Linear distortion and rescaling for quasiregular values

Sobolev mappings exhibiting only pointwise quasiregularity-type bounds have arisen in various applications, leading to a recently developed theory of quasiregular values. In this article, we show that by using rescaling, one obtains a direct bridge between this theory and the classical theory of quasiregular maps. More precisely, we prove that a non-constant mapping f Ω → Rⁿ with a (K, Σ)-quasiregular value at f(x₀) can be rescaled at x₀ to a non-constant K-quasiregular mapping. Our proof of this fact involves establishing a quasiregular values -version of the linear distortion bound of quasiregular mappings. A quasiregular values variant of the small K -theorem is obtained as an immediate corollary of our main result.