Géométrie différentielle et singularités de systèmes dynamiques. (Differential geometry and singularities of dynamical systems). (French) Zbl 0601.58002

Chaperon, Marc

Astérisque, 138/139. Publié avec le concours du Centre National de la Recherche Scientifique. Paris: Société Mathématique de France. 440 p. FF 280.00; $ 37.00 (1986).

The main body of this book constitutes an exhaustive treatise about the classification of differentiable actions of elementary groups (i.e. groups of the form \(F\oplus T^{\ell}\oplus {\mathbb{R}}^ k\oplus {\mathbb{Z}}^ m\) where F is a finite abelian group and \(T^{\ell}\) the \(\ell\)-torus) on differentiable manifolds. The subject is presented very geometrically and more or less from scratch. The book starts with an introduction of the basic concepts of differential topology and then describes the ”classical” results of Hartmann-Grobmann and Sternberg about linearization resp. conjugacy of germs of hyperbolic actions of \({\mathbb{R}}\) and \({\mathbb{Z}}\). In the sequel these results are generalized and extended in various directions, including new and original contributions of the author.
The main lines of generalization are (i) to the above-mentioned elementary abelian groups and (ii) to group actions that preserve an additional structure (like symplectic or contact structure) on the manifold.

Reviewer: H.Knörrer

MSC:

58-02	Research exposition (monographs, survey articles) pertaining to global analysis
37G99	Local and nonlocal bifurcation theory for dynamical systems

Keywords:

hyperbolic sets; singularities of dynamical systems; group action