How a lock comes apart

The staircase that is flat almost everywhere explains a plateau in one sentence and then moves on: “The plateau ends where the slope reaches the boundary and the fixed point disappears.”

A fixed point disappearing is not a thing points do on their own. What happens is a collision.

Two orbits, not one #

Inside the plateau at $p/q$ the map’s $q$-th iterate has a fixed point, and it has two. One attracts nearby orbits and one repels them, and the attracting one is the whole reason the plateau exists: it is what any orbit falls into, so it is what a measured average reports — which is the sense in which three gaps and no more’s rigid rotation is the unperturbed case: it has no attracting orbit at all, and every orbit is as good as every other.

The repelling one is invisible to a measurement. Start an orbit exactly on it and the orbit stays; start anywhere else and the orbit leaves, so no long run ever reveals it. The figures find it by solving $f^q(x) = x + p$ on a fine grid and refining by bisection, which is the only way it can be found at all.

That the two exist together is not a coincidence. The equation $f^q(x) - x - p = 0$ is a continuous function on the circle, and if it has a zero it generically has an even number of them counted with sign — because a continuous function on a circle returns to where it started, so it crosses zero upwards as often as downwards. Attracting and repelling fixed points come in pairs on a circle, and that is a topological fact rather than a dynamical one.

The multiplier is the number that runs out #

The quantity deciding whether an orbit attracts is the derivative of the $q$-th iterate at it — the multiplier — and by the chain rule it is the product of the map’s derivatives at the orbit’s own points:

$$m = \prod_{k=0}^{q-1} \left(1 - K\cos 2\pi x_k\right).$$

An orbit attracts when $|m| < 1$ and repels when $|m| > 1$. The plot shows both across the plateau, and the shape is the same for every plateau of every such family: the attracting orbit’s multiplier is smallest in the middle and climbs to one at each edge, while the repelling orbit’s falls to one from above.

At the edge they meet at $m = 1$ — and $m = 1$ is exactly the condition for the two zeros of $f^q(x) - x - p$ to merge, since a double zero of a function is a zero of its derivative, which is $m - 1$. So the edge of the plateau is a double root, and beyond it the function has no root at all: the graph has lifted clear of the axis.

That collision has a name, the saddle-node bifurcation, and it is the commonest way a fixed point can cease to exist. It is the same mechanism a point that pulls and a point that pushes is about, and the thing to notice is that it is the only way available: a fixed point with $|m| < 1$ cannot simply vanish as the parameter moves, because the function’s sign change either side of it has to go somewhere, and the only place it can go is into another root.

Where the power of K comes from #

An earlier essay computes the width of the zero plateau exactly and states the general rule without deriving it: the plateau at $p/q$ has width of order $K^q$. The collision picture is where that exponent comes from.

The $q$-th iterate’s deviation from the identity is built by composing $q$ copies of a map whose nonlinear part is $K$ times a sine. Expand in powers of $K$: the term that can create a fixed point at a rotation of $p/q$ has to survive averaging round the orbit, and the first such term is of order $K^q$ — every lower order averages to nothing over the $q$ points of the orbit, because those points are nearly equally spaced and a sine summed over equally spaced points cancels.

So the window of parameters in which the two roots exist has width of order $K^q$, which is the hierarchy of denominators the staircase shows. The exponent is a count of how many times the nonlinearity has to be used before it can help, and the equal spacing of the orbit is what forces the count.

The arithmetic is checkable from the figures, which report each plateau’s ends. Dropping $K$ from 1 to 0.6 multiplies $K^2$ by 0.36, and the half plateau narrows from 0.0740 to 0.0278 — a factor of 0.376. That is close to 0.36 and not equal to it, which is what an estimate keeping only the leading term should produce, and the small excess is the next order in $K$. That agreement is closer than the argument deserves — the expansion is only the leading term — and it is what makes the exponent believable rather than merely stated. The staircase reports the widths as measurements from a sweep; here they are computed by bisecting on whether the equation has a root, which is a different instrument reaching the same numbers.

