Relating Hausdorff measures and harmonic measures on parabolic Jordan curves.
Representing real numbers in Möbius number systems
Rigidity and inflexibility in conformal dynamics.
Rigidity of harmonic measure
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...