Green\'s Mapping and Julia Sets

In March 1999, the first named author (Binder) posed the problem of showing that a ``good direction'' $\psi\in [0,2]$ exists, for any Green's mapping $T:H\rightarrow\tilde \Omega$, i.e., \begin{equation}\label{binder} \int\limits_0\limits^{1} |T''(re^{i\pi\psi})|dr <\infty, \quad\text{ for at least one } \quad \psi\in [0,2]. \end{equation} Presently this problem is open even in the special case where $\partial \Omega$ is a uniformly perfect subset of the real line. In this paper we obtain a positive solution when $\Omega = \overline{C} \setminus E_0$ where $E_0 \subset R$ is the Julia set of an expanding quadratic polynomial.