Illinois Journal of Mathematics

Dimension of elliptic harmonic measure of snowspheres

Daniel Meyer

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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.

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Illinois J. Math., Volume 53, Number 2 (2009), 691-721.

First available in Project Euclid: 23 February 2010

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Zentralblatt MATH identifier

Primary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations
Secondary: 37A05: Measure-preserving transformations 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]


Meyer, Daniel. Dimension of elliptic harmonic measure of snowspheres. Illinois J. Math. 53 (2009), no. 2, 691--721. doi:10.1215/ijm/1266934799.

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