Cauchy transforms of self-similar measures

Abstract

The Cauchy transform of a measure in the plane, $$ F(z) = \frac{1}{2\pi i}\int_{\C} \frac{1}{z-w} \,d\mu(w)\hbox{,} $$ is a useful tool for numerical studies of the measure, since the measure of any reasonable set may be obtained as the line integral of $F$ around the boundary. We give an effective algorithm for computing $F$ when $\mu$ is a self-similar measure, based on a Laurent expansion of $F$ for large $z$ and a transformation law (Theorem 2.2) for $F$ that encodes the self-similarity of $\mu$. Using this algorithm we compute $F$ for the normalized Hausdorff measure on the Sierpiński gasket. Based on this experimental evidence, we formulate three conjectures concerning the mapping properties of $F$, which is a continuous function holomorphic on each component of the complement of the gasket.