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