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An introduction to the theory of smooth dynamical systems. Transl. from the Polish by Marcin E. Kuczma. (English) Zbl 0566.58015

A Wiley-interscience publication. Chichester etc.: John Wiley & Sons; Warszawa; PWN-Polish Scientific Publishers. X, 369 p. £29.50 (1984).
[For the Polish original (1982) see below.] As the author states in the preface, ”The methodology (of dynamical systems) is so widespread in mathematics that, for instance, dynamical systems are not given a separate heading in the Mathematical Reviews”. An interesting paradox.
This monograph consists of five chapters. First the basic facts like the stable-unstable manifold theorem for hyperbolic critical points and ideas in suspension mappings and limit sets. The second chapter surveys dynamical systems on manifolds of dimension 1 and 2, for instance homeomorphisms of the circle and automorphisms of the torus \(T^ 2.\)
Chapters 3 and 4 are connected with transversality, generic properties and structural stability, including Morse-Smale systems, Anosov diffeomorphisms, Smale’s counterexample on the existence of structural stability of certain systems, Axiom A systems, horseshoes. The final chapter discusses ergodic theory and entropy.
The theory of dynamical systems is flourishing both as an abstract mathematical theory and as a tool for understanding new phenomena in the applications of nonlinear analysis. This book presents the abstract mathematics but made as concrete as possible i.e. illustrated by many examples and exercises. Another excellent introduction to the field is the book by J. Guckenheimer and P. Holmes [Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1983; Zbl 0515.34001)], which is somewhat more application inclined. To have both books as complementary material in a workshop would be very useful.
Reviewer: F.Verhulst

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
37C75 Stability theory for smooth dynamical systems
58C05 Real-valued functions on manifolds
37A99 Ergodic theory
37D99 Dynamical systems with hyperbolic behavior