The maximum modulus set of a quasiregular map

TL;DR

This paper investigates the structure of maximum modulus sets of quasiregular maps, demonstrating their flexibility in realization and contrasting with the more rigid case of entire maps.

Contribution

It shows that any closed set containing points of all moduli can be realized as a maximum modulus set of a quasiregular map, including transcendental types.

Findings

Any closed set with points of each modulus can be a maximum modulus set.

Such sets can be realized by polynomial-type quasiregular maps.

Under additional constraints, transcendental quasiregular maps can also have prescribed maximum modulus sets.

Abstract

We study, for the first time, the maximum modulus set of a quasiregular map. It is easy to see that these sets are necessarily closed, and contain at least one point of each modulus. Blumenthal showed that for entire maps these sets are either the whole plane, or a countable union of analytic curves. We show that in the quasiregular case, by way of contrast, any closed set containing at least one point of each modulus can be attained as the maximum modulus set of a quasiregular map. These examples are all of polynomial type. We also show that, subject to an additional constraint, such sets can even be attained by quasiregular maps of transcendental type.