The Hausdorff dimension of planar elliptic measures via quasiconformal mappings

TL;DR

This paper establishes new bounds for the Hausdorff dimension of planar elliptic measures using quasiconformal mappings, linking elliptic and harmonic measures to extend prior results.

Contribution

It introduces bounds depending only on ellipticity and relates elliptic measures to harmonic measures via quasiconformal mappings, extending previous research.

Findings

Bounds depend solely on ellipticity constant

Relates elliptic measure to harmonic measure through quasiconformal maps

Extends results of Makarov, Jones, and Wolff

Abstract

In this paper, we obtain new bounds for the Hausdorff dimension of planar elliptic measure via the application of quasiconformal mappings, with these bounds depending solely on the ellipticity constant of the matrix. In fact, in our case studies, we find a quasiconformal mapping that relates the elliptic measure in a domain to the harmonic measure in its image domain, allowing us to deduce bounds for the dimension of the elliptic measure from the known results on the harmonic side. This extends previous works of Makarov, Jones and Wolff.