Mapping problems for quasiregular mappings

TL;DR

This paper investigates the global behavior of certain non-homeomorphic quasiregular mappings, extending classical results from quasiconformal mappings to a broader class of mappings with similar properties.

Contribution

It introduces and analyzes topologically closed quasiregular mappings, expanding understanding beyond the well-studied quasiconformal case.

Findings

Closed quasiregular mappings share many properties with quasiconformal mappings.

The global behavior of these mappings resembles the local behavior within normal domains.

The study extends classical results to non-homeomorphic quasiregular mappings.

Abstract

We study images of the unit ball under certain special classes of quasiregular mappings. For homeomorphic, i.e., quasiconformal mappings problems of this type have been studied extensively in the literature. In this paper we also consider non-homeomorphic quasiregular mappings. In particular, we study (topologically) closed quasiregular mappings originating from the work of J. V\"ais\"al\"a and M. Vuorinen in 1970's. Such mappings need not be one-to-one but they still share many properties of quasiconformal mappings. The global behavior of closed quasiregular mappings is similar to the local behavior of quasiregular mappings restricted to a so-called normal domain.