Accessibility and porosity of harmonic measure at bifurcation locus

Abstract

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 M d ${\cal M}_d$ for unicritical polynomials f c ( z ) = z d + c $f_c(z)=z^d+c$ . It is known that a generic parameter c ∈ ∂ M d $c\in \partial {\cal M}_d$ is not accessible within a John angle and ∂ M d $\partial {\cal M}_d$ spirals round them infinitely many times in both directions. We prove that almost every point from ∂ M d $\partial {\cal M}_d$ is asymptotically accessible by a flat angle with apperture decreasing slower than ( log ∘ ⋯ ∘ log dist ( c , ∂ M d ) ) − 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.