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.

Figures (5)

• Figure 1: The set $S$ and a filtration pair $(N,L)$
• Figure 2:
• Figure 3:
• Figure 4:
• Figure 5:

Theorems & Definitions (38)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Remark 4
• Example 5
• Example 6
• proof