Linear Systems, Exponential Dichotomy, and Structure of Sets of Hyperbolic Points Z. Lin and X.-Y. Lin, World Scientific (2000), 205 pp., price USD 48.- ISBN: 9810242832.
Reviewer: J. Jung, University of Illinois, Urbana-Champaign, USA.

Level: intermediate/advanced.

This book is an introduction to exponential dichotomy theory in dynamical systems. This theory is useful in, for example, extending the Hartman-Grobman linearization theorem to nonautonomous systems. In so doing, this technique provides an alternate proof of the autonomous version of the theorem. The book provides an introduction to such methods, with a particular emphasis on quasiperiodic systems, their Floquet theory, and their spectral properties.

This book is a posthumous work of Zhensheng Lin, completed by the coauthor Yan-Xia Lin. The work represents the first author's perspectives on differential equations and dynamical systems. The reader will find certain viewpoints on hyperbolic dynamics which are perhaps not so well known outside of China. In particular, the final chapter of the book gives the author's alternate formulation of hyperbolicity in terms of the structures proposed in this book. Along with this are presented proofs of the C^r Closing Lemma and also the Kupka-Smale Theorem on generic systems.

The book is written at a level which assumes a familiarity with dynamical systems and linear analysis. It is suitable for graduate level reading.