On singularities of mappings with a majorant integrable by spheres

Abstract

The paper is devoted to the study of the boundary behavior of
mappings with finite distortion, more precisely, open discrete
mappings with moduli conditions similar to Poletsky inequality in
the inverse direction. We study the case when some majorant in the
above inequality is integrable over spheres centered at each finite
point. It is established that the indicated mappings have a
continuous extension to an isolated boundary point of the boundary
of a domain whenever the mapping $f$ omits at least one a point. The
proof of the main results is step-by-step and is based on the
following logic, in line with which we prove that 1) no two
different boundary points of the corresponding mapped domain can
belong to the limit set of the mapping $f$ at the point $x_0,$ 2)
the cluster set of a mapping at a given point is, in principle,
always a singleton, provided that this mapping omits at least one
finite point. The proofs of the these statements are made by the
contradiction. This contradiction, in turn, is ensured by the
property of approaching continua in the preimage under the mapping
on the one hand, and by the upper bound on the mapped families of
paths (taking into account the definition of the mapping through
this upper bound) on the other hand. In order to construct the above
approaching continua, we use some geometric constructions that take
into account the nature (definition) of the class of mappings under
study. In particular, to prove that the cluster set under the
mapping does not contain more than one boundary point, we use
segments joining elements of some approximate sequences with its
limit points, and the corresponding converging continua which are
the pre-images of these segments under the mapping. Note that, the
result obtained in the article was previously obtained by us first
for homeomorphisms, then for open discrete closed mappings, and then
for open discrete mappings but with an integrable majorant in the
inverse Poletsky inequality. We also previously established a
similar result, but in the case where the mapping omits at least two
points.