The uniformization of the complement of the Mandelbrot set (Q1073954)

Let \(p_ c(z)=z^ 2+c\), and let \(p^ k_ c\) be the kth iterate of \(p_ c\). The Mandelbrot set, M, is the set of c's for which \(p^ k_ c\) is bounded, or equivalently, of those c's for which the Julia set of \(p_ c\) is connected. In C. R. Acad. Sci., Paris, Sér. I 294, 123-126 (1982; Zbl 0483.30014)], \textit{A. Douady} and \textit{J. H. Hubbard} construct an analytic bijection \(\Phi\) from \({\bar {\mathbb{C}}}-M\) to \({\bar {\mathbb{C}}}-{\mathbb{D}}\) (where \({\bar {\mathbb{C}}}\) is the Riemann sphere and \({\mathbb{D}}\) the closed unit disc). Their construction does not lend itself to computation. The author gives an alternative construction which does, by exhibiting \(\Phi\) (c) as a limit of \(2^ kth\) roots of \(p_ c^{k+1}(0)\). This construction uses Douady and Hubbard's result that M is simply connected. There is a simple algorithm for finding the coefficients in the Laurent expansion \[ \Phi^{-1}(z)=z+\sum_{0\leq j<\infty}b_ jz^{-j}. \] All the coefficients are dyadic rationals and \(b_{2^ k}=0\) for \(k>1\). The question of local connectedness of M is equivalent to the convergence of this series on \(| z| =1\). By computation, \(| b_ n| <(1/n)\) for the first 4095 terms.