Accessibility and porosity of harmonic measure at bifurcation locus

We study hyperbolic geodesics running from ∞$\infty$ to a generic point, by the harmonic measure with the pole at ∞$\infty$, on the boundary of the connectedness locus Md${\cal M}_d$ for unicritical polynomials fc(z)=zd+c$f_c(z)=z^d+c$. It is known that a generic parameter c∈∂Md$c\in \partial {\cal M}_d$ is not accessible within a John angle and ∂Md$\partial {\cal M}_d$ spirals round them infinitely many times in both directions. We prove that almost every point from ∂Md$\partial {\cal M}_d$ is asymptotically accessible by a flat angle with apperture decreasing slower than (log∘⋯∘logdist(c,∂Md))−1$(\log \circ \dots \circ \log {\rm dist}\,(c,\partial {\cal M}_d))^{-1}$ for any iterate of log$\log$. This is a consequence of an iterated large deviation estimate for exponential distribution. Additionally, for an arbitrary β>0$\beta >0$, the bifurcation locus is not β$\beta$‐porous on a set of scales of positive density along almost every external ray with respect to the harmonic measure.