Hausdorff spectrum of harmonic measure

Ryokichi Tanaka

Published 2014-11-10Version 1

For every non-elementary hyperbolic group, we show that for every finite range admissible random walk, the associated entropy equals to the drift times the logarithmic volume growth if and only if the corresponding harmonic measure is comparable with Hausdorfff measure on the boundary. Moreover, we introduce one parameter family of probability measures which interpolates a Patterson-Sullivan measure and the harmonic measure, and establish a formula of Hausdorff spectrum (multifractal spectrum) of the harmonic measure. We also give some finitistic versions of dimensional properties of the harmonic measure.