On the sets of branch points of mappings more general than quasiregular

Abstract

It is shown that if a point $x_0 ∊ ℝ^n, \; n ≥ 3$, is an essential isolated singularity of an open discrete $Q$-mapping $f : D → \overline{ℝ^n}, B_f$ is the set of branch points of $f$ in $D$; and a point $z_0 ∊ \overline{ℝ^n}$ is an asymptotic limit of $f$ at the point $x_0$; then, for any neighborhood $U$ containing the point $x_0$; the point $z_0 ∊ \overline{f(B_f ∩ U)}$ provided that the function $Q$ has either a finite mean oscillation at the point $x_0$ or a logarithmic singularity whose order does not exceed $n − 1$: Moreover, for $n ≥ 2$; under the indicated conditions imposed on the function $Q$; every point of the set $\overline{ℝ^n}\ f(D)$ is an asymptotic limit of $f$ at the point $x_0$. For $n ≥ 3$, the following relation is true: $\overline{ℝ^n}∖f(D) ⊂\overline{f(B_f ∩ U)}$. In addition, if $∞ ∉ f(D)$, then the set $f B_f$ is infinite and $x_0 ∈ \overline{B_f}$.