The zero plateau, where it can be done by hand #

The plateau at zero is the one case where everything can be verified without a computation, and it is worth doing because it shows the mechanism with no expansion in it.

A fixed point of the map itself needs $\Omega = (K/2\pi)\sin 2\pi x$. The right-hand side runs over $[-K/2\pi, K/2\pi]$ as $x$ goes round, and for $\Omega$ strictly inside that interval the equation has two solutions — one where the sine is rising and one where it is falling. The multiplier is $1 - K\cos 2\pi x$, which is below one at the rising solution and above it at the falling one.

At $\Omega = K/2\pi$ exactly, the two solutions merge at the sine’s maximum, where the cosine is nought and the multiplier is exactly one. Past it the equation has no solution. Two roots, a collision at multiplier one, and a plateau of width $K/\pi$ — the general picture, with every step an identity rather than an estimate.

A longer period, and the same collision #

Nothing about the collision depends on the period. At $p/q$ the two orbits have $q$ points each, they interleave round the circle, and the pair collides point by point as the parameter reaches the edge — every point of the attracting orbit meeting the corresponding point of the repelling one.

That interleaving is forced by the ordering. A circle map that preserves orientation carries the cyclic order of any set of points to itself, so two period-$q$ orbits cannot be arranged arbitrarily: they alternate. So the collision is $q$ simultaneous saddle-node collisions, one in each gap, and they happen at the same parameter because they are $q$ readings of one equation.

The number of points changes and the mechanism does not, which is why one calculation covers every plateau. What does change is the exponent in the width, and the reason it changes is the averaging argument above — more points to average over, more orders of $K$ before anything survives.

What happens after the collision #

Outside the plateau the periodic orbit is gone and the rotation number moves. What the map does instead is worth stating, because the transition is less abrupt than “the orbit vanishes” suggests.

Just outside, the graph of $f^q(x) - x - p$ lifts barely clear of the axis, so the iterate is nearly fixed over a long stretch. An orbit entering that stretch creeps through it slowly, spending many steps almost stationary, and then moves quickly round the rest of the circle. So the rotation number is close to $p/q$ but not equal to it, and the closer to the edge the closer it is — which is why the staircase’s rise near a plateau’s end is steep but not vertical.

That behaviour has a name in the literature and a characteristic arithmetic: the time spent creeping goes like one over the square root of the distance past the edge, which is what an almost-tangency to the axis gives. So the plateau does not end sharply in the orbit’s behaviour, only in the exact value of the average — and a measurement with a finite orbit length reports a plateau slightly wider than it is, which is the standing error in this kind of numerical work.

Above K equal to one #

Another records that the theory needs the map to be invertible and that invertibility fails above $K = 1$. The collision picture says what goes wrong, and it is not that the argument breaks: it is that the plateaus start to overlap.

Each plateau widens as $K$ rises, at its own rate $K^q$. At $K = 1$ they touch — the rationals’ plateaus between them cover the whole axis — and above one they would have to overlap, which means two different periodic orbits with different rotation numbers coexisting at the same parameters. For an invertible circle map that is impossible, because the rotation number is well defined and single-valued.

So $K = 1$ is not an arbitrary cut-off. It is where the arithmetic of widths runs out of room, and above it the map stops being invertible for exactly the reason it has to: the derivative $1 - K\cos 2\pi x$ goes negative somewhere. Two facts that look unrelated — the widths filling the axis, and the derivative changing sign — happen at the same parameter, and they are the same event seen from two sides.

The same collision elsewhere in this collection #

The mechanism is not special to circle maps, and two of its other appearances here are worth naming because they look nothing alike.

A point that pulls and a point that pushes is the mechanism on the interval, where a map’s fixed points are the crossings of its graph with the diagonal and a collision is a tangency. That is the same algebra with one dimension fewer and no wrapping, and it is the argument this one leans on.

