Global stability of dynamical systems. With the collab. of Albert Fathi and Remi Langevin. Transl. from the French by Joseph Christy. (English) Zbl 0606.58003

Shub, Michael

New York etc.: Springer-Verlag. XII, 150 p. DM 74.00 (1987).

This book is the translation from French of notes of a course given by the author at Orsay, France in 1976-1977 and published in Astérisque 56 (1978; Zbl 0396.58014). The exposition starts with the notion of nonwandering points and in the first few chapters most of the machinery for the study of hyperbolic sets is developed. This includes the theory of filtrations, the stable manifold theorem, the invariance of hyperbolicity under perturbations, local product structure, and Markov partitions. These enable the author to present a complete proof of Smale’s \(\Omega\)-stability theorem and the rationality of \(\zeta\)- function of an axiom A diffeomorphism. Each of the ten chapters includes exercises as well as a bibliography. For the English edition the author corrected small errors and added some appendices concerning \(C^ r\) center, center unstable, and strongly unstable manifolds.

Reviewer: Yu.Kifer

MSC:

58-02	Research exposition (monographs, survey articles) pertaining to global analysis
37C75	Stability theory for smooth dynamical systems
37D99	Dynamical systems with hyperbolic behavior
37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
54H20	Topological dynamics (MSC2010)

Keywords:

dynamical systems; global stability; hyperbolic sets

Citations:

Zbl 0396.58014