How smooth the disguise is

The circle map $f(x) = x + \Omega - \frac{K}{2\pi}\sin 2\pi x$, at a drive $\Omega$ where it does not lock, is a rotation in different coordinates. Denjoy’s theorem supplies an increasing continuous function $\varphi$ of the circle with $f(\varphi(\theta)) = \varphi(\theta + \rho)$, where $\rho$ is the map’s irrational rotation number: in the coordinate $\theta$ the map is simply the step $\theta \mapsto \theta + \rho$. That essay built $\varphi$ by sorting one long orbit round the circle, and it closed on a question the construction could not answer. Sorting produces a continuous function. It says nothing about whether $\varphi$ has a derivative, and whether it does turns out to depend on the arithmetic of $\rho$.

The reason is short enough to state at once. To solve for $\varphi$ harmonic by harmonic, the $k$-th Fourier coefficient has to be divided by $e^{2\pi i k \rho} - 1$. That number is small whenever $k\rho$ is close to a whole number — whenever $\rho$ is close to a fraction with denominator $k$ — and since every irrational number has fractions arbitrarily close to it, the divisors come arbitrarily close to nought. This is the small-divisor problem, which Henri Poincaré identified in celestial mechanics in the 1890s and which Andrey Kolmogorov, Vladimir Arnold and Jürgen Moser learned to control sixty years later.

The circle map is the cleanest place to watch it happen, because here the change of coordinates can be solved for directly, on a computer, to fifteen digits, and its Fourier coefficients read off one by one. The figures below do that for the golden rotation number and for a number lying very close to $5/8$, and the difference between the two is the whole subject.

An equation for the change of coordinates #

Write the unknown change of coordinates as the identity plus a periodic correction, $\varphi(\theta) = \theta + u(\theta)$. Substituting into $f(\varphi(\theta)) = \varphi(\theta+\rho)$ and cancelling $\theta$ from both sides leaves one equation for $u$:

$$u(\theta + \rho) - u(\theta) = \Omega - \rho - \frac{K}{2\pi}\sin 2\pi\big(\theta + u(\theta)\big).$$

The left side compares $u$ at two points a rotation apart; the right side is the map’s own departure from a rotation, evaluated at the point $\varphi(\theta)$. The drive $\Omega$ appears as an unknown too. Fix the rotation number at the golden value $\rho = (\sqrt 5 - 1)/2 = 0.618\ldots$ and choose $K$, and there is exactly one drive at which the map has that rotation number — the staircase of the staircase that is flat almost everywhere rises strictly through every irrational value — so the equation determines $u$ and $\Omega$ together.

The figure solves it. The correction $u$ is sampled at 128 equally spaced points, the shift by $\rho$ is carried out exactly on its Fourier series, and the whole system — 128 values and the drive — is handed to Newton’s method, starting from $u = 0$ at small $K$ and carrying each solution up to the next value. At every nonlinearity drawn, the equation is satisfied at every sample to thirteen digits or better.

At $K = 0.3$ the correction is nearly a single sine wave of height about $0.025$: the map is almost a rotation already, and the coordinates barely need adjusting. At $K = 0.6$ it is larger and leans. At $K = 0.9$ it has visible wrinkles, and its slope ranges from $0.29$ to $2.94$ — the new coordinate stretches some arcs of the circle ten times as much as others. It is still smooth. The question is why, and how long it can stay so.

Dividing by numbers close to nought #

Take $K$ small, so that $u$ is small, and drop everything quadratic in small quantities. The equation becomes linear, $u(\theta+\rho) - u(\theta) = g(\theta)$, with $g$ the map’s departure from the rotation. Expanding both sides as Fourier series, $u(\theta) = \sum_k u_k e^{2\pi i k\theta}$, the shift multiplies the $k$-th coefficient by $e^{2\pi i k \rho}$, and the equation separates into one equation per harmonic:

$$u_k\,\big(e^{2\pi i k\rho} - 1\big) = g_k, \qquad\text{so}\qquad u_k = \frac{g_k}{e^{2\pi i k\rho} - 1}.$$

The constant harmonic $k = 0$ has divisor nought, so $g_0$ must vanish; that condition is what fixes the drive. Every other harmonic is solved by division, and the size of the divisor is $\lvert e^{2\pi i k\rho} - 1\rvert = 2\lvert\sin \pi k\rho\rvert$, which is about $2\pi$ times the distance from $k\rho$ to the nearest whole number. It is small exactly when $\rho$ is close to a fraction $p/k$, and the best approximations to a number are what make it small.

