Removal of singularities and analogs of the Sokhotskii–Weierstrass theorem for Q-mappings

Abstract

We prove that an open discrete Q-mapping $f:D \to \overline {{\mathbb{R}^n}}$ has a continuous extension to an isolated boundary point if the function Q(x) has finite mean oscillation or logarithmic singularities of order at most n – 1 at this point. Moreover, the extended mapping is open and discrete and is a Q-mapping. As a corollary, we obtain an analog of the well-known Sokhotskii–Weierstrass theorem on Q-mappings. In particular, we prove that an open discrete Q-mapping takes any value infinitely many times in the neighborhood of an essential singularity, except, possibly, for a certain set of capacity zero.