Towards a characterization of \(\omega\)-limit sets for Lipschitz functions
It is known that any nowhere dense compact subset \(W\) of the interval \(I=[0,1]\) is an \(\omega\)-limit set of a suitable continuous map \(f\) of this interval [cf. \textit{A. M. Bruckner} and \textit{J. Smítal}, Math. Bohem. 117, No. 1, 42-47 (1992; Zbl 0762.26003)]. The author characterizes such sets \(W\) for which the corresponding map \(f\) can be find in a Lipschitz class. Similarly, using a characterization of nowhere dense compact sets \(W\subset I\), which are \(\omega\)-limit sets of continuous mappings of zero topological entropy [cf. \textit{A. M. Bruckner} and \textit{J. Smítal}, Ergodic Theory Dyn. Syst. 13, No. 1, 7-19 (1993; Zbl 0788.58021)], the author gives a sufficient condition for \(W\) to be an \(\omega\)-limit set of a Lipschitz function. The corresponding conditions are too complicated to be displayed here.
scientific article; zbMATH DE number 1095889