A cup product pairing and time-duality for discrete dynamical systems
نویسنده
چکیده
With each isolating block for a homeomorphism of an orientable manifold we associate a cup product pairing. Up to an isomorphism, this is a pairing of the discrete Conley indices of the homeomorphism and its inverse on the isolating block. We show that this pairing is nondegenerate. As a corollary we prove a time duality result for the discrete Conley index. 0. Introduction The observation that the Poincare-Lefschetz duality theorem can be used to establish a relationship between the Conley indices of an isolated invariant set with respect to forward and backward time comes from C.McCord 4]. A simple proof of a result of this type in the continuous setting was given by M.Mrozek and R.Srzednicki in 7]. In the unpublished typescript 9] a discrete counterpart of the time-duality is proved. Roughly speaking, it states that the q-dimensional cohomological Conley index of an isolated invariant set with respect to an orientation preserving homeomorphism is equal to the (n ? q)-dimensional homological Conley index of the same invariant set with respect to its inverse, where n is the dimension of the manifold. If the homeomorphism reverses orientation, the index maps with respect to
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