Abstract:
We prove that sets of zero modulus with weight $Q$ (in particular,
isolated singularities) are removable for discrete open $Q$-maps
$f\colon D\to\overline{\mathbb R}{}^n$ if the function $Q(x)$ has
finite mean oscillation or a logarithmic singularity of order not
exceeding $n-1$ 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 $Q$-map takes any value
infinitely many times, except possibly for a set of values of zero
capacity.
Keywords:maps with bounded distortion and their generalizations, discrete open maps, removing singularities of maps, essential singularities, Picard's theorem, Sokhotskii's theorem, Liouville's theorem.
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