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Introduction to Dynamical Systems

Micahel Brin and Garrett Stuck
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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The goal of this book is to provide a “broad and gentle” introduction to the subject of dynamical systems for graduate students. The authors do this in nine chapters and less than 250 pages. Whole volumes have been written on the subjects of each of those nine chapters. Consequently, the book feels something like a bullet train ride through dynamical systems. So the word “broad” rings true; “gentle”, not so much.

Still, the authors manage do quite a bit very well. While the pace is fast and the book is very concise, the organization and selection of topics has been considered carefully, and the writing is strong enough to support the speedy treatment. The main themes of dynamical systems are introduced by examples followed by discussion of fundamental results and proofs. Not everything is proved and there is no attempt to be comprehensive.

The first chapter introduces basic ideas about dynamical systems with a range of examples. It gets quickly to important constructions like Smale’s horseshoe and the solenoid, establishes connections with differential equations and flows, and then introduces chaos and the Lorenz attractor. The next two chapters explore topological and symbolic dynamics. Topological entropy provides a valuable way to measure the complexity of the structure of orbits for a dynamical system. Symbolic dynamics characterizes a dynamical system by recording its itinerary through a finite collection of disjoint subsets.

Two chapters at the heart of dynamical systems follow. The first addresses ergodic theory. This is a subject with a history going back at least as far as Gibbs and with deep connections to statistical mechanics. It is naturally formulated here in terms of dynamical systems on a measure space; mixing and the various standard ergodic theorems are the main topics. The next chapter treats hyperbolic dynamics; the main topics here are hyperbolic sets, stable and unstable manifolds, Anosov and Axiom A diffeomorphisms and the critical notion of structural stability.

The four remaining chapters explore and amplify extensions of prior topics: ergodicity of volume- preserving Anosov diffeomorphisms, low dimensional dynamics, dynamics on the Riemann sphere, and measure-theoretic entropy.

As they touch on all the main subjects of dynamical systems, the authors present some interesting sidelights along the way: applications of topological recurrence to Ramsey theory, of symbolic dynamics to data storage, and of ergodic theory to number theory and internet search. With their obvious broad command of the subject, one wishes that they had also spent a little bit of time discussing the interrelationships and interplay among the several areas of dynamical systems.

While this text would probably serve as the base textbook for a graduate lecture course, its pace makes it less useful for students new to the subject. It certainly does give a notion of the scope of dynamical systems in a way that few other single books do.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

1. Examples and basic concepts
2. Topological dynamics
3. Symbolic dynamics
4. Ergodic theory
5. Hyperbolic dynamics
6. Ergodicity of Anosov diffeomorphisms
7. Low-dimensional dynamics
8. Complex dynamics
9. Measure-theoretic entropy