§2. Polynomials for which All But One of the Critical Orbits Escape

Introduction The following notes provide an introduction to recent work of Branner, Hubbard and Yoccoz on the geometry of polynomial Julia sets. They are an expanded version of lectures given in Stony Brook in Spring 1992. I am indebted to help from the audience. Section 1 describes unpublished work by J.-C. Yoccoz on local connectivity of quadratic Julia sets. It presents only the " easy " part of his work, in the sense that it considers only non-renormalizable polynomials, and makes no effort to describe the much more difficult arguments which are needed to deal with local connectivity in parameter space. It is based on second hand sources, namely Hubbard [Hu1] together with lectures by Branner and Douady. Hence the presentation is surely quite different from that of Yoccoz. Section 2 describes the analogous arguments used by Branner and Hubbard [BH2] 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 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. Chapter 3 complements the Yoccoz results by describing a family of examples, due to Douady and Hub-bard (unpublished), showing that an infinitely renormalizable quadratic polynomial may have non-locally-connected Julia set.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. We will assume that the reader is familiar with the basic properties of Julia sets and the Mandelbrot set. (For general background, see for example [Be], [Br2], [D1], [D2], [EL], [L1], as well as the brief outline in §3.) In particular, we will make use of external rays for a polynomial Julia set J(f) ⊂ C. Theorem 1. If f c (z) = z 2 + c is a quadratic polynomial such that: (1) the Julia set J(f c) is connected, (2) both fixed points are repelling, and (3) f c is not renormalizable 1 then J(f c) is locally connected. In terms of the familiar parameter space picture for the family of quadratic maps f c (z) = z 2 + c , Condition (1) says that the parameter value c belongs to the Mandelbrot set M , while (2) says that c does …