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Disconnected Julia set and rotation sets. (English) Zbl 0857.30024

Let \(f_c(z) = z^2+c\), and \(M= \{c\in \mathbb{C}:f^n_c(0)\) bounded for \(n\in\mathbb{N}\}\) the Mandelbrot set. Let \(\Psi\) be the conformal map from the complement of the unit disc \(D\) to the complement of \(M\), such that \(\Psi(w) \sim w\) as \(w\to\infty\). The boundary behaviour of this map is of the greatest interest. Let \(w_0\in \partial D\) be a periodic point of the map \(w\to w^2\). Then there is a hyperbolic component \(G\) of \(M\) such that \(w\) is either \(\exp (2\pi it)\) or \(\exp (2\pi it')\), where \(t\) and \(t'\), \(0<t<t'<1\) are the external angles at the root \(c\) of \(G\). If \(G\) is not a primitive component of \(M\) then \(|\Psi (w)-c |/(-\log |w-w_0 |)^{-a}\) is bounded away from \(0,\infty\) as \(w\to w_0\), where \(a=1\). If \(G\) is primitive the same holds with \(a=2\), provided \(\arg (w/2\pi)\) remains in \([t,t']\) as \(w\to w_0\). For \(c\) in \(G\) there is an attracting cycle whose multiplier \(\lambda(c)\) is an analytic function of \(c\) and \(\lambda(c)\) remains analytic in the domain \(W(G)\) which is bounded by the external rays to \(M\) at \(w_0\) and which contains \(G\). The main tools are the so-called hedgehogs, derived from Böttcher’s function, and an inequality extending one of Yoccoz relating the multiplier at a fixed point to a rotation number.
Reviewer: I.N.Baker (London)

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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