Topological stability in set-valued dynamics
type: Item
instance of: scholarly article
title: Topological stability in set-valued dynamics
zbMATH Open document ID: 1367.54022
author: Carlos Arnoldo Morales Rojas
author: Phillipe Thieullen
author: Roger Metzger Alvan
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2017-04-06T00:00:00Z
review text: A definition of topological stability for set-valued maps is proposed. Then, the following basic theorems are stated and proved. Theorem 1. Every continuous single-valued map of a metric space, which is topologically stable as a set-valued map, is also topologically stable in the classical sense. Theorem 2. Every topologically stable single-valued map \(f\) of a compact metric space \(X\) satisfies the property that for each \(\varepsilon > 0\) there exists \(\delta> 0\) such that: for every continuous single-valued map \(g:X\to X\) with \(d_H(f,g)< \delta\), there is a continuous map \(h:(\lim_{\leftarrow}g,d^*)\to (\lim_{\leftarrow}f,d^*)\) such that \(D_H(h,\text{Id}_X)< \varepsilon\), \(\sigma_f\circ h=h\circ \sigma_g\). Theorem 3. Every upper semicontinuous positively expansive closed-valued map with the POTP\(_+\) of a compact metric space is topologically stable. Theorem 4. Every topologically stable upper semicontinuous closed-valued map of a compact metric space has the POTP\(_+\); moreover, \(\text{Per}(f)\) is dense in \(\Omega(f)\). All these results are shown to extend the classical single-valued ones.
reviewed by: Mihai Turinici
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 6701836
zbMATH Keywords: topological stability
zbMATH Keywords: set-valued map
zbMATH Keywords: upper semicontinuous positively expansive map
MaRDI profile type: Publication
full work available at URL: dcdsb.2017115
OpenAlex ID: W2600779251
cites work: Q4297122
cites work: \(\omega\)-limit sets for axiom A diffeomorphisms
cites work: Topological entropy for set-valued maps
cites work: Continuum-wise expansiveness and specification for set-valued functions and topological entropy
cites work: Expansive transformation semigroups of endomorphisms
cites work: Topological entropy on set-valued functions
cites work: Shadowing and inverse shadowing in set-valued dynamical systems. Contractive case
cites work: Shadowing in dynamical systems
cites work: The specification property on a set-valued map and its inverse limit
cites work: Unstable Homeomorphisms
cites work: Q4188476
cites work: Anosov diffeomorphisms are topologically stable
cites work: A Note on Expansive Mappings
DOI: 10.3934/DCDSB.2017115
Recommended article: Q3791876
Recommended article: Abstract topological dynamics involving set-valued functions
Recommended article: Topological properties of asymptotically stable sets
Recommended article: Stability concepts in topological dynamics by nonstandard methods
Recommended article: Topological stability for differential equations
Recommended article: Topological bifurcations of minimal invariant sets for set-valued dynamical systems
Recommended article: Topologically transitive and mixing properties of set-valued dynamical systems
Recommended article: Q3022622
Recommended article: The topological stability of diffeomorphisms
Recommended article: Topological stability of chain recurrent sets
description: scientific article; zbMATH DE number 6701836