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Local connectivity of Julia sets: expository lectures. (English) Zbl 1107.37305
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, 67-116 (2000).
From the introduction: These notes provide an introduction to work of B. Branner, J. H. Hubbard, A. Douady, and J.-C. Yoccoz on the geometry of polynomial Julia sets. They are an expanded version of lectures given in Stony Brook in Spring 1992.
Section 1 describes unpublished work by Yoccoz on local connectivity of quadratic Julia sets [cf. J. H. Hubbard, in Goldberg, Lisa R. (ed.) et al., Topological methods in modern mathematics. Proceedings of a symposium in honor of John Milnor’s sixtieth birthday, held at the State University of New York at Stony Brook, USA, 1991. Houston, TX: Publish or Perish, Inc., 467–511 (1993; Zbl 0797.58049)]. It presents only the ‘easy’ part of his theory, in the sense that it considers only non-renormalizable polynomials, and makes no effort to describe the more difficult arguments which are needed to deal with local connectivity in parameter space. It is based on second-hand sources (Hubbard [“Puzzles and quadratic tableaux (according to Yoccoz)”, Preprint, 1990], together with lectures by Branner and Douady), and uses the language of Branner and Hubbard. The presentation is quite different from that of Yoccoz. (Compare Problem 1-e at the end of \(\S 1\).)
Section 2 describes the analogous arguments used by B. Branner and J. H. Hubbard [Acta Math. 169, No. 3-4, 229–325 (1992; Zbl 0812.30008)] to study higher degree polynomials for which all but one of the critical orbits escape to infinity. In this case, the associated Julia set \(J\) is never locally connected. The basic problem is rather to decide when \(J\) is totally disconnected. This Branner-Hubbard work came before the work of Yoccoz, and its technical details are not as difficult. However, in these notes their work is presented simply as another application of the same geometric ideas.
Section 3 complements the Yoccoz results by describing a family of examples, due to Douady and Hubbard, showing that an infinitely renormalizable quadratic polynomial may have non-locally-connected Julia set. An appendix describes needed tools from complex analysis, including the Grötzsch inequality.
See also the author’s book “Dynamics in one complex variable. Introductory lectures”, Vieweg, Braunschweig (1999; Zbl 0946.30013)].
For the entire collection see [Zbl 0935.00019].

MSC:
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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