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arXiv · 2007.07735

Stretching and Rotation of Planar Quasiconformal Mappings on a Line

Olli Hirviniemi·István Prause·Eero Saksman
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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 $ \overline{B}(1/(1-k^4),k^2/(1-k^4))$. 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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Olli Hirviniemi, István Prause, Eero Saksman. 2020-07-15. Stretching and Rotation of Planar Quasiconformal Mappings on a Line. https://arxiv.org/abs/2007.07735

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