On the Harmonic Measure of Self-Similar Sets on the Plane

A. L. Volberg
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A. L. Volberg: University of Kentucky, Department of Mathematics

A chapter in Harmonic Analysis and Discrete Potential Theory, 1992, pp 267-280 from Springer

Abstract: Abstract Harmonic measure is one of the basic objects of one dimensional complex analysis. Recently the structure of harmonic measure of rather general plane sets became much more comprehensible due to works of Makarov [1], Carleson [2] and Jones, Wolff [3]. The deep analogy between the behaviour of sums of (almost) independent random variables and the behaviour of the Green function of a domain plays a crucial role in this subject. We refer the reader to [15] for more details. This analogy becomes still more conspicuous if the domain for which the harmonic measure is investigated has regular self-similar structure. The methods of ergodic theory turn out to be relevant in this case, see e.g. [2], [4], [5], [6].

Keywords: Green Function; Independent Random Variable; Hausdorff Dimension; Gibbs Measure; Harmonic Measure (search for similar items in EconPapers)
Date: 1992
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DOI: 10.1007/978-1-4899-2323-3_22

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