The staircase that is flat almost everywhere

Turn a circle by a fixed fraction of a turn at every step and the orbit is entirely predictable: a rational fraction closes into a cycle, an irrational one never repeats and fills the circle evenly. The average advance per step is the fraction itself, and there is nothing more to say.

Now let the amount of turn depend on where the point is. The standard version is

$$\theta \mapsto \theta + \Omega - \frac{K}{2\pi}\sin 2\pi\theta,$$

with $\Omega$ the underlying rotation and $K$ how strongly the position pulls it about. At $K = 0$ this is the rigid rotation. Above zero, something remarkable happens to the average.

The number that does not budge #

The rotation number is the limit of the total advance divided by the number of steps, and for the family above it exists for every parameter and rises as $\Omega$ rises — never falling, which the figures check on every consecutive pair of measurements rather than assuming.

What the figures show is that it rises in a peculiar way. Over an interval of parameters around each simple rational the number is exactly that rational and does not move at all. The map has locked onto a periodic orbit, and small changes to the parameter do not shift it.

The mechanism is a fixed point of an iterate. Inside the plateau the map applied twice has a fixed point — an actual solution of $f(f(\theta)) = \theta + 1$ — and that fixed point is attracting, so nearby orbits fall into it. A small change to $\Omega$ moves the fixed point a little and does not destroy it, because an attracting fixed point survives a perturbation whenever the slope at it is strictly inside the unit interval. The plateau ends where the slope reaches the boundary and the fixed point disappears.

That is the whole of mode locking: a rational average is protected by the stability of a periodic orbit, and an irrational average is not protected by anything.

Every rational gets an interval #

The plateaus are not confined to the halves and thirds a figure can label. Every rational gets one, and the widths shrink fast with the denominator.

Magnify any rising portion and it turns out to be another staircase, with the widest plateau at the rational of smallest denominator in the window. The construction is self-similar in the sense that matters: the same description applies at every scale, and there is no scale at which the rise becomes an honest slope.

The plateaus arrive in the order the mediant construction produces. Between the plateaus at $p/q$ and $r/s$, the widest is at $(p+r)/(q+s)$ — which is exactly the rule that generates the Stern–Brocot tree, and it is the same rule for the same reason: the fraction with the smallest denominator between two others is their mediant, and denominators control plateau width.

At $K$ below one, the plateaus have total width less than one and the set where the number is irrational has positive measure. At $K$ exactly one the plateaus fill the axis except for a set of measure zero, and the graph becomes the devil’s staircase proper: continuous, non-decreasing, rising from nothing to one, and with zero derivative almost everywhere.

Why the plateau has the width it has #

The width of a plateau is not arbitrary, and working out where one ends explains why the small-denominator rationals dominate.

Take the plateau at zero, which is the easiest. The rotation number is zero when the map has a fixed point — a $\theta$ with $\theta + \Omega - (K/2\pi)\sin 2\pi\theta = \theta$, which is to say $\Omega = (K/2\pi)\sin 2\pi\theta$. The right-hand side ranges over $[-K/2\pi, K/2\pi]$ as $\theta$ runs round the circle, so a fixed point exists exactly when $|\Omega|$ is at most $K/2\pi$, and the plateau has width $K/\pi$. It opens linearly from a point as $K$ rises, which is the tongue’s shape at its root.

For the plateau at $1/2$ the same calculation is about the second iterate, and the sine has been composed with itself: the parameter window in which the second iterate has a fixed point is narrower, of order $K^2$. For $p/q$ it is of order $K^q$.

That single exponent explains everything about the picture. At small $K$ a plateau of denominator seven is $K^7$ wide, which is nothing, and the staircase looks like a line with a few steps. As $K$ approaches one all the powers approach one another and the plateaus become comparable, which is when they fill the axis. The hierarchy of denominators is a hierarchy of powers of $K$.

The parameter plane #

Sweeping $\Omega$ at one value of $K$ gives one staircase. Sweeping both gives the picture the phenomenon is usually named after.

Each plateau, followed as $K$ increases from zero, opens into a wedge — an Arnold tongue — that starts as a point on the $K = 0$ axis and widens. At small $K$ the tongues are thin and mostly separated; at $K = 1$ they touch, and their total width covers the axis; above one they overlap, which is where two different periodic behaviours are available at the same parameters and the system can be chaotic.

The measured share of the axis that is locked is what the figure reports: a few per cent at the smallest nonlinearity drawn, about half at the largest.

Where the two parameters come from, and what they are not #

The family above has two knobs, and it is worth saying where the shape of it comes from — with the boundary stated in the same breath, because the answer reaches straight out of this subject.

Take any pair of quantities that each return to their starting state periodically, at their own rates, with each nudging the other a little. Record the state of the first every time the second completes a cycle. What remains is a point on a circle and a rule for advancing it — a first-return map — and to lowest order that rule is a rotation plus a term proportional to a sine. $\Omega$ is the ratio of the two rates and $K$ is the strength of the nudge.

Nothing on this page is a claim about any such pair. The reduction is a construction that yields a circle map, and once the map is in hand this essay works on the map: the rotation number is defined from the map, the plateaus are measured from the map, and every number in every figure is generated from the stated rule rather than from anything observed. Which systems reduce this way, and what the reduction leaves out, is a question about those systems and belongs to whoever owns them.

What the map does say, on its own terms, is worth stating precisely, because it is what makes a plateau interesting rather than a curiosity. A plateau is an interval of $\Omega$ on which the rotation number is exactly $p/q$. Locking at a rational is therefore not a property of a single finely tuned ratio; it is a property of a range, and the range has positive width for every $K$ above zero.

