Hausdorff spectrum of harmonic measure

TL;DR

This paper investigates the multifractal spectrum of harmonic measures on hyperbolic groups, establishing conditions for measure equivalence and introducing a parameter family interpolating between Patterson-Sullivan and harmonic measures.

Contribution

It provides a new formula for the Hausdorff spectrum of harmonic measures and introduces a parameter family of measures bridging Patterson-Sullivan and harmonic measures.

Findings

Entropy equals drift times volume growth under measure equivalence

A formula for the Hausdorff spectrum of harmonic measure is established

Finitary dimensional properties of harmonic measure are derived

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.