The silver mean at the flat point

Two constants at the flat point measured the circle map

$$f(\theta) = \theta + \Omega - \frac{K}{2\pi}\sin 2\pi\theta$$

at the setting $K = 1$, where its graph is flat at one point, and at rotation numbers that approach the golden mean through the ratios of Fibonacci numbers, $1/2, 2/3, 3/5, 5/8, \ldots$. The parameters $\Omega$ at which those rotation numbers lock in — at which the flat point’s orbit closes up after $q$ steps having gone round $p$ times — crowd towards the golden parameter, and the gaps between them shrink by a factor of 2.8336 each time. The orbit’s closest returns to the flat point shrink by 1.2886. Both numbers are universal: a second family of maps with the same kind of flat point gives them again.

That essay ended by asking about every other rotation number. The golden mean is the simplest irrational, its continued fraction all ones, and the renormalisation that explains its constants repeats one step for ever. A number like $\sqrt2 - 1 = [0; 2, 2, 2, \ldots]$, the silver mean, also has a continued fraction that repeats a single term, so the same theory applies with a different step. The question is whether it gives the same constants or different ones, and the computation answers it at once.

The plan of the measurement is the golden one, repeated with a different sequence of fractions. Find the parameters at which the flat point locks into each convergent’s rotation, measure how fast they crowd together and how fast the orbit’s returns shrink, and repeat in a second family of maps to see whether the numbers belong to the map or to something deeper. Then change the rotation number again, in two instructive ways, to see what the numbers depend on.

Convergents of the silver mean #

The convergents of $[0; 2, 2, 2, \ldots]$ are $1/2, 2/5, 5/12, 12/29, 29/70, 70/169, \ldots$ — each numerator is the previous denominator, and each denominator is twice the previous one plus the one before. These are the Pell numbers, and they are to the silver mean what the Fibonacci numbers are to the golden: their ratio tends to $1 + \sqrt2 = 2.4142$, the silver ratio, and their fractions approach $\sqrt2 - 1 = 0.41421$ alternately from above and below, each better than any fraction with a smaller denominator.

For each convergent $p/q$ the figures find the parameter $\Omega$ at which the map’s flat point returns to itself after $q$ steps having gone round exactly $p$ times — the superstable parameter in the middle of the corresponding plateau of the devil’s staircase. Each is found by bisection, starting from a guess extrapolated from the previous two, and the search stops when the next gap would be smaller than $10^{-10}$, below which double-precision arithmetic cannot separate the parameters reliably. On the critical line the parameters converge to $\Omega = 0.4188649864$; at $K = 0.5$, to $0.4164263849$. Both are close to the rotation number itself, as they must be for a map close to a rotation.

The same numbers that solve Pell’s equation #

The Pell numbers are named for the equation they solve. The convergents $1/2, 2/5, 5/12, 12/29$ of $\sqrt2 - 1$ become, after adding one, the fractions $3/2, 7/5, 17/12, 41/29$ that approach $\sqrt2$ itself, and their numerators and denominators satisfy $x^2 - 2y^2 = \pm 1$: $9 - 8 = 1$, $49 - 50 = -1$, $289 - 288 = 1$. Every solution of that equation comes from the smallest by repeated multiplication by $3 + 2\sqrt2 = (1 + \sqrt2)^2$, the one solution that makes all the others. That factor is exactly the gap ratio below the critical line.

So the plateaus of the devil’s staircase that crowd towards the silver parameter are indexed by the solutions of Pell’s equation for 2, and below the critical line they crowd at the rate of Pell’s fundamental unit. On the critical line the indexing survives — the plateaus are still those of the convergents — and the rate does not: the flat point replaces $5.8284$, the unit squared, by $6.7992$, a number that has nothing to do with $\sqrt2$ that anyone has found. Arithmetic decides which parameters are special at every $K$; dynamics decides how fast they crowd, and on the critical line dynamics wins.

Finding a parameter on the critical line #

Each superstable parameter is a definite number, and finding it is a small exercise in the intermediate value theorem. For a convergent $p/q$, run the map $q$ times from the flat point and record how far round the circle the orbit has gone, counting whole turns; at the superstable parameter that total is exactly $p$. As $\Omega$ increases the total increases continuously, so it crosses $p$ at a single point, and bisection finds it: halve the interval, keep the half where the crossing is, and repeat until the interval is as small as the arithmetic allows.

