Metrizable extensions of dynamical systems from function algebras
type: Item
instance of: scholarly article
title: Metrizable extensions of dynamical systems from function algebras
zbMATH Open document ID: 0982.46050
DOI: 10.1007/BF02505988
author: Juan J. Font
author: Manuel Sanchis
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2002-03-24T00:00:00Z
review text: Suppose that \(\varphi\) is a homeomorphism of a compact metric space \(X\). Then, one defines a dynamical system \((X,\varphi,{\mathbb Z})\). Another dynamical system \((X',\varphi',{\mathbb Z})\) is called an extension of \((X,\varphi,{\mathbb Z})\) if there exists a continuous surjection \(p\) from \(X'\) onto \(X\) satisfying \(p \circ \varphi' = \varphi \circ p\). The authors study a special kind of extension, called a tame extension, introduced by \textit{J. Peters} and \textit{T. Pennings} in [Proc. Am. Math. Soc. 105, No. 1, 80-86 (1989; Zbl 0674.43003) and J. Math. Anal. Appl. 159, No. 2, 345-360 (1991; Zbl 0734.54024)]. Roughly speaking, a tame extension arises by adding to the commutative \(C^*\)-algebra \(C(X)\) of continuous functions on \(X\) a family of bounded functions on \(X\) having not too many discontinuities and then passing to the spectrum \(X'\) of this new commutative \(C^*\)-algebra. The homeomorphism \(\varphi\) of \(X\) is naturally extended to \(X'\). The authors study how several properties of dynamical systems are or are not preserved through tame extensions. They study source, sink and saddle points, the shadowing property and distallity.
reviewed by: Stefaan Vaes
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 1472030
zbMATH Keywords: metrizable tame extensions
zbMATH Keywords: shadowing property
zbMATH Keywords: distallity
zbMATH Keywords: commutative \(C^*\)-algebra
zbMATH Keywords: source
zbMATH Keywords: sink
zbMATH Keywords: saddle points
MaRDI profile type: Publication
cites work: Q3469889
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full work available at URL: https://doi.org/10.1007/bf02505988
OpenAlex ID: W2052949127
description: scientific article; zbMATH DE number 1472030