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 ${\mathcal{M}}_d$ for unicritical polynomials $f_c(z)=z^d+c$. It is known that these parameters are structurally unstable and have stochastic dynamics. We prove $C^{1+\frac{\alpha }{d}-\epsilon }$-conformality, $α= 2-\mbox{HD}\,({\cal J}_{c_0})$, of the parameter-phase space similarity maps ${{\rm Y}}_{c_0}(z):\mathbb{C}\mapsto \mathbb{C}$ at typical $c_0\in \partial {\mathcal{M}}_d$ and establish that globally quasiconformal similarity maps ${{\rm Y}}_{c_0}(z)$, $c_0\in \partial {\mathcal{M}}_d$, are $C^1$-conformal along external rays landing at $c_0$ in $\mathbb{C}\setminus {\mathcal{J}}_{c_0}$ mapping onto the corresponding rays of ${\mathcal{M}}_d$. This conformal equivalence leads to the proof that the $z$-derivative of the similarity map ${{\rm Y}}_{c_0}(z)$ at typical $c_0\in \partial {\mathcal{M}}_d$ is equal to $1/{\mathcal{T}}^{\prime }(c_0)$, where $\mathcal{T}(c_0)={\sum }_{n=0}^{\infty }(D(f_{c_0}^n)(c_0){)}^{-1}$ is the transversality function. The paper builds analytical tools for a further study of the extremal properties of the harmonic measure on $\partial {\mathcal{M}}_d$. In particular, we will explain how a non-linear dynamics creates abundance of hedgehog neighborhoods in $\partial {\mathcal{M}}_d$ effectively blocking a good access of $\partial {\mathcal{M}}_d$ from the outside.

The second author was supported in part by Narodowe Centrum Nauki - grant 2015/17/B/ST1/00091