Everywhere chaotic homeomorphisms on manifolds and \(k\)-dimensional Menger manifolds
type: Item
instance of: scholarly article
title: Everywhere chaotic homeomorphisms on manifolds and \(k\)-dimensional Menger manifolds
zbMATH Open document ID: 0859.54031
DOI: 10.1016/0166-8641(96)00008-9
author: Hisao Kato
published in: Topology and its Applications
publication date: 1997-04-07T00:00:00Z
review text: A map \(f:X\to X\) of a metric space \((X,d)\) is said to be everywhere chaotic if \(f\) has sensitive dependence on initial conditions and, for any nonempty open sets \(U\) and \(V\) of \(X\) and any \(\varepsilon>0\), there exist points \(x\in U\) and \(y\in V\) and a positive integer \(n\geq 1\) such that \(\text{dist} (f^n(x), f^n(y))< \varepsilon\). The author gives constructive proofs to show that every compact, connected \(n\)-manifold \(M^n\) \((n\geq 2)\) and every compact, connected Menger manifold \({\mathcal M}^n\) \((n\geq 2)\) admits an everywhere chaotic homeomorphism. The construction of such homeomorphisms relies essentially on one-dimensional maps and an idea of \textit{M. Barge} and \textit{J. Martin} [Proc. Am. Math. Soc. 110, 523-525 (1990; Zbl 0714.58036)].
reviewed by: Wayne Lewis
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 37C70
zbMATH DE Number: 937152
zbMATH Keywords: shift map
zbMATH Keywords: attractor
zbMATH Keywords: Menger compactum
zbMATH Keywords: \(UV^ k\)-map
zbMATH Keywords: \(Z\)-set
zbMATH Keywords: near homeomorphism
zbMATH Keywords: sensitive dependence on initial conditions
zbMATH Keywords: Menger manifold
zbMATH Keywords: everywhere chaotic homeomorphism
MaRDI profile type: Publication
full work available at URL: 0166-8641(96)00008-9
OpenAlex ID: W1985763848
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description: scientific article; zbMATH DE number 937152