On boundary properties of spatial non-homeomorphic mappings with an s-averaged characteristic
In this paper, we continue to develop the geometric method of modules of curve families for studying analytical and geometrical properties of nonhomeomorphic mappings with s-averaged characteristic. We consider the question of the erasure of special sets under mappings with s-averaged characteristic. In this work, in contrast to previous results which require that the mapping is homeomorphic or the capacity of singular points is zero, nonhomeomorphic mappings with s-averaged characteristic are considered and a weaker condition is taken as constraints. We generalize the theorem which is known in the case n = 2 as Iversen-Tsuji's theorem for the case n > 3. There are well-known examples demonstrating the existence of essential singularities for which Hausdorffs measure A<sub>B</sub> * 0 at some p * 0 for mappings with an s-averaged characteristic. The work presents some examples which illustrate distinctive properties of the considered class of mappings. A theorem about the module distortion for families of curves under mappings with allowance for multiplicity and, as a consequence, the characteristic property of the spherical module of families of curves asymptotic to a special boundary set is proved. The mappings are extended to continuous ones if the dimension of the set of singular points I dim I < n-2 and s > 1. The results are applicable to many classes of mappings of subclasses W„<sup>1</sup>(U).