Homotopical dynamics of gradient flows

Abstract

In this paper we will be interested in results surrounding the following basic question: what are the homotopy properties that one can extract from a gradient flow? We approach this question by decomposing it into three parts: 1. Identify what are the homotopical objects that are provided by the flow (e.g. critical points, Conley indexes). 2. Discover what are the relations that have to be satisfied by these objects (e.g. Morse inequalities, Lusternik-Schnirelmann type inequalities). 3. (The Realizability Problem.) Given some homotopy objects that satisfy the relations from 2., is there a corresponding flow? Is this flow unique up to some equivalence relation? We will consider only gradient flows induced by functions with isolated critical points, restrict our discussion to the finite dimensional, compact context and concentrate on the non-Morse case. One way to look at these questions in this case is to first treat numerical invariants, then local invariants, and finally, pairwise invariants (that concern pairs of consecutive critical points)... At each level, we can look at the points 1.,2.,3. above.