Attractors for relations in \(\sigma\)-compact spaces
type: Item
instance of: scholarly article
title: Attractors for relations in \(\sigma\)-compact spaces
zbMATH Open document ID: 1073.37014
author: Hahng-Yun Chu
author: Jong Suh Park
published in: Topology and its Applications
publication date: 2005-03-08T00:00:00Z
review text: \textit{C. Conley} [Isolated invariant sets and the Morse index, Regional Conference Series in Mathematics, No. 38, Providence, R.I.: American Mathematical Society (1978; Zbl 0397.34056)] introduced a topological definition of an attractor for a flow on a compact metric space and showed that each of these attractors has an associated Lyapunov function. Among the generalizations of Conley's results there is the following theorem from \textit{M. Hurley} [Topology Appl. 109, 201-210 (2001; Zbl 0968.37005)]: Let \(X\) be a locally compact and \(\sigma\)-compact metric space, \(f\) be a continuous map \(X\to X\) and \(A\subseteq X\) be closed, nonempty and \(f\)-invariant. Suppose that \(h: X\to [0, \infty)\) is a continuous function with the properties: (1) \(h^{-1}(0)=A\), (2) there is a neighbourhood \(W\) of \(A\) in \(X\) such that \(h^{-1}([0,1])\subseteq W\) and (3) \(h(f(x)) <f(x)\) whenever \(x\in W\setminus A\). Then \(A\) is a weak attractor for \(f\). The paper under review extends this Hurley result to the case of compact-valued continuous relations on \(X\).
reviewed by: Ľubomír Snoha
Mathematics Subject Classification ID: 37B25
Mathematics Subject Classification ID: 37B35
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 2142290
zbMATH Keywords: attractor
zbMATH Keywords: Lyapunov function
zbMATH Keywords: relation
MaRDI profile type: Publication
full work available at URL: j.topol.2003.05.009
OpenAlex ID: W2055438089
cites work: Q3135846
cites work: Q4180600
cites work: Q4729054
cites work: Noncompact Chain Recurrence and Attraction
cites work: Lyapunov functions and attractors in arbitrary metric spaces
cites work: Weak attractors from Lyapunov functions
cites work: Q4695268
DOI: 10.1016/J.TOPOL.2003.05.009
description: scientific article; zbMATH DE number 2142290