Topological robustness of non-saddle sets
type: Item
instance of: scholarly article
title: Topological robustness of non-saddle sets
zbMATH Open document ID: 1175.54030
published in: Topology and its Applications
publication date: 2009-07-01T00:00:00Z
review text: A invariant set \(X\) of a flow \(\varphi: M\times{\mathbb R} \rightarrow M\) is a non-saddle set provided every neighborhood \(U\) of \(X\) admits a neighborhood \(V\) of \(X\) such that for every \(x\in V\), \(\varphi (\{x\}\times [0, \infty)) \subset U\) or \(\varphi (\{x\}\times (-\infty, 0]) \subset V\). The authors introduce the notions of dynamical robustness and topological robustness for parametrized families \(\{\varphi_{\lambda}\}_{\lambda\in{\mathbb R}}\) of flows in a locally compact and locally connected metric space \(M\). Roughly speaking, the former means that a non-saddle set has a local continuation made of non-saddle sets, and the latter means that an isolated invariant set has a local continuation made of isolated invariant sets with the same shape type. Motivated by the fact that robustness holds for the property of being an attractor, the authors consider the problem: is every non-saddle set dynamically robust? In the first part of the paper, the authors give a simple example to give a negative answer to this question, and obtain two results concerning the problem: does dynamical robustness imply topological robustness? The first theorem states: If \(\{\varphi_{\lambda}\}_{\lambda\in{\mathbb R}}\) is a parametrized differentiable family of flows in an \(n\)-manifold \(M\), and if a connected isolated non-saddle set \(K_0\) for \(\varphi_0\) is dynamically robust, then \(K_0\) is topologically robust. For the second theorem the authors consider regularly dynamical robustness, which means that a regular non-saddle set has a local continuation made of regular non-saddle sets, where regularity is in the sense of [\textit{J. M. R. Sanjurjo}, Mathematika 47, No.~1--2, 299--305 (2000; Zbl 1128.37302)]. The second theorem states: If \(\{\varphi_{\lambda}\}_{\lambda\in{\mathbb R}}\) is a parametrized family of flows in a locally compact ANR \(M\), and if a connected isolated regular non-saddle set \(K_0\) for \(\varphi_0\) is regularly dynamically robu…
reviewed by: Takahisa Miyata
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 55P55
Mathematics Subject Classification ID: 37B25
zbMATH DE Number: 5573804
zbMATH Keywords: shape theory
zbMATH Keywords: shape equivalence
zbMATH Keywords: dyamical system
zbMATH Keywords: isolated invariant set
zbMATH Keywords: continuation of non-saddle sets
author: Antonio Giraldo
author: José Sanjurjo
MaRDI profile type: Publication
full work available at URL: j.topol.2009.03.020
OpenAlex ID: W2057529256
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DOI: 10.1016/J.TOPOL.2009.03.020
description: scientific article; zbMATH DE number 5573804