Green\'s Mapping and Julia Sets

TL;DR

This paper addresses a longstanding open problem regarding the existence of a 'good direction' for Green's mappings, providing a positive solution in the specific case where the domain is related to Julia sets of expanding quadratic polynomials.

Contribution

The paper proves the existence of a 'good direction' for Green's mappings when the domain is a Julia set of an expanding quadratic polynomial, advancing understanding in complex dynamics.

Findings

Confirmed existence of a 'good direction' for Green's mappings in specific Julia set cases.

Extended the class of domains where the problem is solved to include Julia sets of quadratic polynomials.

Provided new insights into the structure of Green's mappings related to complex dynamical systems.

Abstract

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.