Stretching and Rotation of Planar Quasiconformal Mappings on a Line

TL;DR

This paper investigates the stretching and rotation behavior of planar quasiconformal mappings on a line, providing improved bounds on complex stretching exponents and insights into the dimension of images of line subsets.

Contribution

It introduces a quadratic improvement over existing estimates for the set of complex stretching exponents and establishes a lower bound for the dimension of images of line subsets under quasiconformal maps.

Findings

Set of complex stretching exponents is contained in a specific disk for almost every point.

Provides a quadratic improvement over known estimates for Hausdorff dimension 1.

Establishes a lower bound for the dimension of images of line subsets.

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.