A disc that turns for ever

Take a complex number $\lambda$ of size exactly one, so that multiplication by it is a rotation, and consider the map $f(z) = \lambda z + z^2$. Near nought the square is negligible and the map looks like the rotation. So the question is what the small correction does over infinitely many steps. It might, little by little, spiral points outward or inward. It might tear the neighbourhood of the fixed point apart. Or it might do nothing that matters, and there might be a change of coordinates in which the map is exactly the rotation.

The fraction written on every bulb followed what happens when the rotation is a rational fraction of a turn: at $e^{2\pi i p/q}$ the fixed point sheds a cycle of $q$ points and a bulb of the Mandelbrot set is attached. That left unanswered what happens at the irrational angles between the bulbs. The answer is that at most of them the map really is a rotation in disguise, on a whole disc round the fixed point. The answer depends on how closely the angle can be approached by fractions, and for this one family of maps the exact arithmetic condition is known.

Looking for the coordinates that straighten the map #

Suppose there is a change of coordinates $h(w) = w + h_2 w^2 + h_3 w^3 + \cdots$ which turns the map into a rotation, meaning that $f(h(w)) = h(\lambda w)$. If it exists, its coefficients are forced. Comparing powers of $w$ on the two sides gives

$$h_n = \frac{\sum_{k=1}^{n-1} h_k h_{n-k}}{\lambda^n - \lambda}, \qquad n \ge 2.$$

So there is exactly one candidate. Writing it down is easy. The only question is whether the series converges at all, and that is where the trouble starts. Every coefficient is divided by $\lambda^n - \lambda$, which is $\lambda(\lambda^{n-1} - 1)$, and this denominator has size equal to the distance of $\lambda^{n-1}$ from one. Since $\lambda$ is an irrational rotation, its powers return arbitrarily close to one. These are the famous small divisors, and each near-return divides a later coefficient by a tiny number. If $\lambda$ were a root of unity the denominator would be exactly nought at some $n$, and the series would simply stop existing, which is the arithmetic form of a bulb being born.

Near-returns occur, as how close a fraction can get showed, at the denominators of the continued fraction of the angle $\theta$. The distance is close to $2\pi/q_{k+1}$ when $n$ is the denominator $q_k$. A large digit in the continued fraction therefore means a very deep return.

For the golden number, whose digits are all 1, the deepest return below 200 comes at $n = 144$, a Fibonacci number, where $|\lambda^n - 1|$ is 0.0195. That is as far from one as any irrational rotation stays, and in a precise sense the golden number is the worst approximated of all numbers. For an angle whose continued fraction starts with the digits 12, 1 and 80, the return at $n = 13$ reaches 0.0059, more than three times deeper, much earlier and followed by a coefficient that must absorb the division. The figure contains nothing but those two sets of distances. Everything else that follows depends on them.

Running the series #

The series can simply be computed, as a thousand exact terms of the recursion in double precision, with its radius of convergence read by the root test from the sizes of the later coefficients. To make sure the computed series is really doing what is claimed, it was tested against the map itself. At a point well inside the disc, $f(h(w))$ and $h(\lambda w)$ agree to better than one part in ten billion.

The golden number gives a radius of 0.3284, the silver number $\sqrt 2 - 1$ very nearly the same, 0.3272. An angle with digits 4, 2, 7, 1 gives 0.3134. And the one with digits 12, 1, 80 is smaller at 0.2158. In all of these the root test finds a clear level, since these numbers were deliberately chosen with digits that eventually become all 1. Even with its deep return, then, the disc does not vanish; it just shrinks. The small steps early in each curve are the signature of the divisors. A coefficient jumps at a denominator of the continued fraction, and then the slow averaging of the root test absorbs the jump as $n$ grows.

The result of the calculation is the coloured curves in the first figure. Evaluating $h$ on a round circle in the rotating coordinate draws a closed curve in the original plane, and the map carries each such curve exactly onto itself. Points on one of those curves move round it by the same fraction of a turn every step. They never settle and never leave. Inside the Mandelbrot family, this is a parameter on the edge of the main cardioid with a filled picture of its own, and it is one of the rare pictures in which the interior is not attracted to anything.

