An Introduction to the Orbit Structure of Diffeomorphisms
The subject of differentiable dynamical systems in the form recently developed by the group of mathematicians associated with S. Smale and M. Peixoto in the United States and with Ja. Sinai and D. Anosov in the Soviet Union is evoking great interest among this generation's mathematicians. Specialists teaching courses in this field as well as nonexperts interested in a comprehensive introduction should welcome Differentiable Dynamics, the first work to collect and explain in detail a wide selection of results and techniques which have formerly been scattered in the primary literature.
Approaching this literature directly can, moreover, be somewhat treacherous, since a number of obsolete results are embedded within it. In this regard, it is worth noting that one expert in this area who examined the book in manuscript wrote that “Nitecki has it seems to me made an admirable choice of material – both in what he has put it and what he has left out.” Some of this material is so recent that at present it exists only in preprint form.
The book has already proved itself from the standpoint of text use; it derives from a set of lecture notes prepared by the author for a graduate course in dynamical systems he conducted at Yale. Beyond the seminar room, mathematicians generally working in the area of differential topology and global analysis should find it a useful reference for the recent work in dynamics on manifolds.
The book begins with an introduction to the Smale program and philosophy and goes on to give a detailed account of the subject, developing it from the simple case of the circle, since all the topological complications of higher dimensions are absent, but all the essential features of the subject are clearly visible. With this preparation, the student is able to proceed to the inherently difficult “horseshoe example” and its relation to symbolic dynamics.
Differentiable Dynamics consists of the following chapters: Introduction: Flows and Diffeomorphisms – Preliminaries – The Circle – Periodic Points – Anosov Diffeomorphisms – The Horseshoe – Hyperbolic Sets – the Ω-Stability Theorem – A Survey of Recent Work