Conley index

Definition

Let $X$ be a locally compact metric space and let $\varphi \colon \R \times X \to X$ be a continuous flow. For a set $N \subseteq X$, let $\operatorname{Inv}(N) = \set{x \in X : \varphi_t(x) \in N \text{ for all } t \in \R}$, the largest invariant set in $N$. A compact set $N$ is an isolating neighbourhood if $\operatorname{Inv}(N)$ lies in the interior of $N$. An invariant set $S$ is isolated if $S = \operatorname{Inv}(N)$ for some isolating neighbourhood $N$. An index pair for an isolated invariant set $S$ is a pair of compact sets $L \subseteq N \subseteq X$ such that:

1. The closure of $N \setminus L$ is an isolating neighbourhood with $\operatorname{Inv}(\overline{N \setminus L}) = S$.
2. $L$ is positively invariant in $N$: if $x \in L$, $t \ge 0$ and $\varphi_s(x) \in N$ for all $s \in [0, t]$, then $\varphi_t(x) \in L$.
3. $L$ is an exit set for $N$: if $x \in N$ and $\varphi_t(x) \notin N$ for some $t > 0$, then there is $s \in [0, t]$ with $\varphi_s(x) \in L$ and $\varphi_r(x) \in N$ for all $r \in [0, s]$.

Every isolated invariant set has an index pair. The homotopy type of the pointed space $(N/L, [L])$ does not depend on the choice of index pair. This homotopy type is the Conley index $h(S)$ of $S$.