## Institute for Mathematical Sciences

## Preprint ims92-11

** J. Milnor**
* Local Connectivity of Julia Sets: Expository Lectures*

Abstract: 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Section 1 describes unpublished work by J.-C. Yoccoz on local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of examples, due to Douady and Hubbard (unpublished), showing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 Gr\"otzsch
inequality.

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