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Cauchy transforms of self-similar measures. (English) Zbl 0959.28006

Summary: The Cauchy transform of a measure in the plane, \(F(z)= {1\over 2\pi i} \int_{\mathbb{C}}{1\over z-w} d\mu(w)\), 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 \(\mu\) 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.

MSC:

28A80 Fractals
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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