathematica code for computing stable sets for Newton's method, although it is noted that this code is not time-optimal. In chapter twelve Euler's method for approximating solutions to ordinary differential equations is viewed in a dynamics context. As in the previous chapter, the development is detailed and well- motivated.
The last two chapters introduce complex dynamics. Here one is interested in the behavior of iterations of a complex-valued function of a single complex variable. Complex numbers, functions, and differentiability are reviewed in chapter fourteen. This material is straightforward and self- contained, so it is probably unnecessary for students to have had much prior exposure to complex analysis beforehand in order to appreciate Holmgren's treatment of complex dynamics. The dynamics of Newton's method in the complex setting is also studied In the fifteenth and final chapter the quadratic family of maps is introduced. This family of functions leads naturally to Julia sets and the celebrated Mandelbrot set.
The reader is encouraged to explore this set by experimenting with a computer graphics package. In short summary, this textbook comes well-recommended. Richard Holmgren has obviously taken considerable care in writing a thorough introduction to discrete dynamical systems in a style and with content that students will appreciate. This textbook is very much a mathematics book, although there are many opportunities to explore with a computer. Holmgren's book will be a pleasure to read and to learn from.
Raymond Grinnell , University of the West IndiesAdvanced Analysis on the Real Line
R Kannan and Carole King Krueger, Universitext,
Springer-Verlag, Berlin-New York-London, 1996, 260pp, DM 68.00, ISBN
0-387-94642-X
SPRINGER-VERLAG PUBLICATIONS
Information has been received about the following publications. Anyone interested in reviewing any of these books should contact David Alcorn, Department of Mathematics University of Auckland, (email: alcorn@math.auckland.ac.nz)
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