Asymptotic porosity of planar harmonic measure

We study the distribution of harmonic measure on connected Julia sets of unicritical polynomials. Harmonic measure on a full compact set in ? is always concentrated on a set which is porous for a positive density of scales. We prove that there is a topologically generic set $mathcal{A}$ in the boundary of the Mandelbrot set such that for every $cin mathcal{A}$ , β>0, and λ∈(0,1), the corresponding Julia set is a full compact set with harmonic measure concentrated on a set which is not β-porous in scale λ ⁿ for n from a set with positive density amongst natural numbers.