Recurrence and the shadowing property
type: Item
instance of: scholarly article
title: Recurrence and the shadowing property
zbMATH Open document ID: 0861.54036
DOI: 10.1016/0166-8641(95)00109-3
author: Chinku Chu
author: Ki-Shik Koo
published in: Topology and its Applications
publication date: 1996-10-20T00:00:00Z
review text: Let \(f:X \leftarrow X\) be a homeomorphism on a compact metric space. Let \(\Omega(f)\) denote the set of nonwandering points and \(C(f)= \{x\in X: x\in \alpha (x)\cap \omega (x)\}\), where \(\alpha(x)\) and \(\omega(x)\) denote the positive and negative limit set of \(x\). The main results are: (1) If \(f\) is expansive and shadowing, then \(\overline {C(f)} (= \Omega (f))\) has a local product structure. (2) If \(f\) is shadowing and \(f|_{\Omega (f)}\) is topologically transitive, then \(X= \Omega(f)\). (3) If \(X\) is connected, is not a minimal set and \(C(f) =X\), then \(f\) is not shadowing.
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37D99
Mathematics Subject Classification ID: 37C75
zbMATH DE Number: 937177
zbMATH Keywords: chain recurrence
zbMATH Keywords: minimal set
zbMATH Keywords: shadowing
reviewed by: Jerzy Ombach
MaRDI profile type: Publication
full work available at URL: 0166-8641(95)00109-3
OpenAlex ID: W2025299156
cites work: Homeomorphisms without the pseudo-orbit tracing property
cites work: On homeomorphisms with pseudo-orbit tracing property
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cites work: Q3799382
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description: scientific article; zbMATH DE number 937177