The two sets of divisors behave very differently. For the golden number the smallest divisors come at the Fibonacci numbers $k = 13, 34, 89$, because the ratios of consecutive Fibonacci numbers are its best approximations, and even there the divisor is no smaller than about $1.86/k$. It falls, but only as fast as $1/k$. For the second number, which agrees with the golden number’s continued fraction until a single large entry, the divisor at $k = 8$ is already $0.0078$: eight times the rotation number lies within $0.0012$ of the whole number $5$, because the number is within $0.00016$ of $5/8$. Every multiple of 8 inherits the near-miss, and the divisor at $k = 8m$ is about $m$ times the one at $k=8$ until $m$ is large.

A divisor that falls like $1/k$ multiplies the $k$-th coefficient by at most a constant times $k$, and multiplying by $k$ is what differentiating once does to a Fourier series. So for the golden number the linear equation costs at most one derivative: if $g$ has $r$ derivatives, $u$ has about $r - 1$. A number whose divisors fall like $1/k^{\tau}$ costs about $\tau$ derivatives. A number whose divisors fall faster than every power of $1/k$ — a Liouville number, approached by fractions faster than any polynomial rate — can destroy every derivative the map had.

What the golden number buys #

The circle map is not merely smooth. The sine in it is analytic, and an analytic periodic function is characterised by its Fourier coefficients falling geometrically, like $e^{-2\pi\sigma k}$ for some $\sigma > 0$. The number $\sigma$ is the half-width of the strip round the real axis into which the function extends as an analytic function of a complex variable; the wider the strip, the faster the coefficients fall. The map’s own coefficients are nought beyond $k = 1$, and the only thing limiting its strip is the equation itself: the map stops being invertible where its derivative $1 - K\cos 2\pi z$ vanishes, and for real $z$ that never happens while $K < 1$, but for complex $z$ it happens at height $\operatorname{arccosh}(1/K)/2\pi$.

If the divisors fall like $1/k$ and the right-hand side’s coefficients fall like $e^{-2\pi\sigma k}$, the quotient falls like $k\,e^{-2\pi\sigma k}$, and that is still geometric, at any rate slightly slower than $e^{-2\pi\sigma k}$. So the golden conjugacy should be analytic, with a strip a little narrower than the map’s.

Each series is a straight line on the logarithmic scale, which is geometric decay and so analyticity, read off the computed function rather than inferred. At $K = 0.2$ the coefficients fall by a factor of about $6.4$ per harmonic and reach the arithmetic’s floor near $10^{-16}$ by the eighteenth; at $K = 0.95$ they fall by only about ten per cent per harmonic, and the hundredth coefficient is still near $10^{-7}$. None of the lines kinks upward at a Fibonacci harmonic. The divisors do dip there, but dividing by $1.86/89$ multiplies by about fifty, which is nothing against a series that has already fallen by twenty orders of magnitude.

This is the same arithmetic that made the last circle to break the golden one in Chirikov’s rotor. There the question is whether an invariant curve survives a kick; here it is whether a coordinate change stays smooth as a map is pushed away from a rotation. Both are small-divisor problems and both are hardest to break at the number that is hardest to approximate.

A strip that closes at the critical line #

Reading $\sigma$ off each fitted line and plotting it against $K$ shows the strip closing. At $K = 0.1$ the conjugacy extends about $0.42$ of a period into the complex plane on either side; at $K = 0.95$ it extends $0.015$. The map’s own strip closes too, since its critical point descends towards the real axis as $K$ rises to 1, and the conjugacy’s strip stays inside it at every $K$ measured, by a margin that is widest in relative terms near the critical line.

The two do not close at the same rate. The map’s strip shrinks like the square root of $1 - K$; the conjugacy’s shrinks faster, from $0.025$ at $K = 0.9$ to $0.015$ at $K = 0.95$, closer to the three-quarters power over that range. The conjugacy absorbs the map’s singularity and the rotation number’s arithmetic together, and the two compound.

The same closure is visible in real terms. The ratio of the conjugacy’s steepest slope to its shallowest — how unevenly it stretches the circle — is $1.12$ at $K = 0.1$ and $20$ at $K = 0.95$, and it grows faster the closer $K$ comes to 1. At $K = 1$ it has no finite value. The map then has a point where its derivative vanishes, it is no longer a diffeomorphism, and the conjugacy — which Yoccoz showed in 1984 still exists for every irrational rotation number of such a map — is continuous and increasing but singular: Jacek Graczyk and Grzegorz Świątek proved in 1993 that it carries the rotation’s evenly spread orbit onto a distribution concentrated on a set of length nought. At the critical line the disguise stays on, and it stops being smooth anywhere.

