Two constants at the flat point

The staircase that is flat almost everywhere drew the circle map

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

a rotation by $\Omega$ disturbed by a sine of strength $K$, and its devil’s staircase: the average turn per step, the rotation number, as a function of $\Omega$. For $K$ below 1 the map is smooth and invertible; at $K = 1$, the critical line, its graph goes flat at $\theta = 0$, where both its slope and its curvature vanish. On the critical line the locked steps of the staircase fill the whole parameter axis except for a set of measure nought. That essay mentioned, in passing, a constant found by Scott Shenker in 1982: following the parameters whose rotation numbers are ratios of Fibonacci numbers, the gaps between them shrink by a factor approaching 2.8336.

This essay measures that constant and a second one that goes with it, compares both with what happens off the critical line, and checks them against a different map.

Hunting for the golden parameter #

The rotation number $(\sqrt5 - 1)/2 \approx 0.618$, the golden ratio’s reciprocal, is the hardest number to approximate by fractions, as three gaps and no more explained: its best approximations are the ratios of consecutive Fibonacci numbers, $1/2, 2/3, 3/5, 5/8, \ldots$, and each is only as good as its denominator squared allows. On the staircase every one of those fractions owns a locked step, and the golden rotation number is reached only at a single parameter $\Omega_\infty$ squeezed between them.

To find the steps precisely it is convenient to pick, inside each, the parameter at which the orbit of the flat point is itself periodic: the parameter $\Omega_n$ with $f^{F_{n+1}}(0) = F_n$ exactly, so that after $F_{n+1}$ steps the orbit has gone round $F_n$ times and come back to its start. These are called superstable parameters, because a periodic orbit through the flat point attracts its neighbours faster than any other. Each is found by bisection: the position of the orbit after $F_{n+1}$ steps increases with $\Omega$, so the equation has one solution in any interval where its sign changes, and the interval can be narrowed to the limit of double-precision arithmetic. The search is started from the previous two parameters, extrapolating their gap, and runs to $F_{19} = 4{,}181$ steps. The extrapolation matters more than it looks. The locked step for $F_n/F_{n+1}$ is a sliver of parameter only about $2.8^{-n}$ wide, and a search started from a fixed interval would have to narrow down through all the larger steps around it, landing on a wrong fraction if the bracketing slipped by one. Predicting each new parameter from the last gap divided by a rough guess at the constant puts the search inside the right sliver at once, and the bisection then needs only to find where in that sliver the orbit closes exactly. That the predictions keep landing in the right sliver as $n$ grows is itself a weak form of the result: a guess of $2.7$ for the ratio is good enough only because the true ratio is close to it and settles.

Two rates, one on each side of the line #

For $K = 0.5$, below the critical line, the superstable parameters approach their limit with each gap $\varphi^2 = 2.6180$ times smaller than the last, with alternating sign. That is the rate for a pure rotation. A rotation in different coordinates showed that, below the line, a circle map with an irrational rotation number is a rotation in disguise, and the theorem of Herman and Yoccoz that how smooth the disguise is examined makes the disguise smooth for the golden number. For a rotation by $\Omega$ the rotation number is $\Omega$ itself, and the gaps between the Fibonacci fractions shrink by $\varphi^2$, because the $n$-th fraction differs from the golden number by an amount that shrinks like $\varphi^{-2n}$.

For $K = 1$ the rate is different: $2.8336$, the sequence of ratios settling to four decimal places by the fourteenth fraction. The orbit no longer behaves like a disguised rotation near the golden parameter, because there is no smooth disguise on the critical line; and the new rate, which belongs to the critical map, is the constant Shenker measured.

Why a flat point changes the rate #

The difference between $2.618$ and $2.834$ has a mechanism that can be seen without the renormalisation theory. Near its flat point the critical map behaves like $f(\theta) \approx \Omega + c\,\theta^3$: a small interval around $\theta = 0$ is not moved along rigidly, as a rotation would move it, but squashed, its length cubed. An orbit whose rotation number is close to a fraction $p/q$ passes near the flat point once every $q$ steps or so, and each pass compresses the nearby orbits, so that changing $\Omega$ slightly moves the orbit’s return by a different amount than it would for a smooth map. The parameter needed to move from one Fibonacci lock to the next is therefore not the rotation’s amount, and the ratio of successive amounts settles on a new constant.

