Countable compacta admitting homeomorphisms with positive sequence entropy
type: Item
instance of: scholarly article
title: Countable compacta admitting homeomorphisms with positive sequence entropy
zbMATH Open document ID: 1161.37020
published in: Journal of Dynamics and Differential Equations
publication date: 2008-12-01T00:00:00Z
review text: In a topological dynamical system \((X,T)\) given by a compact metric space \(X\) and a homeomorphism \(T: X\to X\), let \(\Omega (T)\) be the set of non-wandering points and \(h_{top}^s(T)\) be the topological sequence entropy, i.e., \(h_{top}^s(T) = \sup_A h_{top}^A(T)\) where \(h_{top}^A(T)\) is the topological sequence entropy with respect to the sequence \(A\). While the topological entropy equals the topological entropy on the set of non-wandering points, the authors give an example of a countable compact metric space \(X\) and a homeomorphism \(T: X\to X\) such that \(h_{top}^s(T) \neq h_{top}^s(T|_{\Omega (T)})\). Further, for a countable compact metric space \(X\) let \(d(X)\) be the derived degree of \(X\), i.e., the countable ordinal number \(\alpha\) such that for the derived sets of \(X\) of orders \(\alpha\) and \(\alpha +1\) we have \(X^{(\alpha)} \neq 0\) and \(X^{(\alpha +1)} = 0\), respectively. In the paper under review it is shown that \(h_{top}^s(T)=0\) whenever \(d(X)\leq 1\). On the other hand, if \(d(X)\geq 2\) then there exists a homeomorphism \(T\) on \(X\) such that the set \(X^{(1)}\) of all limit points of \(X\) is the so-called sequence entropy set of \((X,T)\).
reviewed by: Ľubomír Snoha
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 5375869
zbMATH Keywords: topological sequence entropy
zbMATH Keywords: sequence entropy set
zbMATH Keywords: derived set
zbMATH Keywords: countable compactum
author: Xiangdong Ye
author: Ruifeng Zhang
MaRDI profile type: Publication
full work available at URL: s10884-008-9115-4
OpenAlex ID: W2022034661
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DOI: 10.1007/S10884-008-9115-4
description: scientific article; zbMATH DE number 5375869