Small Mandelbrot sets among the cubics

An area that never finishes found a cubic with small whole-number coefficients for which Newton’s method fails on a region with area: from every starting point in it, the method falls into a cycle that is not a root and bounces forever. Covering rather than avoiding responded by finding a fixed list of starting points that always works, and closed with a question about the bad cubics themselves: which ones are they, and what does the set of them look like? It suggested that Newton’s method, studied as a family, would have a parameter picture with the same unresolved questions as the Mandelbrot set.

This essay draws that picture for one family of cubics and finds the Mandelbrot set in it — not something resembling it, but the same object, small, turned and very slightly bent, with the same structure down to the periods of its bulbs. The explanation is one of the deepest facts in the dynamics of rational maps, and the route to it begins with the observation that, for these cubics, a single starting point decides everything.

One point decides the whole method #

Newton’s method for a polynomial $p$ is the map $N(z) = z - p(z)/p'(z)$, a rational map of the complex plane. Its critical points — where $N'(z) = 0$ — matter more than any others, because of a theorem from the 1910s of Pierre Fatou and Gaston Julia: every attracting cycle of a rational map attracts at least one critical point. If Newton’s method has some cycle other than the roots that attracts nearby starts, then some critical point must be falling into that cycle.

Now differentiate: $N'(z) = p(z)\,p''(z)/p'(z)^2$. The critical points of $N$ are the roots of $p$, where $N' = 0$ because $p = 0$, and the roots of $p''$. A root is a fixed point of $N$ and attracts its own critical point — itself — so the roots use up their critical points on their own basins. For a cubic, $p''$ is linear and has one root, which is the free critical point: the only critical point not already committed. For the family

$$p_a(z) = (z - 1)(z^2 + z + a) = z^3 + (a - 1)z - a,$$

$p_a'' = 6z$, so the free critical point is $z = 0$ for every $a$. Whether Newton’s method for $p_a$ has a region of starting points that never finish is therefore decided by one orbit: the orbit of $0$. If it reaches a root, there is no extra attracting cycle and almost every start finds a root. If it does not, there may be one.

The family is chosen so that the root $1$ is always present and the other two roots move with $a$. One complex number describes the cubic, so the cubics form a plane, and the opening figure colours each point of that plane by the fate of $0$. Most of the plane is coloured: $0$ reaches a root. The dark specks are the cubics for which Newton’s method has a trap.

Why one complex number describes every cubic #

The family looks special, and it is not. Newton’s method does not care where the origin is or what unit is used: if two polynomials differ by a change of variable $z \mapsto \alpha z + \beta$, their Newton maps differ by the same change, and every picture of one is a moved and scaled picture of the other. Any cubic with three distinct roots can be moved so that the roots add to nought, and then scaled so that one of them is $1$; the other two are then $s$ and $t$ with $s + t = -1$, which makes them the roots of $z^2 + z + st$. So every cubic is, after a change of variable, one of the cubics $(z - 1)(z^2 + z + a)$ with $a = st$ — in fact up to three of them, one for each choice of which root to move to $1$.

That is why the parameter plane is the right place to look. It is not a slice through a larger space of cubics; up to rescaling it is the space of cubics, each appearing at most three times. The free critical point lands at $0$ because the roots were made to sum to nought, which puts the inflection point of the cubic — where $p''$ vanishes — at the origin. Every question about how Newton’s method behaves for cubics is therefore a question about this plane and one orbit in it.

The quadratic has no free point #

The comparison with the quadratic case makes the role of the free critical point vivid. For a quadratic $p$, the second derivative is a constant and has no roots, so the Newton map’s critical points are exactly the two roots and nothing else. Chaos on the line between two roots found the consequence by direct computation: the Newton map is conjugate to squaring, its two basins fill the plane apart from a line, and there is no room for any third attracting cycle, because there is no critical point left over to be attracted to it. Newton’s method can never be trapped on a quadratic.

The cubic is the first degree with a free critical point, and so the first in which the method can fail on an open set. Each further degree adds one more: a polynomial of degree $d$ has $d - 2$ free critical points, and a parameter space of $d - 2$ complex dimensions in which they can be captured independently. The specks of this essay are the one-dimensional beginning of that, and one $c$, one picture is the one-parameter quadratic family whose structure every one of them repeats.

The largest speck, enlarged #

The boxed speck sits near $a = 0.32 + 1.63i$. Enlarged, it has a familiar shape.

