Furstenberg families and sensitivity
type: Item
instance of: scholarly article
title: Furstenberg families and sensitivity
zbMATH Open document ID: 1187.37016
DOI: 10.1155/2010/649348
author: Huoyun Wang
author: Feng Tan
author: Jincheng Xiong
published in: Discrete Dynamics in Nature and Society
publication date: 2010-04-26T00:00:00Z
full work available at URL: https://eudml.org/doc/223272
review text: Summary: We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system \((X,f)\) is \({\mathcal F}\)-sensitive if there exists a positive \(\varepsilon\) such that for every \(x\in X\) and every open neighborhood \(U\) of \(x\) there exists \(y\in U\) such that the pair \((x,y)\) is not \({\mathcal F}\)-\(\varepsilon\)-asymptotic; that is, the time set \(\{n:d(f^n(x),f^n(y))>\varepsilon\}\) belongs to \({\mathcal F}\), where \({\mathcal F}\) is a Furstenberg family. A dynamical system \((X,f)\) is \(({\mathcal F}_1,{\mathcal F}_2)\)-sensitive if there is a positive \(\varepsilon\) such that every \(x\in X\) is a limit of points \(y\in X\) such that the pair \((x,y)\) is \({\mathcal F}_1\)-proximal but not \({\mathcal F}_2\)-\(\varepsilon\)-asymptotic; that is, the time set \(\{n:d(f^n(x),f^n(y))<\delta\}\) belongs to \({\mathcal F}_1\) for any positive \(\delta\) but the time set \(\{n:d(f^n(x),f^n(y))>\varepsilon\}\) belongs to \({\mathcal F}_2\), where \({\mathcal F}_1\) and \({\mathcal F}_2\) are Furstenberg families.
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 37B20
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 58K15
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 5701490
zbMATH Keywords: Furstenberg families
zbMATH Keywords: sensitivity
zbMATH Keywords: symbolic dynamics
Wikidata QID: Q58650856
MaRDI profile type: Publication
OpenAlex ID: W1975136850
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description: scientific article; zbMATH DE number 5701490