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Hausdorff spectrum of harmonic measure

Published online by Cambridge University Press:  21 July 2015

RYOKICHI TANAKA
RYOKICHI TANAKA*
Affiliation:
Tohoku University, 2-1-1 Katahira, Aoba-ku, 980-8577 Sendai, Japan email rtanaka@m.tohoku.ac.jp
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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 Hausdorff 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.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 1 , February 2017 , pp. 277 - 307
Copyright
© Cambridge University Press, 2015 

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