Multivalued maps, selections and dynamical systems
type: Item
instance of: scholarly article
title: Multivalued maps, selections and dynamical systems
zbMATH Open document ID: 1170.54007
published in: Topology and its Applications
publication date: 2008-05-15T00:00:00Z
review text: Let \(X\) be a locally compact space and \((x,t)\mapsto \varphi(x,t)\) a continuous flow on \(X\). Let \(\psi=J^{+}\) be the multivalued map with \(D(\psi)\)=\(\{x\in X\); \(\emptyset \neq J^{+}(x)\)=compact\(\}\) and \(\Psi\) its associated map (from \(2^{D(\psi)}\) to \(2^X\)): \(\Psi(K)=J^{+}(K)\). Theorem. Let \(K\subseteq X\) be compact and invariant. Then (i) \(K\) is stable iff \(\Psi(k)=k\), (ii) \(K\) is a global attractor iff \(\Psi(\text{cl}\{k\})=k\), (iii) \(K\) is an attractor iff \(\Psi(\text{cl} \{k\} \cap W)=k\), for some neighborhood \(W\) of \(k\) in \(2^X\). If, in addition, \(X\) is compact and \(K\) is an attractor then (iv) \(\text{int} (K)= \emptyset\) iff \(\Psi(k^*)=x\) (where \(k^*\) is the dual repeller of \(k\)). Further aspects involving the attractors in \(\mathbb R^n\) are discussed.
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 5275711
zbMATH Keywords: hyperspace
zbMATH Keywords: multivalued map
zbMATH Keywords: continuous flow
zbMATH Keywords: invariance
zbMATH Keywords: attractor
zbMATH Keywords: repeller
zbMATH Keywords: selection
zbMATH Keywords: upper semifinite topology
author: Jaime J. Sánchez-Gabites
author: José Sanjurjo
reviewed by: Mihai Turinici
MaRDI profile type: Publication
full work available at URL: j.topol.2006.10.016
OpenAlex ID: W2022160435
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DOI: 10.1016/J.TOPOL.2006.10.016
description: scientific article; zbMATH DE number 5275711