The Conley index over the circle (Q1591590)

Let \(X\) be a locally compact metric space and \(\xi:X\rightarrow S^1\) a fixed continuous map. The authors consider a flow \(\pi:X\times\mathbb R\rightarrow X\) on \(X\) which satisfies \(\xi\circ\pi(x, t)=e^{2\pi it}\xi(x)\). For an isolated invariant set \(S\subset X\) they define the Conley index \(h_\xi(S,\pi)\) of \(S\) over \(S^1\). This is a special case of the Conley index over a base introduced by the authors in an earlier paper [Trans. Am. Math. Soc. 352, 4171-4194 (2000; Zbl 0967.37011)]. For \(a\in S^1\) the time-one map \(\Pi=\pi(\cdot,1)\) induces a discrete dynamical system \(\Pi_a:X_a:=\xi^{-1}(a)\rightarrow X_a\). The main result of the paper states that the discrete Conley index \(h(S\cap X_a, \Pi_a)\) is completely determined by \(h_\xi(S,\pi)\). It is known that \(h(S\cap X_a,\Pi_a)\) is not determined by the usual Conley index of \(S\).