Linear distortion and rescaling for quasiregular values

Ilmari Kangasniemi; Jani Onninen

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arXiv · 2404.02073

Linear distortion and rescaling for quasiregular values

Ilmari Kangasniemi·Jani Onninen

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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 $f \colon Ω\to \mathbb{R}^n$ with a $(K, Σ)$-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.

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Ilmari Kangasniemi, Jani Onninen. 2024-05-02. Linear distortion and rescaling for quasiregular values. https://arxiv.org/abs/2404.02073

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