On the complement of the Mandelbrot set
The Mandelbrot set \(M= \{c: J_ c\) is connected\}, where \(J_ c\) is the Julia set of \(f_ c(z)= z^ 2+ c\). The complement of \(M\) is conformally equivalent to the exterior of a disc \(D\) by a map \(\Phi\) with \(\Phi(\infty)= \infty\), \(\Phi'(\infty) =1\), \(\Phi(D^ c)= M^ c\). The boundary of \(M\) is extremely intricate and the study of the boundary properties of \(\Phi\) correspondingly difficult. The paper studies this question for certain points of \(\partial M\). A hyperbolic component \(W\) of the interior of \(M\) is a set of \(c\) in which \(f_ c\) has an attracting cycle \(\alpha(c)\) of some fixed period \(m\). Let \(c_ 0\) be a boundary point of \(W\), such that the multiplier \(\lambda(c)\) of \(\alpha(c)\) has the value \(\exp(2\pi i\nu)\), where the rational number \(\nu= p/q\) in lowest terms. Roughly speaking the paper describes the connection between \(\lambda(c)\), the so-called rotation number of the cycle \(\alpha(c)\) and parameters related to the value of the mapping function \(\Phi\) as \(c\) passes from \(W\) to \(M^ c\) through \(c_ 0\). The arguments involve an extension of an inequality of \textit{J. C. Yoccoz} [Sur la taille des membres de l'ensemble de Mandelbrot, Manuscript 1987], and a foliation of \(\widehat{\mathbb{C}}\backslash J_ c\) described as the `hedgehog'.
scientific article; zbMATH DE number 720058