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Conley index theory. (English) Zbl 0847.58062

Johnson, Russell (ed.), Dynamical systems. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 13-22, 1994. Berlin: Springer-Verlag. Lect. Notes Math. 1609, 119-207 (1995).
These lecture notes are designed for a reader wanting to get basic knowledge of the principles of the Conley theory of dynamical systems, in particular the Conley index theory and to grasp a necessary intuition of and motivation for some recent developments of this theory. It consists of eight parts covering a large material: a brief description of basic definitions and a concise introduction to the Conley index theory are given; decomposition techniques are presented in order to study the dynamics of isolated invariant sets, in particular Morse decompositions are studied; index filtrations and connection matrices are discussed in order to study relationships between the Conley indices of the Morse sets and the Conley index of the total invariant set; the existence of periodic orbits of a flow is discussed and established via the Conley index arguments; the local and global semi-conjugacy is presented and used, for instance, to study the set of connecting orbits between Morse sets; finally, the Conley index methods for singularly perturbed dynamical systems are discussed.
The author’s notes give a very neat description of the objectives of the Conley theory: clearly posed questions and problems help to understand the answers, results, their meaning and importance. His arguments and methods of exposition are clear, convincing and straightforward. They also offer a big number of examples, and a vast literature on the subject.
In the reviewer’s opinion the paper is a must for both researchers and beginners in the dynamical systems and the Conley index theory.
For the entire collection see [Zbl 0822.00008].

MSC:

37C10 Dynamics induced by flows and semiflows
54H20 Topological dynamics (MSC2010)
37C80 Symmetries, equivariant dynamical systems (MSC2010)

Software:

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