Hausdorff spectrum of harmonic measure
Probability
2015-07-29 v2 Dynamical Systems
Abstract
For every non-elementary hyperbolic group, we show that for every random walk with finitely supported admissible step distribution, the associated entropy equals 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 finitary versions of dimensional properties of the harmonic measure.
Keywords
Cite
@article{arxiv.1411.2312,
title = {Hausdorff spectrum of harmonic measure},
author = {Ryokichi Tanaka},
journal= {arXiv preprint arXiv:1411.2312},
year = {2015}
}