Conley index at infinity (Q2440842)

The paper deals with flows on \(\mathbb{R}^n\) which can be compactified by adding a sphere \(S^{n-1}\) at infinity. Formally \(\mathbb{R}^n\) is identified with the upper half-sphere \(S^n_+=\{(x,z)\in\mathbb{R}^{n}\times\mathbb{R}:|x|^2+z^2=1, z>0\}\). The flow on \(\mathbb{R}^n\) corresponds to a flow on \(S^n_+\). It is required that, possibly after reparametrization, the flow on \(S^n_+\) extends to a flow on the closure \(\mathcal{H}\) of \(S^n_+\) in \(\mathbb{R}^{n+1}\) which leaves the equator \(\mathcal{E}=\{(x,z)\in\mathcal{H}:z=0\}\), the sphere at infinity, invariant. For this type of flows the author develops Conley index techniques in order to investigate the existence of heteroclinic orbits from isolated invariant sets of the original flow to certain types of invariant sets at infinity.