There is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for nonsimply connected domains in the plane. Because of the close relation to conformal mapping theory, the simply connected case is much better understood, and proving the above statement would give new results concerning the properties of harmonic measure in the general case.
We establish the conjecture in the category of domains bounded by polynomial Julia sets. The idea is to consider the coefficients of the dynamical zeta function as subharmonic functions on a slice of Teichmüller's space of the polynomial and then to apply the maximum principle.
I. Binder. N. Makarov. S. Smirnov. "Harmonic measure and polynomial Julia sets." Duke Math. J. 117 (2) 343 - 365, 1 April 2003. https://doi.org/10.1215/S0012-7094-03-11725-1