The fit is the evidence that this is not a resemblance. The Mandelbrot set is the set of $c$ for which the orbit of $0$ under $z \mapsto z^2 + c$ stays bounded, and it is built from a main cardioid with circular bulbs attached, each bulb carrying an attracting cycle of a definite period. To compare it with the speck, two landmarks are matched: the point $c = 0$, where $0$ is periodic of period one under $z^2 + c$, and the point $c = -1$, the centre of the largest bulb, where $0$ is periodic of period two. In the speck the corresponding points are the parameters where $0$ returns to itself exactly — superattracting centres — and the figure finds them by solving $N_a^k(0) = 0$ for $a$: at $a \approx 0.319 + 1.633i$ the orbit of $0$ returns after two steps, and in the largest bulb after four.

One affine map — a rotation, a scaling and a shift of the complex plane — carries the first pair of landmarks to the second. Under that map the outline of the whole Mandelbrot set lands on the speck, cardioid on cardioid and bulbs on bulbs, with a small but visible mismatch towards the edges. The mismatch is real and expected: the speck is the Mandelbrot set seen through a map that is analytic but not exactly affine, and the more of the speck one compares, the more the bending shows.

What a dark parameter means inside the method #

The parameter picture is a census of cubics; each point of it stands for a whole picture of Newton’s method. At the centre of the copy that picture has a fourth colour.

The three root basins have the fractal boundaries of where Newton’s method goes instead. Inside them sit dark islands, the basin of the 2-cycle: from any start in them the iteration is drawn into bouncing between $0$ and $N_a(0)$, two points that are not roots, and it never stops. This is the phenomenon of an area that never finishes, now located in a family: the bad cubics are not isolated accidents but fill the dark specks of the parameter plane, each speck a continuum of cubics whose methods all fail on open sets of starts.

The dark islands also repeat. Each is surrounded by smaller ones, preimages under the Newton map, so the region of failure is spread across the plane in a fractal pattern of its own, though its total area is small. A random starting guess for this cubic fails about once in a hundred times; one placed near the origin, where a program with no information might well start, fails much more often.

Crossing the edge of the copy #

What happens at the edge of the speck is the same as what happens at the edge of the Mandelbrot set: the attracting cycle loses its stability and disappears, and $0$ escapes. Just outside, the escape is slow.

The two panels look almost identical for the first hundred steps. Inside the speck, the bouncing goes on forever. Outside, the cycle no longer exists, but the map near it is still almost the map that had it, and the orbit lingers in the narrow channel where the cycle used to be before slipping out. The time spent lingering grows without bound as the parameter approaches the edge, in the same way that the escape time of $z^2 + c$ grows as $c$ approaches the boundary of the Mandelbrot set, and for the same reason. Near the edge of a dark speck, Newton’s method is slow before it is wrong: a program that stops after a fixed number of steps and reports failure will misclassify a band of cubics just outside the speck as bad.

The same dynamics, renormalised #

Why the Mandelbrot set, of all shapes? The answer was found by Adrien Douady and John Hubbard in 1985, and it is called renormalisation. Near the trapped cycle, two steps of Newton’s method form a map from a small disc into a slightly larger one that behaves, in every way that matters dynamically, like a quadratic polynomial $z^2 + c$: one critical point inside, mapping the disc over itself twice. Douady and Hubbard called such a map polynomial-like and proved that it is conjugate to a genuine quadratic polynomial on the set of points that never leave the disc. The parameter $c$ of that polynomial moves with $a$, and as $a$ runs over the speck, $c$ runs over the Mandelbrot set.

The figure tests that explanation along a line. Walking down the real axis of the Mandelbrot set, the attracting cycle of $z^2 + c$ has period 1, then 2, then 4 and 8 as the doubling cascade runs. At the corresponding parameters of the copy, the cycle trapping $0$ has period 2, 4, 8 and 16 — exactly double, because one step of the quadratic is two steps of Newton’s method. The two points where the periods are not in that ratio sit at the bifurcations themselves, where neither cycle has settled within the iterations allowed. The copy is not a resemblance; it is the same dynamical system, seen through two steps of the method.

Curtis McMullen proved in 2000 that this is universal: in essentially any analytic family of rational maps in which the dynamics change, small copies of the Mandelbrot set appear wherever a critical point is being captured by a cycle. James Curry, Lucy Garnett and Dennis Sullivan had seen them in exactly this setting — the parameter plane of Newton’s method for cubics — in computer pictures published in 1983, before the theory that explained them existed.