These three pictures show the same map at three angles. The filled sets are similar in general form, with their boundary in grey and the disc in colour, and the sizes run from about 0.33 to 0.22 as the digits grow. What does not show here is what happens for angles nearer the fractions. Each fraction on the cardioid has its own bulb and its own cycle, and an angle extremely close to one begins to look like that cycle: the disc’s edge is pinched in towards a ring of $q$ points long before the angle reaches the fraction. So it makes sense that the radius should fall towards nothing as $\theta$ approaches any fraction.

A radius at every angle #

Sampling the radius across rotation numbers shows just that.

The picture is a sample rather than a function drawn. Arbitrarily close to any angle there are others with a far smaller disc, and even others with none, so the true graph is wildly discontinuous. What the sample does show is a surprisingly flat upper envelope, near 0.33 for much of the range. That envelope is one of the small surprises here. The worst approximable numbers do not make enormous discs, only moderately larger ones than a typical angle. No angle in this sample beats the golden number’s 0.3284, though the nearest comes within a thousandth. Whether the golden number really gives the largest disc of all has been examined numerically, and the sample cannot settle it: a hundred angles are nothing against a function this irregular.

The condition, and why it is exact #

Carl Ludwig Siegel proved in 1942 that the series converges whenever $\theta$ is Diophantine, meaning that $|\theta - p/q| \ge c/q^{\tau}$ for some constants and every fraction. This is true of almost every angle and of every algebraic irrational. Before that nobody had shown that a single such irrational rotation could be linearised at all, and it was the first success against small divisors. It preceded the theory of Kolmogorov, Arnold and Moser, which addressed the same type of denominators in mechanics and yielded the invariant circles that break last at the golden number. The problem there and the problem here have the same difficulty, and in both of them the golden number survives longest.

Alexander Brjuno then weakened the condition. Form the sum

$$B(\theta) = \sum_{k \ge 0} \frac{\log q_{k+1}}{q_k},$$

where the $q_k$ are the denominators of the continued fraction of $\theta$. Each term has a clear meaning, in that $\log q_{k+1}$ is how many digits the next deep return costs, and $1/q_k$ says how often it comes. Brjuno proved that if $B(\theta)$ is finite, the series converges. Then, in 1988, Jean-Christophe Yoccoz proved the converse for this quadratic map: if the sum is infinite, there is no disc. He received the Fields Medal in 1994 for this work among other things.

Yoccoz’s result is stronger than a dichotomy. Using a form $Y(\theta)$ of the same sum, computed from the Gauss map, he showed that $\log r(\theta) + Y(\theta)$ is bounded on the angles where the sum converges. So the radius of the disc is known, up to a constant factor, from the digits.

The figure is a test of that claim and an unusually fair one, since the two axes have been computed by procedures that have nothing to do with each other. One takes digits of a continued fraction and adds logarithms. The other runs a recursion in the complex numbers and looks at how big the terms get. And on 150 random angles, the sum of the two lies between 0.39 and 1.71. Its upper range reflects a real limit of this calculation: 800 terms of the series cannot feel a return at a denominator far beyond 800, whereas the digit sum can, so the computed radius of an angle with a very large later digit is too large and that point lies above the line. A line of slope minus one is a reasonable account of the rest.

Where the disc dies, and a proof by the pigeonhole #

Hubert Cremer had already seen the opposite phenomenon in the late 1920s, and his argument is short enough to give whole. Iterate the map $n$ times. The polynomial $f^n(z) - z$ has degree $2^n$. Its roots are the fixed point at nought and the periodic points of period dividing $n$. Dividing by $z$ leaves a polynomial whose top coefficient is one and whose bottom coefficient is $\lambda^n - 1$, so the product of its $2^n - 1$ roots has size $|\lambda^n - 1|$. If that is extremely small, one root must be small too: at least one periodic point lies within $|\lambda^n - 1|^{1/(2^n - 1)}$ of the origin.

Now choose the digits of $\theta$ to grow so fast that this quantity tends to nought along some sequence. Such $\theta$ exist, and they can be built explicitly, in the same spirit as Liouville’s numbers. Then periodic points accumulate at the fixed point. But no periodic point other than nought can lie in a Siegel disc, because every point of a rotating disc has an orbit going densely round a circle. So at such an angle there cannot be a disc. These are Cremer points, and what the dynamics near one looks like is known only through a strange object, Ricardo Pérez-Marco’s hedgehogs, the invariant sets that replace the disc and that have no interior and no good description by a picture.

