The Conley index for discrete semidynamical systems
type: Item
instance of: scholarly article
title: The Conley index for discrete semidynamical systems
zbMATH Open document ID: 0840.34043
DOI: 10.1016/0166-8641(95)0003J-S
author: Yanyan Li
published in: Topology and its Applications
publication date: 1996-07-04T00:00:00Z
review text: An improved homotopy index of isolated invariant sets of discrete semidynamical systems is introduced. It is based on the concept of so-called category of objects equipped with a morphism defined as follows. For a given category \(K\) let \(K_m\) denote a new category in which the objects are pairs \((X, f)\), where \(X\) is an object in \(K\) and \(f: X\to X\) is a morphism in \(K\). In order to define morphisms in \(K_m\), for objects \((X, f)\) and \((X', f')\) put \[ M((X, f), (X', f'))= \{(g, n): g: X\to X'\text{ is a morphism in }K, n\in \mathbb{Z}^+, g\circ f= f'\circ g\}. \] In \(M((X, f), (X', f'))\) introduce an equivalence relation \(\sim: (g_1, n_1)\sim (g_2, n_2)\) iff there exists \(k\in \mathbb{Z}^+\) such that \(g_1\circ f^{n_2+ k}= g_2\circ f^{n_1+ k}\). Morphisms \((X, f)\to (X', f')\) are defined as the equivalence classes of elements of \(M((X, f), (X', f'))\) in \(\sim\). By \(Htop\) denote the category in which objects are pointed topological spaces and morphisms from \(X\) to \(X'\) are the homotopy classes of continuous maps \(X\to X'\) which preserve the base point. Let \(S\) be an isolated invariant set for a continuous map \(\phi: Z\to Z\), where \(Z\) is a metric space (in practice: locally compact also). If \(N= (N_1, N_0)\) is an index pair (in the Mrozek's sense) for \(S\) then the index consists of the class of all objects isomorphic to \((N_1/N_0, \phi_N)\) in the category \(Htop_m\), where \(\phi_N: N_1/N_0\to N_1/N_0\) is given by \(\phi_N([x])= [\phi(x)]\) if \(x,\phi(x)\in N_1\backslash N_0\) and \(\phi_N([x])= [N_0]\) in the other case. The index defined in this way generalizes both the shape index of Robbin and Salamon, and the Mrozek's algebraic index based on the concept of Leray functor. A proof of the continuation property of the index and a construction of the corresponding connected simple system are also given in the reviewed paper.
reviewed by: Roman Srzednicki
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 18B99
zbMATH DE Number: 823481
zbMATH Keywords: index pair
zbMATH Keywords: Conley index
zbMATH Keywords: morphism
zbMATH Keywords: homotopy index
zbMATH Keywords: isolated invariant sets
zbMATH Keywords: discrete semidynamical systems
MaRDI profile type: Publication
cites work: Q4180600
cites work: Mappings and homological properties in the Conley index theory
cites work: Leray Functor and Cohomological Conley Index for Discrete Dynamical Systems
cites work: Q4312358
cites work: A cohomological Conley index for maps on metric spaces
cites work: Dynamical systems, shape theory and the Conley index
cites work: Connected Simple Systems and The Conley Index of Isolated Invariant Sets
description: scientific article; zbMATH DE number 823481