On the discrete Conley index in the invariant subspace
type: Item
instance of: scholarly article
title: On the discrete Conley index in the invariant subspace
zbMATH Open document ID: 0956.37009
DOI: 10.1016/S0166-8641(97)00164-8
author: Andrzej Szymczak
author: Klaudiusz Wójcik
author: Piotr Zgliczyński
published in: Topology and its Applications
publication date: 2001-03-12T00:00:00Z
review text: 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.
reviewed by: Marian Mrozek
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1307960
zbMATH Keywords: discrete Conley index
zbMATH Keywords: fixed point index
zbMATH Keywords: permanence
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1307960