Linear distortion and rescaling for quasiregular values

Abstract

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 $f:\hat{I}\text{c◯}\to {\mathbb{R}}^n$ with a $(K,\hat{I}\pounds )$-quasiregular value at $f(x_0)$ can be rescaled at $x_0$ 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.

29 pages, 3 figures. v2 fixes a color rendering issue in the included TikZ-generated figures, and makes several technical changes to fix issues in the HTML version