A rotation number that nearly repeats #

Now hold $K$ at $0.6$, where the golden conjugacy is comfortably analytic, and move the rotation number instead. The family of numbers with continued fraction $[0; 1, 1, 1, 1, 1, a, 1, 1, \ldots]$ agrees with the golden number for five steps, whose last convergent is $5/8$, and then has a single entry $a$. For $a = 1$ it is the golden number. As $a$ grows the number moves towards $5/8$ — within $0.0014$ at $a = 10$, within $0.00016$ at $a = 100$, within $0.000016$ at $a = 1000$ — and the divisor at harmonic 8 shrinks in proportion.

The coefficients show the divisor directly. Each tenfold increase in $a$ multiplies the eighth coefficient by about ten, exactly as dividing by a divisor ten times smaller should, and the sixteenth, twenty-fourth and later multiples of 8 rise with it. Between the spikes the series still falls geometrically, so every conjugacy in the figure is analytic. But the decay along the multiples of 8 slows as $a$ grows, and at $a = 1000$ the sixty-fourth coefficient is still above $10^{-5}$. The strip of analyticity is being pinched by one fraction.

In real terms the conjugacy is being crushed. At the golden number its slope ratio at $K = 0.6$ is $2.5$; at $a = 1000$ it is 57, with the slope falling to $0.13$ on one arc. The map is the same sine at the same $K$ in every case; only the drive differs, by $0.0065$ between the golden number and $a = 1000$ and by about $0.0001$ between $a = 100$ and $a = 1000$, and the rotation number is irrational throughout. What has changed is that the orbit very nearly repeats after eight steps — the map is sitting just outside the tongue of the fraction $5/8$, where an attracting and a repelling orbit of period eight are born together — and a map that nearly has an attracting cycle spends a long time on the arcs near it. A conjugacy to an even rotation has to compress those arcs, and the closer the near-cycle comes to existing, the harder it has to compress them.

The limit is where Liouville numbers live. A number whose continued fraction contains ever-larger entries, growing fast enough, comes close to one fraction after another and never recovers. In 1961 Arnold built analytic circle diffeomorphisms with such rotation numbers whose conjugacy to the rotation is not even absolutely continuous: it is continuous, as Denjoy promised, and it carries a set of length nought onto a set of positive length, which no differentiable change of coordinates can do.

Why the corrections do not pile up #

The linear equation is only the first step, and the danger is in what comes after. Solving it produces a correction whose size has been inflated by the divisors; substituting the corrected $u$ back into the full equation leaves an error of roughly the square of that correction, and solving for the next correction divides by the same divisors again. Carried out as a power series in $K$ — the approach of the nineteenth-century astronomers — the $n$-th term contains products of $n$ divisors, and whether such a series converges was the question Poincaré could not settle. Carl Ludwig Siegel settled the analogous question for a fixed point of a complex map in 1942, by estimating the products term by term.

Kolmogorov’s idea, carried through by Arnold for exactly this problem, was to use Newton’s method instead. Each step linearises the equation at the current approximation, solves it harmonic by harmonic, and pays for the divisors with a loss of some strip width. But Newton’s method squares the error at each step, as a double root halves the error shows it does near any simple root, and squaring beats any fixed loss: an error of $10^{-4}$ inflated a hundredfold and then squared is $10^{-4}$ again, but an error of $10^{-8}$ becomes $10^{-12}$. Arnold’s proof loses a little strip at every step, arranges the losses into a convergent sum, and ends with an analytic conjugacy on a narrower strip.

The computation that drew every figure here is that argument made literal. From the identity at $K = 0.6$, the golden case reaches a residual of $2 \times 10^{-16}$ in four steps: $0.1$, then $7 \times 10^{-3}$, $4 \times 10^{-5}$, $9 \times 10^{-10}$. The number near $5/8$ takes nine. For three steps the residual rises instead of falling — each correction overshoots on the eighth harmonic because it was divided by $0.0008$ — and then, once the approximation is close enough that the squared error beats the divisor, the residual falls to $5 \times 10^{-7}$, $9 \times 10^{-10}$ and $3 \times 10^{-16}$ at the same doubling rate as the golden case. The divisor delays convergence without preventing it, which is Arnold’s theorem in a single plot.

