The Topology of Chaos: Alice in Stretch and Squeezeland R. Gilmore and M. Lefranc Wiley Interscience Seires, (2002) 520 pp., price USD 120.- ISBN:0471408166
Reviewer: R. Ghrist, University of Illinois, Urbana-Champaign, USA.

Level: beginning/intermediate.

This book presents itself as an introduction to the topological analysis approach to dynamical systems, as envisioned by the authors and their collaborators in the physics community.

This approach to dynamical systems is briefly summarized as follows. There are a number of dynamical systems that, correctly or not, can be interpreted as a strange attractor for a flow in three-dimensional Euclidean space. For such a flow, the strange attractor can be `approximated' as an embedded branched surface. The canonical example of this is the Lorenz attractor (Lorenz' original paper remarks on this resemblance). The entire strange attractor, with all its attendant periodic orbits, can be described by a few pieces of data which encode the knotting and twisting of the surface and the combinatorics of the branching. If one wants to characterize a complicated three-dimensional flow by a simple set of invariants, then understanding the topology of these branched manifolds (a.k.a. `templates') is a useful endeavor. In particular, many of the systems considered in this text come from a Takens embedding of experimental time-series data, in which case there are a number of interesting issues associated with identifying the `minimal model' of strange attractor present in the data.

As everyone knows, the Lorenz attractor that appears on the computer screen is not really a branched surface; however, as pointed out by Guckenheimer and Williams in the late 1970s, it is possible to approximate it as such and use this approximation to great effect. Neither is the Lorenz attractor uniformly hyperbolic -- a fact that has caused no small amount of grief to mathematicians, while sparing much of the rest of the world. In like manner, Gilmore and Lefranc do not waste time, space, or effort worrying about the details of the dynamics. The goal is to approximate everything in sight with a strange attractor, and then characterize its embedding data succinctly.

This text exhibits an enthusiasm for its subject that, at its zenith, promotes the topological analysis of branched manifolds as a key step in the 13-part model of what the authors describe as a proposed "mature" reformulation of dynamical systems theory (cross-linked with similar components of Lie theory and singularity theory). There does not seem to be much recognition of the sad-but-true fact that three-dimensional flows --- much like their cousins the 1-d maps and 2-d homeomorphisms --- ultimately comprise a small albeit fascinating corner of dynamical systems.

This book makes up for its lack of precision and rigor with an abundance of interesting physically relevant examples. The figures are numerous and illustrative. The writing style of the book is foreshadowed by its title (though, unfortunately, there are no additional references in the text to Alice or to any other interesting denizens). The book is probably not appropriate as a textbook, being rather a breezy introduction to the subject.