For the smooth map at $K = 0.5$ the same passes happen, but every interval is stretched or shrunk by a factor bounded above and below, and those factors average out over the long orbits that the Fibonacci locks require. That averaging is the content of the theorem that the disguise is smooth: the smooth map is a rotation seen through a fixed, gently distorting lens, and a fixed lens does not change ratios measured at ever finer scales. The flat point is a lens that distorts without bound — infinitely at $\theta = 0$ itself — and no averaging removes it.

The same flatness is what how a lock comes apart and a whole interval of speeds followed from the other side, as the strength $K$ was raised towards 1: the locked steps widen, the gaps between them shrink, and at $K = 1$ the gaps have zero total length. The constants here are what that limit looks like at one particular irrational rotation number, measured as a rate rather than as a total.

A different map, the same number #

The sine is not special, and the test of that is to change it.

The second family adds a second harmonic,

$$g(\theta) = \theta + \Omega - \frac{1}{2\pi}\Big(1.2\sin 2\pi\theta - 0.1\sin 4\pi\theta\Big),$$

with weights chosen so that the slope and the curvature at $\theta = 0$ both vanish while the slope stays positive everywhere else. Its flat point is an inflection of order three, like the sine map’s on the critical line. Its superstable golden parameters are entirely different numbers — the limit is $0.600917\ldots$ against the sine map’s $0.606661\ldots$ — and its first few gap ratios differ noticeably. But the ratios converge to the same limit, $-2.8336$, as the figure shows, to four decimal places.

This is the same phenomenon as a constant that does not care which map, where Feigenbaum’s $4.669$ appeared in the period-doubling of every map with a quadratic maximum. The explanation is the same too. Zooming in on the dynamics near the golden parameter, with the right rescaling of space and time, defines a renormalisation operator on circle maps; maps with a cubic flat point are drawn towards a single fixed point of that operator, and the constants are read off the fixed point, not off any map that is drawn towards it. Mitchell Feigenbaum, Leo Kadanoff and Scott Shenker, and independently Stellan Östlund, David Rand, James Sethna and Eric Siggia, gave that explanation in 1982 and 1983.

The figure makes the shared feature visible. At $\theta = 0$ the two critical maps touch a horizontal line, their graphs bending from concave to convex exactly there, while the subcritical map still has slope one half at its flattest. The critical maps differ everywhere else, and everywhere else is what the renormalisation discards.

The second constant: how close the orbit returns #

The first constant measures the parameter. The second measures space: how close the orbit of the flat point comes back to its start.

At the superstable parameter for $F_n/F_{n+1}$, the orbit of $\theta = 0$ closes after $F_{n+1}$ steps; one Fibonacci number earlier, after $F_n$ steps, it lands close to its start but not on it, at a distance $d_n$. The ratio $d_n/d_{n+1}$ alternates in sign — the closest returns fall alternately just past the start and just short of it — and converges to $-1.2886$ for both critical maps. For $K = 0.5$ it converges to $-\varphi = -1.6180$, the ratio for a rotation, whose closest returns are governed by the same Fibonacci numbers and shrink by the golden ratio each time.

The critical constant is smaller in size than the golden ratio: on the critical line the closest returns shrink more slowly. The orbit spends longer near the flat point, where the map barely moves anything, and its returns crowd towards the start more gently than a rotation’s would.

The orbit itself shows the flat point’s influence. Its first 1,597 points do not fill the circle evenly, as a rotation’s would; they bunch on the side where the map is flat and thin out opposite. And the returns, plotted on a logarithmic scale against $n$, fall on a straight line with a slope of $1.29$ per Fibonacci step, the same constant measured a different way, at the limit parameter rather than at the superstable ones.

An orbit that is not evenly spread #

The bunching visible in the orbit has a precise description, and it is one more way the critical map differs from a rotation. For a rotation by an irrational angle every orbit is evenly spread round the circle: the share of its points in any arc tends to the arc’s length, which is the sense in which the histogram an orbit leaves is flat for such a map. A smooth circle map with the golden rotation number is a rotation in smooth disguise, so its orbits are spread according to a smooth density — uneven, but with no part of the circle favoured infinitely more than another.

