Shadowing with chain transitivity
type: Item
instance of: scholarly article
title: Shadowing with chain transitivity
zbMATH Open document ID: 1178.54018
author: Dawoud Ahmadi Dastjerdi
author: Maryam Hosseini
published in: Topology and its Applications
publication date: 2009-08-20T00:00:00Z
review text: Let \(f:X\to X\) be a continuous surjective map on a compact metric space \(X\). A \(\delta\)-chain from \(x\) to \(y\) of length \(n\) is a finite sequence \(x_0=x\), \(x_1, \dots,x_n=y\) such that \(d(f(x_i),x_{i+1}) \leq\delta\) for \(i=0,\dots,n-1\). A sequence \(\xi=\{x_i\}^\infty_{i=0}\) is called a \(\delta\)-pseudo orbit if for any \(i\geq 0\), \(d(f(x_i),x_{i+1})\leq \delta\). A \(\delta\)-pseudo orbit is said to be \(\varepsilon\)-shadowed by a true orbit if there exists \(z\in X\) such that for \(i\geq 0\), \(d(f^i(z),x_i) \geq\varepsilon\). The map \(f\) is said to have the shadowing property if for any \(\varepsilon>0\) there exists \(\delta>0\) such that any \(\delta\)-pseudo orbit is \(\varepsilon\)-shadowed by a true orbit. The system is said to be chain transitive if any two points can be jointed by a \(\delta\)-chain for any \(\delta>0\). The sensitivity of the system is defined by the formula: \(\exists\varepsilon:\forall x,\forall\delta,\exists B(x,\delta),\exists n>0\), \(d(f^n(x),f^n(y))\geq\varepsilon\). The authors' main result is: a chain transitive system with the shadowing property is either sensitive or equicontinuous.
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 28D20
zbMATH DE Number: 5597560
zbMATH Keywords: chain transitivity
zbMATH Keywords: chain recurrent
zbMATH Keywords: shadowing
zbMATH Keywords: distal
reviewed by: Ludvík Janoš
MaRDI profile type: Publication
full work available at URL: j.topol.2009.04.021
OpenAlex ID: W1975421478
cites work: Q4895967
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DOI: 10.1016/J.TOPOL.2009.04.021
description: scientific article; zbMATH DE number 5597560