Dimension of elliptic harmonic measure of snowspheres

Abstract

A metric space $\mathcal{S}$ is called a quasisphere if there is a quasisymmetric homeomorphism $f : S^2\to\mathcal{S}$. 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 $\mathcal{S}$. This result is obtained by representing the self similarity of a snowsphere by a postcritically finite rational map, and showing a corresponding result for such maps. As a corollary a metric characterization of Lattès maps is obtained. Furthermore, a method to compute the dimension of elliptic harmonic measure numerically is presented, along with the (numerically computed) values for certain examples.