The aim of this book is to study some properties of flows, especially those related to Morse theory and the Conley index, by using topological tools. We are also interested in the topological properties of some of the most important objects that appear in dynamics, such as attractors and isolated invariant sets, and in stability-related topological aspects of dynamics. We would like to provide a firm foundation for readers who wish to delve into Morse theory and Conley index theory and discuss some results that may be less well known to experts in these areas.
Chapter 1. Preliminaries
Dynamical systems and flows, stability, attraction, manifolds, flows of tangent fields, homotopy, cohomology, ANRs.
Chapter 2. Gradient flows and elements of Morse Theory
Gradient systems on manifolds and gradientlike flows. Lusternik-Schnirelman category. Morse functions and fields. Morse and Poincaré polynomials. Level sets topology. Morse inequalities. Homotopy type associated to Morse functions.
Chapter 3. Conley index theory of flows
Chapter 4. A survey of selected results