The orbit there is not merely periodic. Because it passes through the flat point, where the map’s derivative is nought, the derivative of the $q$-fold composition along the orbit is nought too, so the periodic orbit attracts its neighbours as strongly as possible: hence superstable. Every plateau of the staircase on the critical line contains such a parameter, inside it rather than at an edge, and the gap between two superstable parameters is a stand-in for the distance between plateaus.

Below the critical line: the square of the ratio #

At $K = 0.5$ the map is a smooth, invertible map of the circle, and at an irrational rotation number it is a rigid rotation in different coordinates. Near such a number the parameter is a smooth function of the rotation number — how smooth the disguise is — so the parameter’s distance from its limit is proportional to the rotation number’s distance from its limit. For a convergent $p_n/q_n$ that distance is about $1/(q_n q_{n+1})$, and consecutive distances shrink by the ratio of consecutive denominators squared: by $\varphi^2 = 2.6180$ for the golden mean and by $(1 + \sqrt2)^2 = 5.8284$ for the silver.

The hero’s dashed line shows exactly that. The gaps between successive superstable parameters at $K = 0.5$ shrink by a factor that settles at 5.8284 to the four decimals measured. Nothing about the sine in the map matters below the critical line; only the arithmetic of the rotation number does.

On the critical line: a new number #

At $K = 1$ the map has a flat point of cubic type — its derivative vanishes at $\theta = 0$ and its graph has an inflection there — and the smooth conjugacy to a rotation breaks down. The gaps shrink faster, and by a different factor.

For the silver mean the ratio settles at $-6.7992$, the sign recording that the parameters approach their limit alternately from either side. For the golden mean it is $-2.8336$, the number from before. The two are not related by any simple formula: $6.7992$ is not $(1 + \sqrt2)^2$, nor the square of anything obvious, nor $2.8336$ raised to a power in proportion to the two ratios. It is a new constant, and it is universal in the same sense as the golden one. The second family of maps,

$$\theta + \Omega - \frac{1}{2\pi}\Big[(1 + b)\sin 2\pi\theta - \frac{b}{2}\sin 4\pi\theta\Big],$$

has the same kind of flat point at $\theta = 0$ but a different shape everywhere else. Its gap ratios start further from the limit — $-8.02$ at the first step against $-7.07$ for the sine map — and approach the same $-6.80$ more slowly.

The second constant measures the orbit rather than the parameters.

At each superstable parameter, the flat point’s orbit comes closest to its starting point after a number of steps equal to the previous denominator, and the distance of that closest return shrinks from one convergent to the next by a factor that settles at $-1.5868$ for the silver mean and $-1.2886$ for the golden. Below the critical line the corresponding factors are the ratios themselves, $1 + \sqrt2$ and $\varphi$, since a rigid rotation’s closest returns shrink exactly that way. On the critical line the flat point squeezes the orbit near itself, and the returns shrink more slowly.

Why there is a constant at all #

The constants come from renormalisation, the same idea that explains the period-doubling cascade. Take the map near its flat point and look at the first return of points to a small interval round it: points go round the circle, and the map that sends each point of the interval to where it next lands in the interval is a new map of a smaller piece. Rescale that piece to unit size and the new map is again a circle map with a flat point, with a rotation number obtained from the old one by removing the first term of the continued fraction. For the golden mean, removing a 1 from $[0; 1, 1, 1, \ldots]$ leaves the same number, so the renormalised map has the same rotation number as the original, and repeating the operation converges to a fixed point — a single map, the same for every family with a cubic flat point, whose features are the universal constants. For the silver mean, removing a 2 from $[0; 2, 2, 2, \ldots]$ also leaves the same number, so again there is a fixed point. But the operation is different — it composes the map with itself twice before rescaling, where the golden step composes once — and so is the fixed point, and so are its constants.

The gap ratio is the expanding eigenvalue of the renormalisation at its fixed point, and the return ratio is the scale of the rescaling. Both are properties of the fixed point alone, which is why every map with a cubic flat point that can be renormalised towards it shows the same numbers. Stellan Ostlund, David Rand, James Sethna and Eric Siggia, and independently Mitchell Feigenbaum, Leo Kadanoff and Scott Shenker, set out this picture in 1982 and 1983, and Michael Yampolsky proved in 2002 that renormalisation of critical circle maps is hyperbolic, which is what makes the numbers well defined.

A tail that takes over #

If the constants belong to a fixed point of renormalisation, and renormalisation removes the continued fraction one term at a time, then the constants should depend only on the tail of the continued fraction. A rotation number that begins with a few twos and then has ones for ever should look silver at first and golden in the end.

