Stretching and Rotation of Planar Quasiconformal Mappings on a Line

2020/07/15 by Olli Hirviniemi, Hirviniemi, Olli, István Prause +3
Engineering · Mathematics · #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2007.07735

openalex publication_date 2020/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk B(1/(1-k⁴),k²/(1-k⁴)). This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension 1. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a 1-dimensional subset of a line under a quasiconformal mapping.

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• Existence of quasiconformal maps with maximal stretching on any given countable set
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• Sharp Pointwise Contraction of Mappings with Integrable Distortion Which Are Quasiconformal in a Disk