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.