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Dynamical Systems on 2- and 3-Manifolds

Viacheslav Z. Grines, Timur V. Medvedev, and Olga V. Pochinka
Publication Date: 
Number of Pages: 
Developments in Mathematics
[Reviewed by
William J. Satzer
, on

This book addresses the topological classification of smooth structurally stable discrete dynamical systems induced by a diffeomorphism on closed and orientable two- and three-dimensional manifolds. The authors call their book an introduction. While there is introductory material in the early chapters, however, some background in dynamical systems theory at the level of an advanced undergraduate course is a practical necessity just to get started. The bulk of the book is a careful compilation and exposition of known results in the area. Since the main results exist in widely scattered papers, reports, notes and surveys, the authors have done a real service in bringing all the results together and presenting them in a coherent way.

The topological classification problem here is to identify topological invariants of the decomposition of the manifold into trajectories. That means finding the topological invariants, proving that the set of invariants is complete, and constructing canonical representatives. A topological invariant of a dynamical system is a property that is preserved under conjugacy. Most of the results in this book are for dynamical systems that satisfy Smale’s Axiom A. (This means that the set Ω of non-wandering points is hyperbolic and that the periodic points are dense in Ω.) The Axiom A condition is necessary for structural stability.

The categories of main results presented here include the following:

  • Topological classification of gradient-like diffeomorphisms on two- and three-manifolds;
  • Construction of smooth global Lyapunov functions for Morse-Smale diffeomorphisms;
  • Exploration of the connection between the dynamics of Morse-Smale discrete dynamical systems and the topology of the ambient space; and
  • Topological classification of basic sets of diffeomorphisms on two-manifolds.

This is not a textbook, and there are no exercises. The book is directed at experts, those trying to track down details of prior work, and others who wish to expand their expertise. The references are extensive. The authors have attempted to point the reader to more readily available sources when the original work is more difficult to find.

The authors use some terminology unique to the Russian school of dynamical systems. So the term “cascade” refers to a discrete dynamical system on a manifold induced by a diffeomorphism, and the word “rough” means structurally stable. The authors often use one term with the other in parenthesis, so this is not usually a problem, but the reader should be cautious. There are also a variety of stylistic quirks — some of which require careful reading to resolve. In particular, the use of the indefinite article — appearing where it’s not expected and absent when it is — can be rather distracting.

Bill Satzer ( was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

See the table of contents in the publisher's webpage.