The histogram an orbit leaves

A chaotic orbit cannot be predicted and does not repeat, and the honest response to that is not to give up but to change the question. Individual steps are unavailable; where the orbit spends its time is available, is stable, and turns out to have a closed form.

Not uniform, and the shape is the point #

The first thing the histogram says is that the orbit does not spread evenly. It piles up at the two ends of the interval and thins in the middle, by a factor of several.

The density is

$$\rho(x) = \frac{1}{\pi\sqrt{x(1-x)}},$$

the arcsine distribution, and it is unbounded at both ends. A quantity that is unbounded and still has total mass one is a reminder that a density is not a probability: the mass of a bin is the integral, and the integral of $1/\sqrt{x}$ near zero is finite even though the function is not.

That is why the figure compares bin masses rather than sampling the curve. The exact share of the bin from $a$ to $b$ is $F(b) - F(a)$ with $F(x) = (2/\pi)\arcsin\sqrt{x}$, which is finite at every bin including the two on the ends — where a midpoint sample of $\rho$ would be a number the drawing has no business printing.

Why that density and no other #

A density is invariant when pushing it forward by the map returns it unchanged: the mass that lands in a region equals the mass that was already there. For a map that folds two branches onto the interval, the condition is

$$\rho(y) = \sum_{f(x) = y} \frac{\rho(x)}{|f'(x)|},$$

each preimage contributing its own density divided by the stretching there.

The arcsine density satisfies that for the logistic map at four, which is a calculation. It is worth doing the other way instead, because the other way is one line.

The conjugacy carries the tent map to the logistic map by $h(y) = \sin^2(\pi y/2)$. The tent map at slope two stretches uniformly, so it preserves the flat density — every region’s mass is its length. Carrying a flat density through a change of variable multiplies it by the derivative of the inverse, and

$$\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{2}{\pi}\arcsin\sqrt{x}\right) = \frac{1}{\pi\sqrt{x(1-x)}}.$$

The arcsine density is a flat density seen in the wrong coordinates. Nothing about it is special to the parabola; it is what uniformity looks like after the substitution the conjugacy makes.

Why the tent map is not iterated directly #

The second figure says it is showing the tent map’s density and it iterates the logistic map. That is a deliberate substitution and the reason is worth a section, because it is the sort of thing a figure can hide.

The tent map at slope two is a shift of the binary places, exactly. Doubling a number moves every bit one place left; the fold handles the carry. A double-precision number holds fifty-three bits, so after about fifty steps there are no bits left, the orbit reaches zero, and zero is fixed.

Measured while this figure was being written: sixty thousand steps from $0.3183$, and 59,948 of them in the first bin. The histogram would have been a picture of the arithmetic running out, and it would have looked like an ordinary histogram to anybody who did not check the first bar.

So the orbit is computed where the arithmetic survives — the logistic map’s parabola is not a bit shift, and its rounding errors do not accumulate in one direction — and read in the tent coordinate. The figure asserts that the first twenty-five values so obtained satisfy the tent map’s own recurrence, so the claim that it is a tent orbit is checked rather than argued.

A figure of a chaotic system looks the same whether or not the arithmetic behind it is right. That is this field’s standing hazard, and this is the sharpest case of it on the site.

The average along an orbit is the average over the interval #

The density supports a stronger statement than a histogram.

Birkhoff’s ergodic theorem, applied here, says that for almost every starting point and any reasonable function $g$,

$$\lim_{n\to\infty}\frac{1}{n}\sum_{k<n} g(x_k) = \int_0^1 g(x)\,\rho(x)\,\mathrm{d}x.$$

The left side is a time average along one orbit; the right is a space average over the interval, weighted by the density. They are equal, and the left side does not depend on where the orbit started.

That is the property called ergodicity, and it is what makes a statistical description of a chaotic system worth having. One long orbit is a fair sample of the whole interval. The figure checks the simplest instance — the average of $x$ itself along sixty thousand steps, against the half that symmetry gives — and finds them agreeing to two decimal places.

The theorem’s almost every is doing real work and cannot be dropped. The fixed points, the periodic orbits and their preimages all have time averages that are nothing like the space average, and there are infinitely many of them. They form a set of measure zero, so a starting point picked by throwing a dart misses them all — the same situation as the periodic points a wandering orbit passes, and the same resolution as the orbit that must come back needs for its own measure-theoretic step.

How fast the histogram settles #

The theorem is a limit and says nothing about how long the limit takes. That is a separate question, it has an answer, and the answer is what makes the statistical description usable rather than merely true.

The operator taking a density to its push-forward — the transfer operator — has the invariant density as its fixed point, and how fast an arbitrary starting density approaches it is set by the operator’s next eigenvalue. For the logistic map at four that decay is geometric and fast: a histogram of a few thousand steps is already close, and the error falls like the reciprocal square root of the count from there on, which is sampling noise rather than approach.

Which of the two errors dominates is the practical question. A map that mixes slowly needs a long transient discarded before any counting begins; a map that mixes quickly needs only a large sample. The logistic map at four is in the second class, which is why the figures here discard nothing and simply count.

That distinction is the same one a walk that samples a distribution has to make, in a setting where the map is chosen rather than given: the whole art there is designing a chain that mixes fast, because the transient is wasted work and the theorem does not bound it.

