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.

我们首次研究了准正则映射的最大模数集。不难看出，这些集合必须是封闭的，并且每个模数至少包含一个点。Blumenthal表明，对于整个地图，这些集合要么是整个平面，要么是可数的分析曲线的并集。我们证明，在准规则的情况下，作为对比，包含每个模数至少一个点的任何封闭集都可以作为准规则映射的最大模数集获得。这些例子都是多项式的。我们还表明，在其他约束条件下，此类集合甚至可以通过先验类型的准规则映射来获得。