It is one of the striking things about this result that a disc of positive radius is decided by something as small as which digits occur far out in a continued fraction, digits that no calculation to fixed precision can see. There is a further twist. Xavier Buff and Arnaud Chéritat used exactly this delicate dependence in the 2000s, tuning the rotation number to make the filled set bulge at its edge, to prove that there exist quadratic Julia sets of positive area. That had been an open problem for decades. The Siegel disc’s fragile dependence on digits was the lever for showing a fractal boundary can be fat. The area of the Mandelbrot set has the same difficulty one level up, though it is not resolved there.

What a floating-point picture cannot decide #

Nothing in a computed image can distinguish a Siegel angle from a Cremer angle. To the machine’s precision, any double-precision number is a rational number, a fraction with a denominator of about $2^{53}$, and strictly speaking the map at that number has a bulb and no disc. The images are trustworthy for the same reason that ordinary numerics on KAM curves are: at low order and at moderate radius, the series genuinely agrees with the calculated orbits, and the orbit diverges from the true disc only after a time set by the first enormous denominator. The digits drawn here were chosen with that in mind, with continued fractions that end in a long run of 1s.

That the critical point’s orbit runs along the disc’s edge is a theorem in this special case and not something visible at all. It says that for rotation numbers of bounded type, those whose digits are bounded, the edge is a quasicircle containing the critical point, as Michael Herman and Grzegorz Świątek established. For unbounded digits, whether the critical point lies on the boundary has been open for a long time, and the question had a real surprise in it. In 2004 Artur Avila, Buff and Chéritat proved that quadratic Siegel discs with perfectly smooth boundaries exist, for rotation numbers tuned with care. So the picture’s rough edge is a feature of this golden case, and not of all discs. A smooth edge cannot carry the critical point, and no picture could have revealed such a case, since the tuned digits lie far beyond anything double precision can represent.

What was computed and what was quoted #

The series coefficients were computed from the recursion, in double precision, with a scale factor to keep them in range, and the conjugacy was checked to ten digits at one point. The radii are root-test estimates from the upper half of 800 to 1,200 terms: they are finite-sample approximations, generally good to about two or three digits, and for angles with large far-out digits they are overestimates. The small divisors are exact arithmetic of an elementary function. The bound that the pairs of sum and radius satisfy was measured on the sample only, and the 150-angle figure is an illustration of Yoccoz’s theorem and not a verification.

The theorems of Siegel, Brjuno, Cremer and Yoccoz, those of Herman and Świątek, and those about smooth boundaries and positive area are quoted from the literature, though Cremer’s argument was given in full above. Facts about the location of the maximum of $r$ were also quoted.

Still open: does the edge have to be rough #

For which rotation numbers does the critical point lie on the boundary of the disc? For bounded digits it does, and for carefully chosen fast digits it does not. Nobody can state the exact arithmetic condition that separates the two cases, and this is the condition Yoccoz’s result has not delivered. The question has been asked since the 1980s, and it is still unsettled for a wide range of rotation numbers.

The biggest open question connected to this is the one the bulbs led to already, whether the Mandelbrot set is locally connected. Yoccoz himself proved local connectivity at a large class of parameters, by a method whose reach stops at the set’s small copies. The points of the cardioid’s edge are among the places where a proof needs fine control of exactly this dependence on digits. It is also unknown whether all quadratic polynomials with Siegel discs have a Julia set that is locally connected. In Cremer’s case, the Julia set is known not to be, by an argument of Sullivan and Douady. Beyond quadratic maps, the sharpness of the Brjuno condition is only proved in restricted classes: for a general rational map or a general holomorphic germ, nobody has shown that the condition is necessary. For the quadratic family it is a theorem, and one of the few places where a dynamical property of a whole family has an exact statement in terms of digits alone.

One sum of digits #

A rotation with a quadratic correction is secretly an exact rotation on a disc, if the series for the straightening coordinates converges, and that is decided by whether the sum of $\log q_{k+1}/q_k$ over the continued-fraction denominators is finite. When it is finite the disc exists and its size is known up to a constant. When it is infinite, periodic points or something stranger crowd in, and no disc can be drawn at all.

The figures show the mechanics of one such test: a list of denominators on one hand, a series whose terms each carry one of them, and a rotating disc that fits inside the filled set and shrinks as the digits grow. The result is one of the sharpest in dynamics. It is striking that a condition about how well a number can be approximated by fractions, which a table of convergents can compute by hand, is exactly the difference between a stable centre and an orbit structure that nobody can draw.