The road paved with doublings is a different bifurcation entirely and the contrast is instructive. There a stable orbit loses stability at multiplier $-1$ rather than $+1$, and instead of vanishing it is replaced by an orbit of twice the period. Multiplier $+1$ is a collision and an end; multiplier $-1$ is a split and a continuation. The sign of the multiplier at the moment stability is lost decides which of the two happens, and the two cascades that follow are the two structures the parameter plane of these families is made of.

Which is why the circle maps’ parameter plane and the interval maps’ look different at a glance and are built from the same two events. Tongues rooted at rationals and widening are what a family of saddle-node collisions draws; a cascade of splittings is what the other draws; and a family with both has both.

What the pictures cannot show #

The periodic points are found by scanning a grid of four thousand and refining, so an orbit living in a window narrower than the grid step would be missed — and near the edge of a plateau the two orbits are exactly that close. The figures place their panels at stated fractions of the plateau’s width and report the gap, so how near the edge they get is on the page rather than implied.

The multiplier plot samples a hundred and twenty parameters across the plateau, and the collision happens in the last fraction of a per cent of it. What the plot shows is the approach; the equality at the edge is the algebra above, and it is what the bisection that located the edge is testing.

And the repelling orbit is not observable. Nothing in a long run of the map ever visits it, so no measurement of the kind those averages are built on would find it — it is there because the equation has two roots, and the figures draw it because they solve the equation rather than run the map.

Still open: how the collisions organise themselves #

The plateaus are arranged by the mediant rule and the collisions at their edges are all of the same kind, which raises the question of whether the whole arrangement is explained by one local mechanism repeated. The answer is that it is not quite, and the gap is where the subject’s hard results live.

What is understood is each plateau on its own: its width to leading order, the collision at its edges, and the creeping behaviour just outside. What is not explained by any local argument is the self-similarity — that magnifying the gap between two plateaus reveals the same structure — which needs a renormalisation argument acting on the whole family of maps at once, and which produces the universal constants the flat staircase quotes — the same kind of argument behind a constant that does not care which map.

The other open direction concerns what the collision looks like at $K = 1$ exactly. There the plateaus touch and the map has a point where the derivative is nought, so the multiplier calculation degenerates; the behaviour at criticality is where the universal exponents come from and is understood by renormalisation rather than by the elementary argument on this page.

The two ends are not the same collision #

One asymmetry is worth noticing because it is easy to assume away. The plateau’s two edges are both saddle-node collisions and they are not mirror images of each other.

At the lower edge the two orbits collide where the map’s graph is tangent to the line $y = x + p$ from one side; at the upper edge, from the other. The orbits involved swap roles in the sense that the point at which the collision occurs is different — at one end the attracting orbit sits at a place where the map’s derivative is smallest and at the other where it is largest — and the creeping behaviour just outside the plateau happens at different places on the circle.

What is symmetric is the arithmetic, for this particular family, because $\sin 2\pi x$ is odd about its zeros: the standard circle map has a symmetry sending $x$ to $-x$ and $\Omega$ to $-\Omega$, which maps the plateau at $p/q$ to the one at $-p/q$ and, applied to the zero plateau, exchanges its two ends. So the widths either side of the plateau’s centre match to the precision the figures report, and that is a property of this family rather than of the mechanism.

A different family would break it. Replace the sine by a function with no such symmetry and the plateau is no longer symmetric about anything; the two collisions still happen, the widths of the two halves differ, and every statement on this page except the symmetry survives. Which is the usual situation with a worked family: most of what is seen is the mechanism and a little of it is the example.

What the pair was for #

A plateau is a stretch of parameters over which a measured average does not move, and the explanation is a single word — stability. What this essay adds is that stability is a number with a range, and a range has ends.

The attracting orbit does not weaken and fade; it is destroyed by a collision with an orbit nobody was watching. That is worth carrying because it generalises: wherever a stable state exists over a range of conditions and ceases to exist outside it, the usual mechanism is a collision with an unstable state, and the unstable one is invisible right up until the moment it matters.

The instrument that sees both is the equation rather than the experiment. A long run of the map reveals the attracting orbit and nothing else; solving $f^q(x) = x + p$ reveals the pair, and only the pair explains where the plateau ends.