Dimension of elliptic harmonic measure of Snowspheres

TL;DR

This paper investigates the elliptic harmonic measure on certain self-similar quasispheres called snowspheres, showing that its dimension can be strictly less than the Hausdorff dimension of the space.

Contribution

It demonstrates that for specific self-similar quasispheres, the elliptic harmonic measure's dimension is strictly less than the space's Hausdorff dimension, revealing new geometric measure properties.

Findings

Elliptic harmonic measure dimension is less than Hausdorff dimension on snowspheres.

Self-similar quasispheres exhibit unique measure-theoretic properties.

The study links quasisymmetric maps with measure dimension discrepancies.

Abstract

A metric space $\mathcal{S}$ is called a \defn{quasisphere} if there is a quasisymmetric homeomorphism $f\colon S^2\to \mathcal{S}$. We consider the elliptic harmonic measure, i.e., the push forward of 2-dimensional Lebesgue measure by $f$. It is shown that for certain self similar quasispheres $\mathcal{S}$ (snowspheres) the dimension of the elliptic harmonic measure is strictly less than the Hausdorff dimension of $\mathcal{S}$.