On the openness and discreteness of mappings with unbounded characteristic of quasiconformality

Abstract

The paper is devoted to the investigation of topological properties of space mappings. It is shown that orientation-preserving mappings f W D ! R n in a domain D R n ; n 2; which are more general than mappings with bounded distortion, are open and discrete if a function Q corresponding to the control of the distortion of families of curves under these mappings has slow growth in the domain f .D/; e.g., if Q has finite mean oscillation at an arbitrary point y 0 2 f .D/:

On some properties of generalized quasiisometries with unbounded characteristic

“…For p D n; properties of mappings of the form (3) such as the removal of isolated singularities and the normality of families were studied by the author (see, e.g., [14,15]). For this reason, in the present paper, we are interested in inequality (3) mainly for unbounded Q and for p < n: Also note that the case p > n is associated with some degeneration, which is not considered here in detail.…”

“…As in the case of removal of singularities of mappings with unbounded characteristic for p D n (see, e.g., [14,15]), Statement 2 yields several corollaries, which are presented below. Since they are proved by analogy with the case p D n; we omit their proof in most cases.…”

On the order of growth of ring Q-homeomorphisms at infinity

“…In this case, the result formulated above can be generalized to a mapping f defined in domains G n ⊂ R containing ∞ (see, e.g., [26]). …”