Computing the Laurent series of map \(\Psi:C-\overline D\to C-M\) (Q1802341)

Let \(\Psi(z)=z+b_ 0+b_ 1/z+\cdots\) be the conformal map from the exterior of the unit circle onto the complement of the Mandelbrot set. In this paper, first the existence proof for \(\Psi\) due to \textit{A. Douady} and \textit{J. H. Hubbard}, i.e. the proof that the Mandelbrot set is connected, is outlined. Next it is shown that if \(P\) is a polynomial of degree \(d\), then the coefficient of \(z^{-(2j+1)2^ n}\) in the Laurent series of \(P(\Psi(z))\) is zero when \(d+j\leq 2^ n-2\). This implies (for \(P(x)=x\)) that \(b_{(2j+1)2^ n}=0\) for \(0\leq j\leq 2^ n-3\). It is an open question whether \(\Psi\) has a continuous extension to the unit circle. In the final section it is shown that if this is the case, then the extension is not Hölder continuous there. The results of this paper have some overlap with those obtained by \textit{G. M. Levin} [J. Sov. Math. 52, 3512-3522 (1990; Zbl 0716.30017)] and \textit{J. H. Ewing} and \textit{G. Schober} [Mich. Math. J. 37, 315-320 (1990; Zbl 0719.30011)].