Cauchy transforms of self-similar measures: Starlikeness and univalence
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AbstractFor the contractive iterated function system Skz = e2πik/m + ρ(z − e2πik/m) with 0 <ρ< 1, k = 0, ··· , m − 1, we let K ⊂ C be the attractor, and let μ be a self-similar measure defined by μ = 1/m m−1 k=0 μ◦S−1k .
We consider the Cauchy transform F of μ. It is known that the image of F at a small neighborhood of the boundary of K has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate
the behavior of F away from K; it has nice geometry and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures.
All Author(s) ListXIN-HAN DONG, KA-SING LAU, HAI-HUA WU
Journal nameTransactions of the American Mathematical Society
Year2017
Month7
Volume Number369
Issue Number7
Pages4817 - 4842
ISSN0002-9947
eISSN1088-6850
LanguagesEnglish-United States

Last updated on 2021-18-01 at 00:57