The Morse index, repeller-attractor pairs and the connection index for semiflows on noncompact spaces

AbstractThis paper extends the Morse index theory of C. C. Conley to semiflows π on a noncompact meric space X. π is assumed to satisfy a hypothesis related to conditional α-contraction. We collect background material, define quasi-index pairs and the Morse index of a compact, isolated invariant set K, and prove that the Morse index is a connected simple system. We study repeller-attractor pairs in K, define index triples, and prove their existence and several properties leading to the concepts of the connection index, the connection map and the splitting class. Finally, we consider paths (continuous families) of pairs (π, K) and study continuations of the Morse and the connection indices along such paths. The present paper is a sequel to the author's previous work: On the homotopy index for infinite-dimensional semiflows (Trans. Amer. Math. Soc. 269 (1982), 351–382)