The Conley Index for Decompositionsof Isolated Invariant Sets 1

Let f be a continuous map of a locally compact metric space X into itself. Suppose that S is an isolated invariant set with respect to f being a disjoint union of a xed nite number of compact sets. We deene an index of Conley type for isolated invariant sets admitting such a decomposition and prove some of its properties, which appear to be similar to that of the ordinary Conley index for maps. Our index takes into account the existence of the decomposition of S and therefore carries more information about the structure of the invariant set. In particular, it seems to be a more accurate tool for the detection of periodic trajectories and chaos of the Smale horseshoe type than the ordinary Conley index. 0. Introduction The Conley index has become an important tool in the study of the qualitative behaviour of dynamical systems, with both discrete and continuous time. The results concerning attractor-repeller decompositions ((1], 15], 18]), the connection matrix theory ((3], 4], 5]) as well as recent papers by Ch.McCord, K.Mischaikow and M.Mrozek 7] and the latter two authors 9] (see also 20]) show that the Conley index reeects the structure of an isolated invariant set. In this paper we are mainly interested in the Conley index as a tool for the detection of chaos and periodic orbits. Comparing the results of 9] and 20] with the criterions for chaos based on the xed point index in 19] or 23] shows that the ones based on the Conley index