## Institute for Mathematical Sciences

## Preprint ims95-5

** G. Levin and S. van Strien**
* Local Connectivity of the Julia Set of Real Polynomials.*

Abstract: One of the main questions in the field of complex dynamics is the question whether the Mandelbrot set is locally
connected, and related to this, for which maps the Julia set is
locally connected. In this paper we shall prove the following
Main Theorem:
Let $f$ be a polynomial of the form $f(z)=z^d +c$ with
$d$ an even integer and $c$ real. Then the Julia set of
$f$ is either totally disconnected or locally connected.
In particular, the Julia set of $z^2+c$ is locally connected
if $c \in [-2,1/4]$ and totally disconnected otherwise.

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