Stability in the family of \(\omega \)-limit sets of alternating systems
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 \((x,g)\in\mathcal D\).
scientific article; zbMATH DE number 6015277
scientific article