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.
Citation
Daniel Meyer. "Dimension of elliptic harmonic measure of snowspheres." Illinois J. Math. 53 (2) 691 - 721, Summer 2009. https://doi.org/10.1215/ijm/1266934799
Information