On the discrete Conley index in the invariant subspace
Let \(f: \mathbb{R}^{n-1}\times [0,\infty)\rightarrow \mathbb{R}^{n-1}\times [0,\infty)\) be a continuous map and let \(E\subset \mathbb{R}^{n-1}\times [0,\infty)\) be invariant under \(f\). In the paper the Conley index of an isolated invariant set \(S\subset E\) is studied. \(S\) is said to be of attracting (repelling) type if the unstable (stable) set of \(S\) is contained in \(E\). The main result states that the Conley index of \(f\) in \(S\) coincides with the Conley index of \(f\) restricted to \(E\) in \(S\) if \(f\) is of attracting type and is trivial if \(S\) is of repelling type. This allows the authors to generalize a cirterion for permanence in terms of the fixed point index studied in mathematical ecology from flows induced by a vector field to arbitrary discrete dynamical systems.
scientific article; zbMATH DE number 1307960