On a generalization of the Conley index
In this note we present the main ideas of the theory of the Conley index over a base space introduced in the papers [7, 8]. The theory arised as an attempt to solve two questions concerning the classical Conley index. In the definition of the index, the exit set of an isolating neighborhood is collapsed to a point. Some information is lost on this collapse. In particular, topological information about how a set sits in the phase space is lost. The first question addressed is how to retain some of the information which is lost on collapse. As an example, is it possible that two repelling periodic orbits which are not homotopic in the punctured plane are related by continuation? Clearly one cannot be continued to the other as periodic orbits, but the index of such a periodic orbit is the same as the index of the disjoint union of two rest points, so the question of continuation as isolated invariant sets is far less obvious.