A dynamical interpretation of the connection map of an attractor-repeller decomposition

Abstract

In Conley index theory one may study an invariant set S by decomposing it into an attractor A, a repeller R, and the orbits connecting the two. The Conley indices of S, A and R fit into an exact sequence where a certain connection homomorphism Γ plays an important role. In this paper we provide a dynamical interpretation of this map. Roughly, R “emits” an element of its Conley index as a “wavefront”, part of which intersects the connecting orbits in S. This subset of the wavefront evolves towards A and is then “received” by it to produce an element in its Conley index.