The Conley index for decompositions of isolated invariant sets

Let f be a continuous map of a locally compact metric space X into itself. Suppose that S is an isolated invariant set for f and a disjoint union of a fixed finite number of compact sets. We define 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 behavior 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 last two authors [9] (see also [20]) show that the Conley index reflects 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 fixed point index in [19] or [23] shows that the ones based on the Conley index are, in some sense, weak. They only guarantee that some iteration of the map restricted to the isolated invariant set is semiconjugate to the shift map. Thus, they provide information about the dynamics of some iteration of the map rather than the map itself. The information about the number of periodic orbits is also not as accurate as that provided by the methods based on the fixed point index. The aim of this paper is to define an index of Conley type which fills this gap. Our index is defined for a decomposition 1991 Mathematics Subject Classification: 54H20, 34C35. Research supported by UG grant BW 5100-5-0092-4.