The Lipschitz structure of continuous self-maps of generic compact sets
The main result: In the space of compact nowhere dense subsets of the interval, equipped with the Hausdorff metric, there is a residual subset \({\mathcal K}^*\) with the following properties: Let \(E\in{\mathcal K}^*\), and let a map \(f\) be continuous on \(E\), not the identity on any portion of \(E\), and such that \(f(E)= E\). Then there is a Cantor set \(K_ \mu\subset E\) such that \(f| K_ \mu\) is not Lipschitz on any portion of \(K_ \mu\). Furthermore, \(f(K_ \mu)= E\) and if \(P= (a,b)\cap E\) is a portion of \(E\) contiguous to \(K_ \mu\), then \(f(P)\) is nowhere dense in \(E\). Finally, if \(f\) is Lipschitz on a subset \(C\) of \(E\), then \(f(C)\) is nowhere dense in \(E\). This result has a number of applications, like the following one: If \(E\in{\mathcal K}^*\) is an \(\omega\)-limit set for a continuous map \(f\) of the interval then, among others, \(f\) is not a Lipschitz function, and the set of points of \(E\) at which \(f\) is not differentiable is mapped onto a residual subset of \(E\).
scientific article; zbMATH DE number 720402