The Conley Index and Symbolic Dynamics

Suppose that N is an isolating neighborhood for S with respect to a continuous map f which decomposes into a disjoint union of nitely many compact sets. We establish a relationship between the Conley indices of subsets of S which appear in a natural way when studying the behavior of f from the viewpoint of symbolic dynamics. The relationship is then used to prove a simple criterion for chaos in an isolated invariant set. The Conley index has been an important tool in the study of the qualitative properties of dynamical systems. Recently K.Mischaikow and M.Mrozek 4] observed that it can be applied for the study of chaotic behavior of a dynamical system on an isolated invariant set. The assumptions of the criterion for chaos proved there were successfully checked in the case of the Lorenz equations by means of numerical computations 3]. The aim of this paper is to provide another method of detecting chaos based on the Conley index.