CAUCHY TRANSFORMS OF SELF-SIMILAR MEASURES

JOHN-PETER LUND, ROBERT S. STRICHARTZ, AND JADE P. VINSON

Abstract. The Cauchy transform of a measure in the plane,

F (z) = \frac{1}{2πi} ∫_{ℂ} \frac{1}{z − w} dμ(w),

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 μ 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 μ. 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.