On the critical line the density does not exist. The orbit’s long-run distribution is a measure concentrated on a set of length nought, piling infinitely many points, in proportion, into arbitrarily small neighbourhoods of the flat point’s backward orbit. It is still a continuous distribution — no single point carries positive weight — but it is singular with respect to length, the circle-map counterpart of a Cantor-like distribution. The 1,597 points drawn are a finite glimpse of that: already visibly thinner on one side, and with more points the contrast between the crowded and empty parts of the circle would only sharpen. The spectrum of local scaling exponents of that measure was among the first such spectra to be computed and measured in experiments, and its extremes are set by the same renormalisation fixed point as the two constants.

Self-similarity, checked by magnification #

The two constants together say that the picture near the golden parameter repeats itself under magnification: shrink the parameter window by 2.8336 and the rotation-number window by $\varphi^2$ for each Fibonacci step, and the staircase should look the same.

The figure computes the rotation number at 161 parameters in each of three windows around $\Omega_\infty$, each window $2.8336^2$ times narrower than the last, and rescales the rotation number by $\varphi^4$ each time. The three staircases lie on top of one another; the two deeper ones differ by at most $0.001$ on a vertical scale of $0.12$. The locked steps — the plateaus for $5/8$, $8/13$, $13/21$ and their neighbours — appear in the same places at every magnification. That is what it means for the staircase to be self-similar at the golden point, and the pair of constants is its scale: Shenker’s along the parameter, the golden ratio’s square along the rotation number.

The rotation number is rescaled by $\varphi^2$ rather than by anything critical because the rotation numbers at the golden point are fixed by arithmetic alone: the Fibonacci fractions approach the golden number at the golden ratio’s rate whatever map is used. What the map controls is how much parameter it takes to move between them, and that is where the critical constant enters.

Two routes to chaos, two sets of constants #

Period doubling has its own pair of universal constants, which a constant that does not care which map measured: Feigenbaum’s $4.669$ for the parameter and $2.503$ for space. The circle map’s pair, $2.834$ and $1.289$, belong to a different route. Period doubling is the way a system with one rhythm becomes chaotic, by halving its frequency again and again; the circle map describes a system with two competing rhythms — an oscillator driven at a frequency not its own — whose ratio is irrational, and the critical line is where that quasi-periodic motion breaks down.

Both pairs were measured in laboratories within a few years of being computed. Experiments on fluid convection, on electrical oscillators and on Josephson junctions driven at two frequencies found staircases of locked steps whose gaps near the golden ratio shrank at rates consistent with Shenker’s constant, and whose fractal structure matched the dimension $0.87$ that the staircase that is flat almost everywhere quoted for the critical line. The constants are numbers about an operator on maps, but they are the numbers a real system with a flat point in its return map exhibits, because the real system’s return map is, near the golden point, drawn towards the same fixed point as the sine.

The standard map of the last circle to break, a twist map of the plane rather than a map of the circle, has its own golden-mean constants at the breakup of its last invariant circle, found by John Greene and explained by renormalisation by Robert MacKay. They are different numbers again, because the setting is area-preserving rather than invertible-on-a-circle, and they reinforce the pattern: each class of map has its own fixed point, and each fixed point its own constants.

What the measurements show and what they do not #

Every number here is computed in double precision, by bisection to the last representable digit and iteration of the map. The constants are stable to four decimal places by the fifteenth Fibonacci fraction, which is as far as double precision allows before rounding in an orbit of thousands of steps begins to matter. More digits require extended precision and a better method: the renormalisation operator itself can be computed, and its fixed point and the constants read off to many places, which is how the published values $\delta = -2.833610\ldots$ and $\alpha = -1.288575\ldots$ were obtained.

The universality is tested here on two maps, which is evidence and not proof. The proof that every analytic circle map with a cubic critical point has the same constants at the golden rotation number is a theorem about the renormalisation operator, established in stages by computer-assisted proofs and then by the work of Edson de Faria, Welington de Melo and Michael Yampolsky in the 1990s and 2000s, who showed that the renormalisations of such maps converge to a unique limit.

Still open: every rotation number, every order #

The golden number is the simplest case, because its continued fraction is all ones and the renormalisation repeats the same step for ever. For other rotation numbers whose continued fractions eventually repeat — the quadratic irrationals — there are analogous constants, one pair for each periodic pattern of continued-fraction digits, also universal. For general irrational rotation numbers the renormalisations do not repeat, and the right statement is about the whole sequence of renormalised maps converging together rather than about two numbers. How the constants depend on the order of the critical point — three here, but any odd order keeps the map invertible — is understood for the golden number and computed for small orders, and whether there is a usable closed description of how they vary with the order is a question the computations have explored more than the theory has answered.