On boundary discreteness of mappings with a modulus condition (Q6061158)

For a domain \(G \subset \mathbb{R}^n, n\ge 2,\) consider the class \(F(G)\) of discrete, open and continuous mappings \(f:G \to \mathbb{R}^n.\) The study of the boundary behavior of mappings of the class \(F(G)\) includes, in particular, questions of the following types: (1) Under which conditions a mapping of class \(F(G)\) admits a continuous extension to the closure of \(G.\) (2) If a continuous extension exists, under which conditions the extension is discrete.\par Caratheodory's classical work proves results of this type for conformal mappings of plane domains and these results have been extended to the case of quasiconformal homeomorphisms and closed quasiregular mappings in \(\mathbb{R}^n\).\par The author extends these results to a wider subclass of \(F(G)\) than quasiregular mappings, to mappings satisfying a technical condition, so called inverse Poletsky inequality, which ensures that the same proof methods as those in case of quasiregular mappings, still work. The modulus of a curve family is the key tool. Working out the technical details, the author has found a way to proceed to new territories beyond the known cases.\par One can observe that for the proofs suitable assumptions about the topological and analytic properties of the domains \(G\) and \(f(G)\) have a crucial role. These properties include local connectedness at a boundary point of a domain and the notion of a uniform domain, expressed in terms of moduli of curve families.