One c, one picture

The last essay fixed the starting point at zero and varied the parameter. Swap the two.

Fix $c$. Now ask, of every starting point $z_0$ in the plane, whether the orbit under $z \mapsto z^2 + c$ stays bounded. The points that do are the filled Julia set of that $c$, and there is one for every complex number.

The two pictures are made by the same code with one flag changed. Everything that distinguishes them is a consequence of which variable is on the axes — and the swap is not a symmetry, because the two questions are not symmetric. The Mandelbrot set is one picture, drawn once, about a whole family of maps. A Julia set is one of uncountably many pictures, each about a single map, and each one is the object the map actually acts on rather than a chart of parameters. What follows is what that difference costs and what it buys.

The dichotomy #

The striking fact about these sets is that there are only two kinds, and nothing in between.

Either the set is connected — one piece, however elaborate its outline — or it is totally disconnected, a dust of uncountably many points no two of which are joined. There is no $c$ giving a set in three pieces, or in forty.

That is a theorem, and it is old: Fatou and Julia proved it independently around 1918, six decades before anyone saw one of these sets. Their work was done entirely by hand, on objects they could describe but not draw, and the reason the sets carry Julia’s name and the pictures Mandelbrot’s is precisely that gap.

The theorem also says which is which, and the criterion is the one the last essay was about: the set is connected exactly when the orbit of the critical point $0$ stays bounded — that is, exactly when $c$ is in the Mandelbrot set.

So the Mandelbrot set is not a fractal that happens to resemble Julia sets. It is the index of them: a map of the parameter plane in which the black region marks the connected ones and the white region the dusts.

Crossing the boundary #

The dichotomy has a consequence that is hard to accept until it is stated carefully: an arbitrarily small change in $c$ can destroy the set.

Take a $c$ just inside the Mandelbrot boundary and one just outside, a millionth apart. The first gives a connected set with interior; the second gives a dust with no interior at all. Nothing continuous happens in between, because there is no in between.

Two hundredths is not a millionth, and the reason for the gap is a limitation of the drawing rather than of the claim. A parameter a millionth outside gives a dust so nearly connected that no grid of samples could tell it from the first figure — the theorem is about the object, and a picture of it needs the change to be large enough to survive rendering. The two figures above differ by a change small enough to be a rounding error in most contexts, and they are pictures of qualitatively different objects. This is sensitivity again — the same phenomenon as two orbits parting — but now applied to the shape of a set rather than to the position of a point, and the discontinuity is exact rather than a matter of degree.

Why there is nothing in between #

The dichotomy sounds like a fact that would need a delicate proof, and the reason for it is close to a counting argument.

Drawing the set backwards rather than forwards makes it visible. Every point has exactly two square roots, so the inverse of $z \mapsto z^2 + c$ is two-valued: $z \mapsto \pm\sqrt{z - c}$. Start anywhere on the set and iterate backwards, taking both branches every time, and after $n$ steps there are $2^n$ points, all of them on the set, and they fill it in.

That construction is a binary tree, and what it produces depends on one thing: what the two branches do to the region they are applied to. If the two preimages of the region overlap or touch, the tree’s images stay joined and the result is connected. If they are disjoint — if the two halves separate cleanly at every level — the result is a set built by repeatedly splitting a region in two and discarding the middle, which is a Cantor dust.

The critical point is what decides between the two, because it is the one place where the two branches of the square root meet. If the critical orbit escapes, the splitting is clean at every level and the dust follows; if it does not, the branches keep touching and everything stays joined. There is no third option because a binary tree has no third option: either the branches separate or they do not.

This is the same argument that produces the ordinary Cantor set’s relatives — repeated splitting, uncountably many points, measure zero — and it explains why the disconnected case is always a dust rather than, say, four pieces. Every level of the tree splits everything or nothing.

What the boundary is the boundary of #

The word “Julia set” strictly means the boundary, not the filled region, and the distinction is worth keeping because the boundary is what the theorems are about.

For a point strictly inside, the orbit is bounded and stays bounded under a small perturbation. For a point strictly outside, the orbit escapes and keeps escaping. Both of those are stable behaviours: nearby points do the same thing.

The boundary is where they are not. Every point of it has points of both kinds arbitrarily close, so however precisely a boundary point is specified, the behaviour of nearby points is not determined. That is the definition Fatou and Julia actually used: the Julia set is the set where the iteration fails to be stable, and the complement — where it is — is the Fatou set.

That definition explains a property the pictures make plain and no drawing proves. The Julia set is where the whole dynamic lives. Orbits starting on it stay on it, are dense in it for most $c$, and are chaotic there in the technical sense; orbits starting off it are boring, going either to an attracting cycle or to infinity. The interesting behaviour is confined to a set of area zero, and the two-dimensional-looking regions are precisely the parts where nothing happens.

The simplest case, which is not simple #

Set $c = 0$. The iteration is $z \mapsto z^2$, which doubles the angle and squares the length.

Inside the unit circle, lengths shrink to zero. Outside, they grow without bound. The filled Julia set is the closed unit disc, and the Julia set proper is the unit circle — the one case in this whole subject where the answer is a shape with a name.

