Harmonic Measure on Fractal Sets

Abstract

Many problems in complex analysis can be reduced to the evaluation of the universal spectrum: the supremum of multifractal spectra of harmonic measures for all planar domains. Its exact value is still unknown, with very few estimates available. We start with a brief survey of related problems and available estimates from above. Then we discuss in more detail estimates from below, describing the search for a fractal domain which attains the maximal possible spectrum.