Local connectivity of the Julia set of real polynomials

TL;DR

This paper investigates the local connectivity of Julia sets for real polynomials of the form z^d + c, establishing conditions under which they are either totally disconnected or locally connected, with specific results for quadratic polynomials.

Contribution

It proves that for polynomials with even degree and real parameters, the Julia set is either totally disconnected or locally connected, clarifying the structure for a broad class of real polynomials.

Findings

Julia set of f(z)=z^d + c is either totally disconnected or locally connected for even d

For quadratic polynomials, Julia set is locally connected if c in [-2, 1/4]

Outside this range, the Julia set is totally disconnected

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.