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The space of \(\omega\)-limit sets of a continuous map of the interval. (English) Zbl 0860.54036
Let \(\omega_f\) denote the class of all \(\omega\)-limit sets of a continuous self-mapping of a real compact interval \(I\). If \(W\in \omega_f\), then \(W\) is a closed nonempty subset of \(I\) and \(f(W)=W\). Consider \(\omega_f\) as a subspace of the compact metric space \(K\) of all closed nonempty subsets of \(I\) furnished with the Hausdorff metric. It is shown that \(\omega_f\) is a closed and therefore compact subspace of \(K\). Results are then applied to other dynamical systems.

MSC:
54H20 Topological dynamics (MSC2010)
26A18 Iteration of real functions in one variable
37E99 Low-dimensional dynamical systems
37B99 Topological dynamics
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