Towards a theory of removable singularities for maps with unbounded characteristic of quasi-conformity

Abstract

We prove that sets of zero modulus with weight (in particular, isolated singularities) are removable for discrete open -maps if the function has finite mean oscillation or a logarithmic singularity of order not exceeding on the corresponding set. We obtain analogues of the well-known Sokhotskii-Weierstrass theorem and also of Picard's theorem. In particular, we show that in the neighbourhood of an essential singularity, every discrete open -map takes any value infinitely many times, except possibly for a set of values of zero capacity.