Harmonic measure of some Cantor type sets. (English) Zbl 0849.31005

Batakis, Athanassios

Ann. Acad. Sci. Fenn., Math. 21, No. 2, 255-270 (1996).

We show that for a class of Cantor-type sets (not necessarily self-similar ones), the Hausdorff dimension of the harmonic measure is strictly smaller than the dimension of the set (the case of self-similar Cantor sets was studied by Carleson, Makarov and Volberg). We also provide some examples of Cantor sets without this property. The method does not involve ergodic or probabilistic tools.

Reviewer: A.Batakis (Paris)

MSC:

31A15	Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
28A80	Fractals

Keywords:

harmonic measure; Hausdorff measure; Hausdorff dimension; fractals