The parameters where there is no such density #

The invariant density above is a fact about $r = 4$, and most parameters do not have one of this kind.

Below the onset of chaos the map settles onto a cycle, and almost every orbit ends there. The invariant measure describing almost every orbit is then a set of point masses — one at each point of the cycle, each of weight $1/k$ — and there is no density at all, because a point mass is not the integral of anything.

Inside the chaotic range the situation alternates. There are parameters with an absolutely continuous invariant density, and there are the windows — intervals of $r$ in which a cycle is attracting again — where there is not. Both kinds are dense, which is the disquieting part: arbitrarily close to any chaotic parameter there is a window, and arbitrarily close to any window there is chaos.

Jakobson’s theorem is the result that rescues the statistical picture: the set of parameters with an absolutely continuous invariant density has positive measure, so a parameter picked at random has one with positive probability. That is much weaker than all and it is the correct statement, and the windows are why.

What the density is not #

Three misreadings are worth heading off, because each one turns a correct statement into a false one.

It is not a probability distribution over starting points. Nothing random happens. The map is deterministic, the orbit is determined by its start, and the density describes how one orbit distributes itself over time. Nothing here is order arriving out of genuine noise; the rule has no randomness in it at all.

It is not unique. Every periodic orbit supports its own invariant measure — a point mass at each of its points — and there are infinitely many. What singles out the arcsine density is that it is the one absolutely continuous with respect to length, and equivalently the one describing almost every orbit rather than a measure-zero set of them.

It says nothing about any particular step. The histogram is silent on where the orbit is at step ten thousand, which is the question the field cannot answer. Exchanging that question for this one is the whole manoeuvre, and pretending the exchange was free is the standing error.

What this buys, and where the idea goes #

The same substitution runs through the parts of mathematics and physics that deal with systems too complicated to follow.

Statistical mechanics rests on exactly this exchange: a gas’s molecules are followed by nobody, and the ergodic hypothesis — that time averages equal averages over the energy surface — is what licenses computing with the second. Whether it holds for realistic systems is a hard and largely open question, and Boltzmann assumed it.

Monte Carlo methods run the exchange backwards. A walk that samples a distribution is a deterministic-or-random process engineered so that its invariant density is the one wanted, and the answer is read off as a time average along one long run. The theorem in this essay is the guarantee that the reading is correct.

Number theory uses it for equidistribution: the fractional parts of the multiples of an irrational number fill the interval evenly, which is what three gaps and no more is a finite-stage picture of, which is the same statement about a rotation of the circle — a map whose invariant density is flat and whose time averages are its space averages, with no chaos anywhere in it.

The same picture in three other places on this site #

The histogram is a shape and the shape has turned up before under different names, which is worth noticing because the coincidences are not coincidences.

The arcsine law for a fair walk. The proportion of time a random walk spends on one side of the axis has exactly this density — piled at nought and one, thinnest at a half — so the least likely outcome is the balanced one. The two arcsine distributions come from genuinely different arguments and share a name because they share a formula; whether there is a common reason is a real question and the honest answer is that the projection of uniform motion on a circle produces both.

The projection of a circle. A point moving at constant speed round a circle, projected onto a diameter, spends its time according to this density — it lingers at the turning points and hurries through the middle. That is the same substitution $x = \sin^2\theta$ that the conjugacy uses, seen kinematically, and it is the most direct picture of where the piling-up at the ends comes from.

The eigenvalue distribution of a large random matrix. The semicircle and the arcsine are neighbours in that subject, and the same trigonometric substitution relates them.

The common thread is the change of variable rather than the dynamics. Wherever something uniform is seen through $\sin^2$, this density appears; the logistic map is one such case, the projected circle is another, and the walk is a third by a longer route.

What the picture cannot show #

A histogram is a finite sample of an infinite orbit, and the theorem is a limit. The figure’s assertion is a comparison against exact shares within six standard errors, which is a statistical statement and not the theorem.

It cannot show that the density is invariant, which is the defining property. Invariance is a statement about pushing a density forward, and the drawing shows one histogram rather than a histogram and its image.

And it cannot show the orbits that disobey. Every periodic orbit has a completely different time average, they are dense in the interval, and none of them can appear in a picture drawn from a randomly chosen starting point — which is the situation the theorem’s almost every describes and no drawing can display.

Where the ladder goes next #

Above: what a computer’s arithmetic does to an orbit it cannot represent, which this essay has already had to work around and which deserves its own rung. Beyond that: the transfer operator, whose eigenfunctions the invariant density is the first of, and whose spectrum controls how fast a histogram settles.

One debt. Ergodicity is quoted here and not proved. For the logistic map at four it follows from the conjugacy plus the ergodicity of the tent map, and that in turn from the doubling map’s — which is a genuinely provable statement about binary expansions and would make a good rung of its own.

What replaced prediction #

When no step of an orbit can be reported, the distribution of the whole orbit can be, and it is the same for almost every starting point.

That is a strictly weaker kind of answer and it is a real one. It is computable in closed form, it is stable against changing the start, and — because the average along one orbit is the average over the interval — a single run answers questions about the whole system. What was given up is any statement about where the orbit is; what was gained is every statement about where it has been.