Analytic structures and harmonic measure at bifurcation locus

Abstract

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