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Baby Mandelbrot sets are born in cauliflowers. (English) Zbl 1107.37303

Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press (ISBN 0-521-77476-4/pbk). Lond. Math. Soc. Lect. Note Ser. 274, 19-36 (2000).
The aim of the paper under review is to consider a sequence of Mandelbrot-like families. In the neighborhood of a cusp point \(c_0\not={1\over4}\) of the Mandelbrot set \(M\), there is a sequence \((M_n)\) of small quasiconformal copies of \(M\) in \(M\), tending to \(c_0\). The Julia set \(J_{1/4}\) for \(z^2+{1\over4}\) is called the cauliflower and the Julia set \(J_{1/4+\epsilon}\) for \(z^2+{1\over4}+\epsilon\) an imploded cauliflower. Each copy \(M_n\) is contained in a set \(\Gamma_n\) which resembles \(J_{1/4+\epsilon}\), in fact in the nested sequence of the sets \((\Gamma_{n,m})_m\), homeomorphic to the preimage of \(J_{1/4+\epsilon}\) by \(z\mapsto z^{2^m}\) and accumulating to \(M_n\).
The authors show that the distance from \((M_n)\) to \(c_0\) tends to 0 like \(1/n^2\) and that its diameter, with or without the decorations \(\bigcup_m\Gamma_{n,m}\), tends to 0 like \(1/n^3\). Given two compact sets \(X\) and \(Y\) in \(\mathbb C\), \(X\) appears quasiconformally in \(Y\) iff there is a quasiconformal homeomorphism \(\phi\) from \(\mathbb C\) to itself with \(\phi(X)\subset Y, \phi(\partial X)\subset\partial Y\). For an imploded cauliflower and the map \(z\mapsto z^{2^m}\), they construct a model \(\Upsilon\) for this situation.
The main result is Theorem 1: If \(c_0\) is the root point of a primitive component of \(M^\circ\), the set \(\Upsilon\) appears quasiconformally in \(M\), with 0 (in \(\Upsilon\)) corresponding to \(c_0\) (in \(M\)). They show a more general situation in Theorem 2, in which \(\Upsilon\) appears quasiconformally in some set \(M_F\) defined in the parameter space.
For the entire collection see [Zbl 0935.00019].

MSC:

37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37F25 Renormalization of holomorphic dynamical systems
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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