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 ${\cal M}_d$ 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, $\alpha = 2-\mbox{HD}\,({\cal J}_{c_0})$, of the parameter-phase space similarity maps $\Upsilon_{c_0}(z):\mathbb{C}\mapsto \mathbb{C}$ at typical $c_0\in \partial {\cal M}_d$ and establish that globally quasiconformal similarity maps $\Upsilon_{c_0}(z)$, $c_0\in \partial {\cal M}_d$, are $C^1$-conformal along external rays landing at $c_0$ in $\mathbb{C}\setminus {\cal J}_{c_0}$ mapping onto the corresponding rays of ${\cal M}_d$. This conformal equivalence leads to the proof that the $z$-derivative of the similarity map $\Upsilon_{c_0}(z)$ at typical $c_0\in \partial {\cal M}_d$ is equal to $1/{\cal T}'(c_0)$, where ${\cal T}(c_0)=\sum_{n=0}^{\infty}(D(f_{c_0}^n)(c_0))^{-1}$ is the transversality function. The paper builds analytical tools for a further study of the extremal properties of the harmonic measure on $\partial {\cal M}_d$. In particular, we will explain how a non-linear dynamics creates abundance of hedgehog neighborhoods in $\partial {\cal M}_d$ effectively blocking a good access of $\partial {\cal M}_d $ from the outside.