It does. For $[0; 2, 2, 2, 2, 1, 1, 1, \ldots]$ the first two gap ratios are $-7.07$ and $-6.74$, near the silver number; then come $-4.13$ and $-2.54$ as the expansion switches; and from there the ratios settle on the golden $-2.8337$. Every rotation number whose continued fraction ends in ones — the noble numbers — has the golden scaling in the end, which is why the golden constants turned up so persistently in early experiments that were not looking for them: the last circle to break in the kicked rotor is golden, and so are the invariant circles that survive longest near it, because noble numbers are the hardest to approximate by fractions.

A block of two terms #

A rotation number whose continued fraction repeats a block of two different terms, such as $[0; 1, 2, 1, 2, \ldots]$, renormalises to a different number after one step and back after two. There is no fixed point, but there is a cycle of period two, and the constants should alternate.

They do: the ratios alternate between $-4.7886$ and $-3.6899$ from the third convergent on, and their product, 17.669, is the factor by which the gaps shrink each time the block repeats. That product is the analogue of $6.7992$ for this number, a property of the period-two cycle of renormalisation. Every quadratic irrational — every number whose continued fraction is eventually periodic — has such a cycle, and with it a set of universal numbers, one for each place in its repeating block.

Above the line #

The measurements stop at $K = 1$ because the question changes above it. For $K > 1$ the map folds the circle back on itself and is no longer invertible, and a whole interval of rotation speeds is possible, depending on where an orbit starts; the single rotation number that organised everything below the line no longer exists. The critical line is the boundary between the two regimes, and the universal constants are properties of that boundary in the same way that Feigenbaum’s constants are properties of the boundary between order and chaos in the period-doubling cascade. Both are measured at the edge, where a map is about to lose a property it had, and both are numbers that the map’s formula does not contain.

The numbers side by side #

The table makes the contrast plain. Below the critical line, the golden and silver gap ratios are $\varphi^2$ and $(1 + \sqrt2)^2$, read straight off the continued fractions. On it, they are 2.8336 and 6.7992, numbers with no known closed form, and the return ratios 1.2886 and 1.5868 likewise. The flat point replaces simple arithmetic by a fixed point of an infinite-dimensional operator, and that fixed point has been computed numerically to many digits but never written down.

Four decimals from double precision #

The digits are limited by the arithmetic. Every parameter is found in double precision, and the search stops when successive parameters are $10^{-10}$ apart. The silver gap ratio is therefore known here to about four decimals, and the second family’s ratio has not yet converged when the search stops. High-precision computations of the renormalisation fixed point give these constants to many more digits; the figures give what an honest double-precision experiment can.

Parameters, not plateaus. The gaps measured are between the superstable parameters, one per plateau. The plateaus’ widths and edges — where, as the essay on how a lock comes apart showed, two periodic orbits collide and vanish — are a second set of measurements, with their own scaling, and the figures do not make them. The universality argument predicts that they too shrink by the silver factor on the critical line, since a plateau’s width is a feature of the same renormalised map; the figures test the prediction only for the parameters.

Only cubic flat points. A map whose graph flattens more sharply at its critical point — with a vanishing derivative of higher order — belongs to a different universality class, with different constants for the same rotation number. The two families here both have the cubic kind.

Only quadratic irrationals. The constants exist because the continued fraction repeats. For a rotation number whose continued fraction does not — $e - 2$, say, or $\pi - 3$ — renormalisation never settles into a cycle, and the gap ratios wander for ever without converging. They can still be measured, convergent by convergent, but there is no single number to report.

Still open: every rotation number at once #

The golden and silver constants are two points of a much larger structure. Renormalisation acts on critical circle maps by shifting the continued fraction of the rotation number one term to the left, which is exactly what the Gauss map $x \mapsto \{1/x\}$ does to numbers. So the renormalisation dynamics contains a copy of the Gauss map, and its periodic orbits — the quadratic irrationals — carry the constants measured here. What is the distribution of the gap ratios for a typical rotation number, chosen at random rather than with a repeating expansion? Each ratio depends on the coming terms of the continued fraction, which for a random number follow the Gauss–Kuzmin law, so the ratios themselves should follow a law. That law, and how it relates to the universal numbers of the periodic cases, is understood in outline and not computed. And one dimension further, whether the plateaus of locked rotation on the critical line — which fill the whole parameter line except for a set of measure nought and dimension 0.87 — have a structure organised by these same constants, rotation number by rotation number, is the question that ties this measurement to the staircase it started from.