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From the introduction: The aim of this article is to present a short proof, due to M. Shishikura (unpublished), of the following theorem first obtained by J.-C. Yoccoz: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].

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].

##### MSC:

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 |