Greenś Mapping and Julia Sets

Abstract

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