Even here, the dynamics on the boundary are the interesting kind. On the circle the map is $\theta \mapsto 2\theta$, the doubling map, which this collection has already met as the cleanest example of a chaotic system: periodic points dense, sensitive dependence, an orbit that visits everywhere. So the circle looks tame and behaves exactly like everything else in this field.

Perturb $c$ away from zero by any amount and the circle immediately grows structure at every scale. There is no gradual roughening; the smoothness at $c=0$ is a coincidence of one parameter value, and the surrounding parameters have nothing like it.

Reading a set for its cycle #

A connected Julia set’s shape encodes the period of the attracting cycle its parameter belongs to, and that makes the pictures readable rather than merely decorative.

If $c$ is in the main cardioid, the map has an attracting fixed point, the interior of the filled set is one region, and the boundary is a single closed curve — deformed, but not pinched. If $c$ is in the period-two bulb, the interior has infinitely many components meeting in pairs. In a period-$q$ bulb, the components meet $q$ at a time, and the number of filaments meeting at each pinch point is the period.

The hero figure is the period-three case, which is where its name comes from: $-0.123 + 0.745i$ sits in the largest bulb on the upper edge of the cardioid, the orbit of zero cycles through three points, and the set has three lobes meeting at each pinch — one large and two smaller, which is enough for a rabbit’s head and ears.

So the count that the last essay read off the Mandelbrot set’s bulbs — which period sits where — can be read again off any individual Julia set, from a completely different feature. The parameter plane and the dynamical plane carry the same information in different form, which is the general shape of what this subject discovered.

The pictures are drawn forwards anyway #

The backwards construction proves the theorem and makes a poor algorithm, which is worth saying because it is the reverse of what usually happens.

Iterating backwards visits the set itself, so it draws exactly the object rather than an approximation of it. But the tree grows by a factor of two per level, the points crowd unevenly — some regions get exponentially more of them than others — and after twenty levels there are a million points concentrated almost entirely in the wrong places. Getting an even covering requires choosing branches at random and running for a long time, and a long time still misses the thin parts.

The escape-time method draws the complement and gets the set as what is left over. It visits every pixel exactly once, the work is bounded and uniform, and the answer at each point is a definite number rather than a sample. Its cost is the one the last essay’s honesty section named: the black region is always slightly too large, since it is “did not escape within $N$”, not “does not escape”.

So the two methods have opposite errors. Backwards iteration draws only true points of the set and cannot draw all of them; forwards iteration covers everything and draws a set slightly too big. Every picture in this essay is the second kind, which is the standard trade in numerical work and the same one behind approximating an area with rectangles — a systematic error with a bound beats a sampling error without one.

The escape time is a potential #

There is a reading of the contour bands that turns them from a rendering convenience into a quantity with a meaning, and it is the reason this site draws them as lines.

Outside the set, define $G(z)$ as the limit of $2^{-n}\log|z_n|$ along the orbit. Far away, where the iteration is essentially $z \mapsto z^2$, that limit is just $\log|z|$; near the set it goes to zero. The function is harmonic everywhere outside the set, it is zero exactly on the boundary, and it grows like $\log|z|$ at infinity — which is to say it is the electrostatic potential of a charged conductor in the shape of the Julia set.

That is not a metaphor. Charge the set, look at the field around it, and the equipotential lines are these contours. The escape-time bands are a coarse, integer-valued approximation of them: a point takes one more step to escape roughly when it is one level further in.

Two things follow. The capacity of the set — an electrostatic quantity — turns out to be exactly $1$ for every one of these, connected or dust, which is a strong statement about how little the shape matters to some questions. And the field lines perpendicular to the equipotentials, the external rays, land on the boundary at specific points and are the main tool for proving things about it; the correspondence between rays and boundary points is how the period arithmetic of the bulbs was established.

So the bands are not a rendering choice dressed up. They are a level set of a function with a physical interpretation, which is why crowding them towards the boundary is informative rather than decorative — the same reason contour lines on a map carry information that shading does not.

Why anyone cares beyond the pictures #

Three reasons, in increasing order of how far they reach.

They are the model case. Almost every general theorem about iterating a rational map on the sphere was first proved, or first conjectured, for $z^2 + c$. It is the simplest family with the phenomenon in it, in the same way the logistic map is on the interval.

They connect to number theory. The set of $c$ for which the critical orbit is pre-periodic — eventually repeating rather than escaping or converging — turns out to be exactly the algebraic integers of a certain kind, and these Misiurewicz points are dense in the Mandelbrot boundary. Statements about which parameters give which dynamics turn into statements about polynomials with integer coefficients.

They are where the word “chaos” is provable rather than observed. On the interval, showing that a specific map is chaotic takes work and is often only numerical. Here the Julia set of almost any $c$ carries a map that is conjugate to a shift on symbol sequences, and every property in the definition — dense periodic points, a dense orbit, sensitive dependence — follows from that conjugacy in a line each. It is the one family in this collection where the adjective is a theorem.

They gave the subject its method. Fatou and Julia had the theorems and no pictures for sixty years; the pictures produced a decade of new theorems immediately. What changed was not the mathematics available but the ability to look, and the same thing had happened with the Feigenbaum constant a few years earlier and with the four-colour theorem at almost the same moment.

The natural next question is what happens when the iterated map is not chosen for its dynamics at all but arrives from somewhere else — from an algorithm meant to find roots, for instance, whose failure modes turn out to have the same geometry.