Lipschitz regularity in some geometric problems
The starting fact for the recent paper is the following: Let \(U\) be an open subset of \(\mathbb{R}^2\). Let \(\mathcal{F}\) be a family of disjoint (straight) line segments which are connected components of intersections \(l\cap U\) with \(l\) being a stright line. Then \(\mathcal{F}\) is locally Lipschitz, that is the map \(u:S\to S^1\) defined in the support \(S\subset U\) of \(\mathcal{F}\) which associates to a line segment its unit tangent vector is locally Lipschitz. Moreover, the Lipschitz constant (which is a function on \(U\)) depends only on \(U\) and not on \(\mathcal{F}\). The aim of the article is to analyse to what extent the above fact generalizes to codimension one foliations by 'rigid' leaves. In particular, the author answers the followig question: What is the required regularity for a homeomorphism of \(\mathbb{R}\) to be a holonomy map of a foliation of an open set of \(\mathbb{R}^2\) by graphs of polynomials with bounded degree.
scientific article; zbMATH DE number 2145824