Analytic structures and harmonic measure at bifurcation locus

We study conformal quantities at generic parameters with respect to the harmonic measure on boundary of connectedness loci Md for unicritical polynomials fc(z)=zd+c. It is known that these are structurally unstable and have stochastic dynamics. prove C1+α2d+α−ϵ-conformality, α=2−HD(Jc0), parameter-phase space similarity maps ϒc0(z):C↦C typical c0∈∂Md establish globally quasiconformal ϒc0(z), c0∈∂Md, C1-conformal along external rays landing c0 in C∖Jc0 mapping onto corresponding Md. This equivalence leads proof z-derivative map ϒc0(z) equal 1/T(c0), where T(c0)=∑n=0∞(D(fc0n)(c0))−1 transversality function. The paper builds analytical tools a further extremal properties ∂Md, [27]. In particular, we will explain how non-linear dynamics creates abundance hedgehog neighborhoods ∂Md effectively blocking good access from outside.