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The persistence of \(\omega\)-limit sets under perturbation of the generating function. (English) Zbl 1056.26019
Summary: We consider the set-valued function \(\Omega\) taking \(f\) in \(C(I,I)\) to its collection of \(\omega\)-limit sets \(\Omega(f)= \{\omega(x,f):x\in I\}\), and consider how \(\Omega(f)\) is affected by perturbations of \(f\). Our main result characterizes those functions \(f\) in \(C(I,I)\) at which \(\Omega\) is upper semicontinuous, so that whenever \(g\) is sufficiently close to \(f\), every \(\omega\)-limit set of \(g\) is close to some \(\omega\)-limit set of \(f\) in the Hausdorff metric space. We also develop necessary and sufficient conditions for a function \(f\) in \(C(I,I)\) to be a point of lower semicontinuity of the map \(\Omega\).

26E25 Set-valued functions
54C60 Set-valued maps in general topology