Accessibility and porosity of harmonic measure at bifurcation locus

Abstract

We study accessibility of generic parameters with respect to the harmonic measure with the pole at ∞ on the boundary of the connectedness loci M d for unicritical polynomials f c (z) = z d + c. It is known that a generic parameter c ∈ ∂M d is not accessible within a John angle and ∂M d spirals round them infinitely many times in both directions. We prove that almost every point from ∂M d is asymptotically accessible by a flat angle with apperture decreasing slower than (logfor any iterate of log. This is a consequence of an iterated large deviation estimate for exponential distribution.Additionally, for an arbitrary β > 0, the bifurcation locus is not β-porous on a set of scales of positive density along almost every hyperbolic geodesic with respect to the harmonic measure. * Supported in part by Narodowe Centrum Nauki -grant UMO-2018/29/B/ST1/00013. the set G ω := G(c ω ) becomes a random set and log diam U n ∼ n log d because of estimate (1) for large n. After conditioning that n and m are large we have the following density estimate, Lemma 3.4 of [11],with probability 1 − exp(−P m) for some P > 0 locally constant. Therefore, the waiting time for the density of univalent passages to the large scale starting at any large time n to attain a significant level (> 1/3) is given by an exponential distribution.Passages to the large scale are responsible for a self-similar structure of M d around a randomly chosen c ω ∈ ∂M d while a various new shapes are added in complementary scales. The normal distribution discribes an overall growth of complexity of ∂M d . The normal and exponential distributions coexist and lead to new geometric estimates for M d around a random c ω .

Slowly recurrent Collet–Eckmann maps with non‐empty Fatou set

“…M. Benedicks and J. Graczyk also have a preprint (unpublished) on perturbations on such (quadratic or, more generally, unicritical) maps. The maps there and in the recent papers [14,15] are also slowly recurrent, and hence the results in this paper are partially a generalisation of some of those results. We will not use a harmonic measure, but develop the classical Benedicks-Carleson parameter exclusion techniques and combine it with strong results on transversality, by G. Levin [18].…”

“…We are going to study perturbations of such maps in the complex setting. For the quadratic family and other unicritical families, J. Graczyk and G. Świaţek recently made an extensive study of perturbations of typical Collet-Eckmann maps with respect to harmonic measure, in a series of papers [12][13][14][15]. M. Benedicks and J. Graczyk also have a preprint (unpublished) on perturbations on such (quadratic or, more generally, unicritical) maps.…”