Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems
type: Item
instance of: scholarly article
title: Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems
zbMATH Open document ID: 1172.37006
author: Yangeng Wang
author: Guo Wei
author: William H. Campbell
published in: Topology and its Applications
publication date: 2009-03-06T00:00:00Z
review text: If \((E,f)\) is a dynamical system, then the hyperspace dynamical system \((\widehat{E},\widehat{f})\) is defined by \(\widehat{f}(A):=f(A)\) on the collection \(\widehat{E}\) of all subsets of \(E\). The relation of different concepts of chaotic behaviour on \((E,f)\) and \((\widehat{E},\widehat{f})\) has been investigated in several papers. A map is said to depend sensitively on initial conditions (this property is briefly called sensitivity), if there is a \(\delta>0\) such that for any \(x\in E\) and any \(\varepsilon>0\) there is a \(y\in E\) with \(d(y,x)<\varepsilon\) and an \(n\in{\mathbb N}\) with \(d(f^{n}(y),f^{n}(x))\geq \delta\). In this paper the authors introduce the notion of collective sensitivity. This means that there is a \(\delta>0\) such that for finitely many \(x_{1},x_{2},\dots ,x_{k}\in E\) and any \(\varepsilon>0\) there are \(y_{1},y_{2},\dots , y_{k}\in E\) with \(d(y_{j},x_{j})<\varepsilon\) for all \(j\in\{1,2,\dots ,k\}\) and there is an \(n\in{\mathbb N}\) and a \(u\in\{1,2,\dots ,k\}\) such that \(d(f^{n}(y_{j}),f^{n}(x_{u}))\geq\delta\) for all \(j\in\{1,2,\dots ,k\}\) or \(d(f^{n}(x_{j}),f^{n}(y_{u}))\geq\delta\) for all \(j\in\{1,2,\dots ,k\}\). It is proved that \((\widehat{E},\widehat{f})\) is sensitive if and only if \((E,f)\) is collectively sensitive. Here \(\widehat{E}\) is endowed with the hit-or-miss topology. Moreover, also the conditions \(({\mathcal C},\widehat{f})\) is sensitive and \(({\mathcal F},\widehat{f})\) is sensitive are equivalent to \((\widehat{E},\widehat{f})\) is sensitive, where \({\mathcal C}\) is the collection of all nonempty compact subsets of \(E\) and \({\mathcal F}\) is the collection of all nonempty finite subsets of \(E\), both endowed with the Hausdorff metric (which is equivalent to the Vietoris topology in this case). The authors also prove that weak mixing implies collective sensitivity.
reviewed by: Peter Raith
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 5526278
zbMATH Keywords: hyperspace dynamical system
zbMATH Keywords: sensitive dependance on initial conditions
zbMATH Keywords: collective sensitivity
zbMATH Keywords: hit-or-miss topology
MaRDI profile type: Publication
full work available at URL: j.topol.2008.10.014
OpenAlex ID: W2016078491
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DOI: 10.1016/J.TOPOL.2008.10.014
description: scientific article; zbMATH DE number 5526278