An attraction result and an index theorem for continuous flows on ${\displaystyle {ℝ}^n\times [0,\infty )}$

We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on ${\displaystyle E={ℝ}^{n+1}}$ for which ${\displaystyle \partial E={ℝ}^n\times \left\{0\right\}}$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in E\∂E such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on ${\displaystyle {ℝ}^n\times [0,\infty )}$.