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 $\Gamma$ 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.