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Dynamical Systems: An Introduction

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When I was a graduate student in the 1970s there was a remarkable burst of activity in the theory of dynamical systems. What I learned at the time was extracted from an assortment of papers, lectures and conferences and assembled with some difficulty into a semi-coherent picture. So it is a bit unsettling now to find a slim textbook — one aimed toward advanced undergraduates no less — that provides a rather complete introduction to the theory of dynamical systems. It is an ambitious book. The list of topics is extensive: topological dynamics, low dimensional, hyperbolic and symbolic dynamics, and finally a quick introduction to ergodic theory. It is not a shallow survey; all results are carefully proved (although sometimes with restricted hypotheses). The book (with barely 200 pages) is also remarkably concise.

The text focuses largely on dynamical systems with discrete time, but continuous time systems and flows such as those that arise from autonomous ordinary differential equations are not neglected. The authors include a nice introductory chapter that surveys both the topics they will cover and references for further study. They then begin the technical work gently with some definitions and basic discrete and continuous examples: rotations and expanding maps of the circle, endomorphisms of the torus and flows of some autonomous differential equations.

Things accelerate as we move into topological dynamics; the authors introduce limit sets, topological recurrence, and topological entropy and establish some of their basic properties. This is followed by a chapter on a collection of topics in low dimensional dynamical systems including the Poincaré-Bendixson theorem, Sharkovsky’s theorem on the existence of periodic points in one-dimensional dynamics and standard dynamical examples on the circle.

The going gets a bit more difficult with the next two chapters on hyperbolic dynamics, one introductory and one more advanced. There’s quite a bit here, and even with the introductory chapter, it would be quite challenging for a newcomer. One of the key pieces is the Hartman-Grobman theorem that says a diffeomorphism is locally topologically conjugate to its derivative in the neighborhood of a hyperbolic fixed point. The Smale horseshoe example points to the existence of considerable complexity within the framework of hyperbolic dynamics. The final section introduces the geodesic flow in hyperbolic geometry with associated discussions of isometries and the role of the projective special linear group.

The book concludes with chapters on symbolic dynamics (the shift map, revisiting topological entropy and the Smale horseshoe, and topological Markov chains). Finally, a chapter on ergodic theory introduces just enough measure theory to discuss recurrence and prove Birkhoff’s ergodic theorem.

This book is remarkable for its completeness and thoroughness. The authors have taken great care taken in presenting and proving a significant number of results compactly and succinctly. However, there is a great deal of material here, some of it quite challenging, and the pace is unrelenting. This would be quite a hurdle for undergraduates. It’s a better match for graduate courses, yet there are still difficulties: the book is too flat in the sense that all results look to be equally important, the connections between results are not very clear, and the motivation for the various approaches is largely absent. Of course, a good instructor could remedy that. Yet the book would be significantly improved with more attention to establishing context and connections between its parts. Having said that, it is also important to say that the text does provide the foundations for what could be an excellent course.

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.

Date Received: 
Tuesday, January 8, 2013
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Luis Barreira and Claudia Valls
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William J. Satzer
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Sunday, March 31, 2013