Arnold, Herman and Yoccoz #

Arnold’s theorem of 1961 is local. It says that an analytic circle diffeomorphism close enough to a rotation, with a rotation number satisfying a Diophantine condition — the distance from $k\rho$ to the nearest whole number at least $c/k^{\tau}$ for some constants — is analytically conjugate to the rotation. “Close enough” depends on the constants, and it is exactly the regime the Newton iteration above lives in.

The global question — whether every smooth diffeomorphism with a Diophantine rotation number is smoothly conjugate, however far from a rotation it is — was Arnold’s conjecture, and Michael Herman proved it in 1979 for almost every rotation number. Jean-Christophe Yoccoz extended it in 1984 to all Diophantine numbers. For the circle map it means that the golden conjugacy is analytic at every $K$ below 1, however close — the strip in the figure narrows but never closes before the critical line, and no Newton iteration is needed to know it.

The sharp statements came later. For finitely smooth maps, Yitzhak Katznelson and Donald Ornstein showed in 1989 how many derivatives are lost for a given exponent $\tau$, close to the count the linear equation suggests. And in 2002 Yoccoz identified the exact arithmetic condition under which every analytic diffeomorphism with rotation number $\rho$ is analytically conjugate to the rotation: a condition on the continued fraction, stronger than the one that suffices for maps close to a rotation. The division in the linear equation turned out to be the whole story, sharpened into a theorem about which continued fractions it tolerates.

What the pictures cannot show #

Every figure is computed from 128 or 256 samples of the conjugacy, which means its Fourier series is truncated at the 64th or 128th harmonic. The truncation is checked, not assumed: the near-$5/8$ case at $a = 1000$ was solved again with 512 samples, and every coefficient in the figure agreed to three figures. It is still a truncation. A conjugacy with a singular point, as at $K = 1$, cannot be represented this way at all, which is why the figures stop at $0.95$.

The strip widths are fitted slopes. A straight line through the logarithms of coefficients from the third harmonic to the rounding floor gives a single decay rate, and a real conjugacy may decay at slightly different rates in different ranges of $k$ — as the near-$5/8$ cases do, with one rate along the multiples of 8 and another between them. The fitted number is the rate that dominates the computed range.

And the rotation numbers are floating-point numbers. The golden number is represented to sixteen digits, and a sixteen-digit number is a fraction. The equation cannot see the difference as long as the harmonics it uses are far fewer than the denominator, which holds by fifteen orders of magnitude; but no computation distinguishes a Diophantine number from a Liouville number, and that distinction is exactly what the theorems are about.

Still open: how fast the strip closes #

The strip of analyticity of the golden conjugacy vanishes as $K$ rises to 1, and the figure suggests it does so faster than the map’s own strip — something like the three-quarters power of $1 - K$ over the last range measured, against the map’s one-half. Whether it follows a power law at all as $K \to 1$, and if so with what exponent, is not something these computations settle, and it is not a result with a standard statement.

The renormalisation theory of critical circle maps describes what happens exactly at $K = 1$. Its fixed point governs the golden conjugacy’s singularity there, and its scaling constants — measured by Scott Shenker in 1982 and explained the same year by Mitchell Feigenbaum, Leo Kadanoff and Shenker himself, and by Stellan Östlund, David Rand, James Sethna and Eric Siggia — describe how the orbit’s gaps contract. Whether that theory also predicts the rate at which analyticity is lost on the approach to the critical line is a natural question, and the answer would connect the two regimes these essays on the circle map have treated separately: the smooth, analytic side where Arnold’s and Herman’s theorems hold, and the critical line where only topology survives.

A second question is more classical. Yoccoz’s condition says exactly which rotation numbers force an analytic conjugacy for every analytic diffeomorphism. For a specific family like this one, with its specific sine, the set of rotation numbers at which the conjugacy fails to be analytic is not known to coincide with the complement of Yoccoz’s set; a particular family can be better behaved than the worst case, and which rotation numbers are genuinely bad for the circle map is a finer question than the theorem answers.

What smoothness is a measure of #

The conjugacy is the same object in every figure: the map’s dynamics, rewritten as a rotation. Its existence is a fact about orders — Denjoy’s theorem needs only that the map preserves the circular order of points and that its rotation number is irrational. Its smoothness is a fact about arithmetic. How well the rotation number can be approximated by fractions decides the sizes of the divisors, the divisors decide how much of the map’s smoothness survives the division, and Newton’s squaring decides that the loss, for well-behaved numbers, stops at a finite amount.

So the golden number is doing the same job here as at the top of the rotor’s phase portrait and in three gaps and no more. It is the number every fraction approximates worst, and wherever a process has to divide by the quality of an approximation, it is the number that pays least.