Stability in the family of \(\omega \)-limit sets of alternating systems
type: Item
instance of: scholarly article
title: Stability in the family of \(\omega \)-limit sets of alternating systems
zbMATH Open document ID: 1246.37060
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-03-16T00:00:00Z
review text: Let \(C(I)\) be the space of all continuous maps from the unit interval \(I=[0,1]\) into itself endowed with the uniform topology. For \(f,g\in C(I)\) and \(x\in I\) the paper studies the \(\omega\)-limit sets \(\omega(x,[f,g])\) generated by alternating trajectories of the form \(\gamma(x,[f,g])=\{x,f(x),g(f(x)),f(g(f(x))),\dots\}\), as well as the sets \(\Lambda([f,g])=\cup_{x\in I}\omega(x,[f,g])\) and \(\mathcal L([f,g])=\{\omega(x,[f,g]):x\in I\}\). In particular, it is shown that (1) If \(g\) is constant on no interval \(J\subseteq I\), then there exists a residual set \(\mathcal S\subseteq C(I)\) so that the maps \(\Lambda:C(I)\times C(I)\to \mathcal K\) and \(\mathcal L:C(I)\times C(I)\to \mathcal K^\star\) taking \((f,g)\) to \(\Lambda([f,g])\) and \(\mathcal L([f,g])\), respectively, are both continuous at \((f,g)\) whenever \(f\in\mathcal S\). (2) The map \(\omega:I\times C(I)\times C(I)\to\mathcal K\) taking \((x,f,g)\) to \(\omega(x,[f,g])\) is in the second class of Baire, and for any \(g\in C(I)\) there exists a residual set \(\mathcal T\subseteq I\times C(I)\) so that \(\omega\) is continuous at \((x,f,g)\) whenever \(f\in\mathcal S\). (3) If \(f\) is constant on no interval \(J\subseteq I\), then there exists a residual set \(\mathcal D\subseteq I\times C(I)\) so that \(\omega(x,[f,g])=\omega(x,g\circ f)\cup\omega(f(x),f\circ g)\), where both \(\omega(x,g\circ f)\) and \(\omega(f(x),f\circ g)\) are adding machines of type \(\infty\) whenever \((x,g)\in\mathcal D\).
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 37B20
zbMATH DE Number: 6015277
zbMATH Keywords: alternating system
zbMATH Keywords: \(\omega\)-limit set
zbMATH Keywords: typical behavior
author: Emma D'Aniello
author: Timothy H. Steele
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.12.056
OpenAlex ID: W2058960324
cites work: On the periodic logistic equation
cites work: An extension of Sharkovsky's theorem to periodic difference equations
cites work: Dynamics in one dimension
cites work: A characterization of adding machine maps
cites work: The space of $\omega $-limit sets of a continuous map of the interval
cites work: Stability in the family of \(\omega \)-limit sets of continuous maps of the interval
cites work: Q4331836
cites work: Chaos in terms of the map \(x\to\omega(x,f)\)
cites work: The structure of $\omega$-limit sets for continuous maps of the interval
cites work: Periodic structure of alternating continuous interval maps
cites work: Global Dynamics of Some Periodically Forced, Monotone Difference Equations
cites work: The space of adding machines generated by continuous self maps of manifolds
cites work: Ubiquity of odometers in topological dynamical systems
cites work: Pitchfork bifurcation for non-autonomous interval maps
cites work: The ω-limit sets of alternating systems
cites work: Approximating \(\omega\)-limit sets with periodic orbits
cites work: Q5700950
cites work: Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures
cites work: Dynamics of Typical Continuous Functions
cites work: Stability of dynamical structures under perturbation of the generating function
cites work: The essential point set of a continuous function.
cites work: Continuity structure of \(f\mapsto \cup _{x\in I}\omega (x,f)\) and \(f\mapsto \{\omega (x,f) : x\in I\}\)
cites work: Q3370119
cites work: Continuity and chaos in discrete dynamical systems
DOI: 10.1016/J.JMAA.2011.12.056
description: scientific article; zbMATH DE number 6015277