Let J be the Julia set of a conformal dynamics f. Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out to be equivalent to the existence of a conformal change of variable which reduces the dynamical system to another one for which the harmonic measure equals the measure of maximal entropy.
@article{bwmeta1.element.bwnjournal-article-fmv150i3p237bwm,
author = {I. Popovici and Alexander Volberg},
title = {Rigidity of harmonic measure},
journal = {Fundamenta Mathematicae},
volume = {149},
year = {1996},
pages = {237-244},
zbl = {0870.30019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv150i3p237bwm}
}
Popovici, I.; Volberg, Alexander. Rigidity of harmonic measure. Fundamenta Mathematicae, Tome 149 (1996) pp. 237-244. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv150i3p237bwm/
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