Branched folded maps and alternating Beltrami equations (Q1357382)

The paper under review deals with the Beltrami equation \(w_{\overline z}(z)=\mu(z)w_z(z)\) in a plane domain \(D\) of \(C\). In the classical case \(|\mu|_\infty<1\) an existence and representation theorem is well-known. The authors consider the so-called relaxed classical case \(|\mu|<1\) a.e., \(|\mu|_\infty=1\) and the so-called alternating case \(|\mu|<1\) in parts of \(D\) and \(|\mu|>1\) in other parts of \(D\). In the relaxed classical case sufficient conditions for existence and uniqueness of \(\mu\)-homeomorphisms are given. In order to study the alternating case the authors use, the so-called branched folded maps (BF-maps) (see for instance: \textit{V. I. Arnol'd} [Singularities of caustics and wave fronts (1990; Zbl 0734.53001)]). The local behavior of a BF-map in isolated critical points is considered. For the alternating case a kind of counter part of the classical uniqueness and representation theorem is proved.