Where the irrational rotations went #

The unlocked parameters have not disappeared and their behaviour is worth naming, because it is what the plateaus are stealing from.

At an unlocked parameter the orbit fills the circle without repeating, and the closer the rotation number is to a rational with a small denominator, the more nearly it locks. The parameters that hold out longest as $K$ rises are the ones whose rotation numbers are hardest to approximate by rationals — and the hardest of all is the golden ratio’s fractional part, which is why the last curve to break as the nonlinearity passes one is the golden one.

That is the same fact about the golden ratio that three gaps and no more is about, arriving from a completely different direction: a number badly approximated by rationals is a number whose rotation resists being captured by a nearby cycle.

What is being measured, and how it could be wrong #

Every rotation number on this page is a measurement, and the discipline that makes the measurement trustworthy is worth stating because a plateau is exactly the kind of thing a lazy computation invents.

The number is taken as the total advance of the lift — the map read on the line rather than on the circle, so that the advance accumulates instead of wrapping — divided by the number of steps, after several hundred steps have been discarded to let the orbit settle. Nothing is rounded, and nothing is compared against a rational until the number is in hand.

Three checks then run against it. The measured numbers are required to be non-decreasing in the parameter, which is a theorem and which fails immediately if an orbit has not settled. A plateau claimed at one half is confirmed by restarting the orbit at three different points and requiring the same number back, which is what an attracting cycle guarantees and a badly converged average does not. And the number of plateaus found is required to be more than one when the nonlinearity is on and no more than a couple when it is off, so a figure that found steps in a straight line would refuse to draw.

The reason for all three is that a staircase is easy to fake. Round a slowly varying quantity to three decimal places and it becomes constant over stretches; measure an unsettled average and it wanders in a way that looks like structure. Neither of those is mode locking, and neither survives being asked to start somewhere else.

Where it fails, and what it needs #

The rotation number is measured, not solved for. Every number in these figures comes from running an orbit two thousand steps and dividing. That is an approximation to a limit, and near the edge of a plateau — where convergence is slowest — a coarse measurement puts the boundary in the wrong place. The figures use long orbits and check monotonicity, which catches an unsettled measurement; they do not compute the exact endpoints.

Above $K = 1$ the map stops being invertible. The whole theory of the rotation number needs the map to be a homeomorphism of the circle, which fails when the derivative goes negative — at $K$ above one. Beyond that the rotation number may not exist, tongues overlap, and the behaviour is a different subject.

Widths are measured on a lattice of parameters. A plateau narrower than the spacing between sampled parameters is invisible, so the figures show the wide ones and silently omit the rest. Every rational has a plateau; the pictures show a few dozen.

And the devil’s staircase is a limiting object. At $K$ strictly below one the rise is real and has positive measure; only at exactly one does the flat part fill the axis. A picture drawn at $K = 1$ looks the same as one drawn at $K = 0.99$ and the two are different in the property that matters.

Where it came from #

The oldest recorded observation of the phenomenon is Huygens’s, in 1665, and it is worth naming as provenance rather than as evidence: two clocks on a shared beam were seen to fall into step, two hundred and fifty years before there was a theory to say why. Nothing about that observation is used below, and nothing on this page is a statement about clocks.

Poincaré introduced the rotation number in the 1880s for circle maps arising from flows on a torus, and proved that it exists and is continuous. Denjoy showed in 1932 that a smooth enough map with irrational rotation number is conjugate to the rigid rotation — so an irrational rotation number really does mean nothing but a rotation, in different coordinates.

Arnold’s work in the 1960s produced the tongues and the name, and the connection to the devil’s staircase is Jensen, Bak and Bohr’s from 1983. The subject has been busy ever since, in two directions that do not overlap much: the mathematics of the map, which is this essay’s, and the identification of the same picture in one measured system after another, which is not.

What the pictures cannot show #

The staircase is drawn from four hundred and twenty-one measurements, so what appears as a smooth rise between plateaus is a sequence of points. The rise is genuinely there at $K$ below one and genuinely absent in measure at $K = 1$, and no finite sampling distinguishes the two.

The tongue picture marks a point when the measured rotation number is within a tolerance of a simple rational. That tolerance is a choice, the list of rationals is a choice, and a tongue too narrow to contain a sampled parameter is missing rather than absent. What the figure shows is the shape of the locked set at the resolution drawn.

And nothing here shows that the staircase is continuous, which is the least obvious of its properties: a function that is constant on a dense set of intervals and still climbs is hard to believe in, and the belief comes from a theorem rather than from a plot.

The ladder from here #

Below: three gaps and no more, the rigid rotation this family perturbs, and the staircase that shows the whole orbit, where iteration is first drawn. Sideways: the road paved with doublings, another parameter sweep whose structure is self-similar, and a point that pulls and a point that pushes, which is the stability argument the plateaus rest on. Above: Denjoy’s theorem, the breakup of the golden invariant curve, and renormalisation as the explanation of the self-similarity.

What is worth carrying away #

A quantity can vary continuously, never decrease, climb from nothing to one, and still be constant on a set that fills almost everything. That combination sounds impossible on first hearing and is the ordinary behaviour of this family.

The reason it happens is worth more than the curiosity. Rational behaviour is stable — a periodic orbit that attracts survives being nudged — and irrational behaviour is not, so the rationals get intervals and the irrationals get single points. Wherever something is locked in place by stability rather than by exact tuning, the same staircase is waiting to be found.