Julia Sets and the Mandelbrot Set

This post explores the relationship between filled Julia sets and the Mandelbrot set.

Throughout we consider the quadratic map

\[f_c: z \mapsto z^2+c,\]

where \(c\) is a complex constant, and write \(f_c^n\) for its \(n\)-th iterate.

Filled Julia sets

The filled Julia set \(K_c\) is the set of starting points whose orbits stay bounded under iteration:

\[K_c=\left\{z \in \mathbb{C} \;\middle|\; \left(f_c^n(z)\right)_{n \in \mathbb{N}} \text{ is bounded}\right\}.\]

To draw pictures we use the escape time. For a cut-off radius \(R\) (we take \(R=1000\)), let \(n_{\max}(z)\) be the number of iterations before \(\lvert f_c^n(z)\rvert\) first exceeds \(R\). Points of \(K_c\) never escape, so \(n_{\max} = \infty\); points outside escape after finitely many steps.

The Mandelbrot set

The Mandelbrot set \(M\) is the set of parameters \(c\) for which the orbit of \(z = 0\) stays bounded, that is, for which \(0 \in K_c\).

To plot it, we treat each point of the complex plane as a value of \(c\), iterate \(0\) under \(f_c\), and colour the point by its escape time: darker for slower escape, with points that never escape belonging to \(M\).

Figure 1. The Mandelbrot set.

Some of its properties:

• \(M\) is compact. It lies inside the disc \(\lvert c\rvert \le 2\), since for \(\lvert c\rvert > 2\) the orbit of \(0\) escapes, and it is closed, being an intersection of closed sets. By the Heine–Borel theorem, a closed and bounded subset of \(\mathbb{C}\) is compact.
• \(M\) is connected, a theorem of Douady and Hubbard (1982), and it has no holes: its complement is connected too.
• Zooming in on the boundary reveals ever finer detail. Whether \(M\) is also locally connected is the famous open MLC conjecture.

Connectedness of filled Julia sets

We can colour filled Julia sets in the same way: fix \(c\), compute \(n_{\max}(z)\) for each starting point \(z\), and colour accordingly (here, lighter for slower escape).

Figure 2. The filled Julia set for \(c = 0.9i\).

The dichotomy theorem

If \(c \in M\), the filled Julia set \(K_c\) is connected. If \(c \notin M\), it is totally disconnected, a Cantor-like “dust”.

The Julia set in Figure 2 is not connected, so its parameter \(c = 0.9i\) is not in the Mandelbrot set. Iterating directly confirms this: the orbit of \(0\) under \(f_{0.9i}\) leaves the disc \(\lvert z\rvert \le 2\) after eight steps.

Summary

• Filled Julia sets: fix \(c\) and ask which starting points \(z\) have bounded orbits.
• The Mandelbrot set: fix the starting point \(z = 0\) and ask which parameters \(c\) give a bounded orbit.

The dichotomy theorem links the two: the Mandelbrot set is a map of which Julia sets are connected. For example, if \(\lvert c\rvert > 2\) then \(c \notin M\), so the filled Julia set is disconnected.

Exercises

• Write a Python program that plots the Mandelbrot set and a filled Julia set. (Undergraduate, year 1)
• Let \(T=\left\{\left(x, \sin \frac{1}{x}\right): x \in(0,1]\right\} \cup\{(0,0)\}\). Show that \(T\) is connected but not locally connected. Why is this relevant to the MLC conjecture? (Undergraduate, years 2–3)

Changes on 25 October 2026

What was wrong. Nothing in the results. The dichotomy theorem was stated loosely (“otherwise it is disconnected”); when \(c\) is outside the Mandelbrot set, the filled Julia set is in fact totally disconnected.

What changed. The theorem is stated precisely, the escape criterion \(\lvert c\rvert > 2\) is explained and the notation is simpler.