Experimental Mathematics

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, $$ F(z) = \frac{1}{2\pi i}\int_{\C} \frac{1}{z-w} \,d\mu(w)\hbox{,} $$ 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.

Article information

Source
Experiment. Math., Volume 7, Issue 3 (1998), 177-190.

Dates
First available in Project Euclid: 14 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1047674203

Mathematical Reviews number (MathSciNet)
MR1676691

Zentralblatt MATH identifier
0959.28006

Subjects
Primary: 28A80: Fractals [See also 37Fxx]
Secondary: 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx]

Citation

Lund, John-Peter; Strichartz, Robert S.; Vinson, Jade P. Cauchy transforms of self-similar measures. Experiment. Math. 7 (1998), no. 3, 177--190. https://projecteuclid.org/euclid.em/1047674203


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