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Hyperbolic Flows

Todd Fisher and Boris Hasselblatt
European Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Maulik A. Dave
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This book focuses on the study of continuous-time flows in dynamical systems. The authors suggest that books describing discrete-time systems in dynamics receive a lot of attention while continuous-time systems and flows are often treated as an afterthought. Here they present flows and the theory of continuous-time dynamical systems as the primary subject, and they do so from topological, smoothness and measure-theoretic perspectives. Their main emphasis is on uniformly hyperbolic dynamics. (A diffeomorphism on a manifold is called uniformly hyperbolic if there is a splitting of the tangent space at each point of the manifold into stable and unstable subspaces.) 
The book includes both introductory material as well as discussions of recent developments in uniformly hyperbolic dynamics. It includes background material for all the dynamical concepts the authors discuss, but it is not a comprehensive introduction. Readers should have at least prior experience with dynamical systems, calculus on manifolds and measure theory. The book is best suited for graduate study, either as textbook for a course or as a tool for self-study. It would also be an ideal reference for specialists studying hyperbolic flows.
The first part of the book concentrates on the general concept of flows. It treats the basic ideas of topological dynamics first. It reviews fundamental material on conjugacy, attractors and repellers, recurrence properties, transitivity and mixing. Then it takes up the more specialized topic of hyperbolic geodesic flows before reviewing measure-preserving transformations, ergodic theory, and the various notions of entropy. The authors have a bias toward topics that are relevant to hyperbolic flows, but they provide a solid background of known results on the general theory of flows.
In the second part the emphasis turns to purely hyperbolic flows. Here there is a mixture of known results, known results understood in a new way, and entirely new results. The first two chapters are the foundation of the theory. The treatment begins with a general discussion of hyperbolicity, examples of physical flows that are naturally uniformly hyperbolic, the shadowing theorem and structural stability. The other important piece is an extended discussion of invariant foliations for uniformly hyperbolic flows.
The final three chapters treat more specialized topics starting with the ergodic theory of hyperbolic sets and continuing with the theory of Anosov flows, their topology, dynamics and rigidity.
Each of the main chapters contains a number of exercises. Hints and solutions are provided in an appendix. There is an excellent index and a very thorough bibliography. 


 Dr. Maulik A. Dave is an experienced reviewer of research papers, and books in computer science, mathematics, and engineering. Email: