Julia set

The Julia set in this visualization

For a fixed complex number $c$, the filled Julia set $K_c$ is the set of starting values $z_0$ for which the sequence below stays bounded; once $|z_n| > \max(2, |c|)$ it runs off to infinity. The Julia set $J_c$ is the boundary of $K_c$: arbitrarily close to each of its points there are starting values that escape and starting values that do not.

What the two pictures show

The z-plane, the large picture: every point is a starting value $z_0$, coloured by how soon its orbit escapes. The smooth colouring is for display only. Black means “no escape within the iteration limit $N$” — a numerical verdict. The dust highlights and the dendrite $c = i$ have no area, so almost every pixel there escapes; points within half a pixel of $K_c$ by a distance estimate are shaded dark as well, so you can see it: also display only.

The c-plane, the screen in the panel: the Mandelbrot set, with a red cross at the page’s $c$. Drag the cross, or click a point, to change $c$. The yellow dots in the large picture are the orbit of 0 (or of a starting point you click).

Why it looks like this

The red key draws the starting values with $|z_n| \le R$, $R = \max(2, |c|)$, for $n = 1, 2, 3, \ldots$: each region inside the one before, closing in on $K_c$. Each region is the preimage of the previous one under $z \mapsto z^2 + c$, and the preimage of a disc is one piece exactly when the disc contains the critical value $c$. So the regions stay in one piece as long as the orbit of 0 stays within $R$; from the step it leaves, they split into 2, then 4, 8, … pieces.

That is the theorem of Fatou and Julia [1, 2] for $z^2 + c$: $K_c$ is connected when the orbit of 0 is bounded, and a totally disconnected “dust” (a Cantor set) when it escapes. The Mandelbrot set is exactly the set of $c$ with $K_c$ connected.

An attracting cycle always attracts the orbit of 0, so $z^2 + c$ has at most one: the black inside of $K_c$ drains into it. At the cusp $c = 1/4$ and at $c = -3/4$ the cycle is parabolic instead; at the golden-mean point of the main cardioid the fixed point is surrounded by a Siegel disc, where orbits circle forever [3]. In those cases no attracting cycle exists; at the default settings the page’s period readout says “undetermined” (with a very large $N$ the numerical check can mistake a slow approach for a cycle).

Easy to misread

• Black is “did not escape within $N$ steps”. Near the edge, a larger $N$ can turn black pixels into colour.
• The dark shading of dust and branches is a distance estimate, not points that stay bounded: for $c$ outside the Mandelbrot set, $K_c$ has area 0.
• Dust that still looks like a familiar shape (just past the cusp) is in infinitely many pieces nonetheless.
• The c-plane holds the parameter; the orbit lives in the z-plane.

History

Gaston Julia and Pierre Fatou developed the theory of iterating rational functions in 1918–1920, before there were computer images [1, 2]. Carl Ludwig Siegel showed in 1942 that some neutral fixed points are surrounded by discs of circling orbits [3]. Computer pictures from 1980 on, and the work of Douady and Hubbard, made the sets widely known [4].

Related concepts

Mandelbrot set · Newton fractal (page planned) · Complex numbers

References

1. G. Julia, “Mémoire sur l’itération des fonctions rationnelles”, Journal de Mathématiques Pures et Appliquées, série 8, 1 (1918) 47–245.
2. P. Fatou, “Sur les équations fonctionnelles”, Bulletin de la Société Mathématique de France 47 (1919) 161–271; 48 (1920) 33–94, 208–314.
3. C. L. Siegel, “Iteration of analytic functions”, Annals of Mathematics 43 (1942) 607–612. doi:10.2307/1968952
4. J. Milnor, Dynamics in One Complex Variable, 3rd ed., Annals of Mathematics Studies 160, Princeton University Press (2006).

Further reading: Wikipedia: Julia set; MacTutor History of Mathematics: Gaston Julia; MacTutor History of Mathematics: Pierre Fatou.