Where the other specks come from #

The opening figure circles three smaller specks, with periods 3, 6 and 6. Each is another copy, renormalised by three or six steps of the method instead of two, and each is surrounded by smaller copies still. The parameter plane of the cubics is coloured almost everywhere, and the dark set is a dust of Mandelbrot sets of every size, concentrated along the boundaries between colours, where the fate of $0$ is changing.

That placement is the deepest part of the picture. A cubic is bad when $0$ is captured by a cycle, and capture can happen only where $0$ is not decisively heading for any particular root — that is, on the boundaries of the coloured regions. Those boundaries are fractal, and they carry the copies the way the boundary of the Mandelbrot set carries its own smaller copies. The bad cubics form a set of small but positive area, threaded through the plane of all cubics along the lines where the method changes its mind about which root to find.

Whether a better method could avoid them is a question with a sharp answer. Curtis McMullen proved in 1987 that for polynomials of degree four or more there is no generally convergent purely iterative algorithm — no rational map, built from the coefficients alone, that converges to a root from almost every start for almost every polynomial. For cubics there is one, though it is not Newton’s method: McMullen constructed a rational map from the symmetries of the cubic’s three roots that never gets captured. Peter Doyle and McMullen then showed in 1989 that towers of such iterations solve the quartic and the quintic, the quintic through the symmetries of the icosahedron, and that nothing of this kind solves equations of degree six or more. For Newton’s method itself, a practical guarantee exists of a different kind: John Hubbard, Dierk Schleicher and Scott Sutherland showed in 2001 that a finite, explicit set of starting points — a few times $d(\log d)^2$ of them for degree $d$ — is enough to find every root of every polynomial of that degree.

What the pictures cannot settle #

The parameter pictures are computed on grids of a few hundred cells a side, and each cell is decided by following the orbit of $0$ for a fixed number of steps: three hundred in the plane, six hundred in the enlargement. A cell is dark if $0$ has not reached a root by then. The orbits figure shows why that is not the same as being trapped: just outside a speck, $0$ can take more than a hundred steps to escape, so a thin rim of cells drawn dark lies in fact outside the copy. The periods figure avoids this by following orbits for four thousand steps before reading off a period, and its two disagreements are at the bifurcation points, where no finite number of steps suffices.

The match between copy and Mandelbrot set is shown, not proved, by the figures; the proof is Douady and Hubbard’s theory, and the figures check its predictions — the outline under one affine map, and the doubled periods along a line. Whether the copy is exactly the Mandelbrot set up to a continuous deformation, boundary and all, is a further question that the theory addresses for copies of particular kinds; the figures do not test it, and the bending visible at the edges of the outline is a reminder that “copy” means the same structure, not the same shape.

Still open: what the pictures inherit #

The copies carry the Mandelbrot set’s open problems with them. Whether the Mandelbrot set is locally connected — the conjecture known as MLC, open since the 1980s, which would give a complete topological description of the set — is not known, and the same question for each copy is the same question. Whether its interior consists entirely of components where a critical point is attracted to a cycle, the density of hyperbolicity, is likewise open; for the cubic Newton family it would say that every bad cubic in the interior of a speck has a trap of the kind drawn here.

There are questions native to Newton’s method too. The fraction of the parameter plane occupied by dark specks — the probability that a randomly chosen cubic from a region has a trapped critical point — has been estimated numerically for this family and not computed exactly, and its behaviour for higher-degree polynomials, whose Newton maps have several free critical points and parameter spaces of several complex dimensions, is far less explored. Covering rather than avoiding answered the practical question — a guaranteed list of starting points — by sidestepping it; the geometric question of what the bad polynomials look like, in any degree above three, is still mostly a matter of pictures.

The shape that keeps turning up #

The Mandelbrot set was defined by asking a question about one family of maps, $z^2 + c$. It turns up here as the answer to a question about a different family, Newton’s method for cubics, asked for a practical reason: for which cubics does a standard algorithm fail? The answer is that the failing cubics are arranged as copies of that one set, and the reason is that near any trapped critical point, enough steps of any analytic map behave like a quadratic polynomial.

So the set is not a curiosity of one formula. It is the shape of the boundary between order and failure whenever a single critical point decides a system’s fate — which is why it appears among the cubics, why it appears in families nobody designed to contain it, and why a numerical analyst choosing a starting guess for Newton’s method is, without knowing it, choosing a point in a picture that also contains the set drawn on the cover of every book about fractals.