Fixed point index in hyperspaces: A Conley-type index for discrete semidynamical systems.

Abstract

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