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Holomorphic motions and puzzles (following M. Shishikura). (English) Zbl 1063.37042
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, 117-131 (2000).
From the introduction: The aim of this article is to present a short proof, due to M. Shishikura (unpublished), of the following theorem first obtained by J.-C. Yoccoz:
Main theorem (Yoccoz): The Mandelbrot set \(M\) is locally connected at parameters \(c\) for which \(f_c(z)= z^2+ c\) is a nonrenormalizable map whose fixed-points are repelling and whose critical point is recurrent.
For parameters \(c\) satisfying the above conditions, another theorem of Yoccoz (see the paper by J. Milnor [ibid. 274, 67–116 (2000; Zbl 1052.34073)]) asserts that the Julia set \(J_c\) of \(f_c\) is locally connected, and in particular exhibits a fundamental system of neighborhoods \(P_n(c)\) of \(c\) – in the dynamical plane of \(f_c\) – whose closures have connected intersections with \(J_c\). The author derives the main theorem from this dynamical result in the following way. Given a prescribed parameter \(c_0\in M\), he uses the Douady-Hubbard conformal representation of \(\mathbb{C}\setminus M\) to construct neighborhoods \({\mathcal P}_n\) of \(c_0\) – in the parameter plane – whose closures have connected intersections with \(M\). Then, to show that \(\{c_0\}= \bigcap_n{\mathcal P}_n\), he considers the annuli \({\mathcal A}_n={\mathcal P}\setminus \overline{{\mathcal P}_{n+1}}\) and proves that the sum of their moduli is infinite. To get this estimate, he compares the modulus of \({\mathcal A}_n\) with the modulus of the “corresponding” annulus \(A_n(c_0)= P_n(c_0)\setminus \overline{P_{n+1}(c_0)}\). In M. Shishikura’s proof, this key comparison relies on quasiconformality properties of holomorphic motions and the observation that, for \(c\in{\mathcal P}_n\), the dynamical piece \(P_n(c)\) moves holomorphically.
For the entire collection see [Zbl 0935.00019].

37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable