Mappings of finite distortion: Size of the branch set

Abstract

We study the branch set of a mapping between subsets of {\mathbb{R}^{n}} , i.e., the set where a given mapping is not defining a local homeomorphism. We construct several sharp examples showing that the branch set or its image can have positive measure.

A self-contained proof to Martio's conjecture in the class of BLD-maps

“…We close the article with further observations on rigidity and Martio's conjectures. We start by pointing out that in the literature several constructions on quasiregular mappings [BH04, GV73, KTW05], on mappings of bounded length distortion [MV88], and on mappings of finite distortion [GHT20] have the conjugated form h • w m • g , (6.1) where w m stands for the standard m-to-1 winding map, and both h and g are some suitable homeomorphisms. In addition to this, by the results of Church and Hemmingsen [CH60, Theorem 4.1], Martio, Rickman, and Väisälä [MRV71, Lemma 3.20], and Luisto and Prywes [LP21, Theorem 1.1] quasiregular maps with reasonably regular branch sets or branch set images are indeed topologically equivalent to the standard winding map.…”