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Harmonic measure versus Hausdorff measures on repellers for holomorphic maps


Author: Anna Zdunik
Journal: Trans. Amer. Math. Soc. 326 (1991), 633-652
MSC: Primary 58F11; Secondary 30D05, 31A15
MathSciNet review: 1031980
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Abstract: This paper is a continuation of a joint paper of the author with F. Przytycki and M. Urbański. We study a harmonic measure on a boundary of so-called repelling boundary domain; an important example is a basin of a sink for a rational map. Using the results of the above-mentioned paper we prove that either the boundary of the domain is an analytically embedded circle or interval, or else the harmonic measure is singular with respect to the Hausdorff measure corresponding to the function $ {\phi _c}(t)= t\;\exp \,\left(c\sqrt {\log \frac{1} {t}\,\log \,\log \,\log } \frac{1} {t}\right)$ for some $ c > 0$.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1031980-0
Article copyright: © Copyright 1991 American Mathematical Society