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Zbl 0857.30024
Levin, Genadi
Disconnected Julia set and rotation sets. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 29, No. 1, 1-22 (1996). ISSN 0012-9593

Let $f_c(z) = z^2+c$, and $M= \{c\in \bbfC:f^n_c(0)$ bounded for $n\in\bbfN\}$ 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.
[I.N.Baker (London)]

MSC 2000:: *30D05 Functional equations in the complex domain

Keywords: disconnected Julia set; Mandelbrot set; boundary behaviour; hyperbolic component; hedgehogs

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