Fixed Point Index in Hyperspaces: A Conley-Type Index for Discrete Semidynamical Systems

Let X be a locally compact metric absolute neighbourhood retractfor metric spaces, U X be an open subset and f: U X be a continuousmap. The aim of the paper is to study the fixed point indexof the map that f induces in the hyperspace of X. For any compactisolated invariant set, K U, this fixed point index produces,in a very natural way, a Conley-type (integer valued) indexfor K. This index is computed and it is shown that it only dependson what is called the attracting part of K. The index is usedto obtain a characterization of isolating neighbourhoods ofcompact invariant sets with non-empty attracting part. Thisindex also provides a characterization of compact isolated minimalsets that are attractors.