Every window is the whole diagram again

The window that opens with a stutter followed the logistic map $x \mapsto r x (1 - x)$, whose road paved with doublings leads into chaos, to the left edge of its widest periodic window, at $r = 1 + \sqrt 8$, and watched the orbit stutter as a cycle of three was born. It ended by noting what the window does next: the cycle of three does not stay a cycle of three. It forks.

This essay is about what the fork leads to, and the answer is the most striking picture in the subject. Take one of the window’s three strands, the one through the middle of the interval, and magnify it.

It is the whole bifurcation diagram again. One value, splitting into two, then four, then eight, faster and faster, a cascade that runs out at a finite parameter, then grey chaos with white windows cut through it and dark curves threading it — and then, abruptly, the end, where the window closes and the orbit escapes to fill the interval. The copy is squeezed into a parameter range less than three hundredths wide and a band of values a tenth wide, and it is upside down. Otherwise it is the same picture.

Three strands, one of them inverted #

The period-three window holds three strands, not one, because a cycle of three visits three places, and each strand is a copy.

At the window’s left edge the orbit settles on three points — near $0.16$, $0.51$ and $0.96$. As $r$ increases each forks at the same parameter, because a fork of a cycle is a fork of every point on it: the cycle of three becomes a cycle of six, with two points near each of the old three. The strands fork again together, run into chaos together, and together meet the edge of the window, where the orbit breaks out of all three and fills the interval.

Only the top strand is the right way up.

The top strand’s copy opens upward, as the whole diagram does. The middle and bottom strands open downward, and their copies are inverted — the middle one is drawn at the head of this essay turned back over. The inversion is not an accident of drawing; it is the first clue to why copies exist at all. Each strand is the diagram of the map’s third iterate near one point of the cycle, and whether that copy is upright or inverted is decided by the order in which the orbit meets the top of the hump and the stretch where the map is decreasing, which is different for each of the three points.

The same cascade, ninety-seven times narrower #

The copy is not only the same shape. It has the same arithmetic, and the arithmetic can be measured.

The cleanest markers of a cascade are its superstable parameters: the values of $r$ at which the top of the hump, $x = \tfrac12$, is itself on the periodic orbit. At those parameters the orbit is as stable as an orbit can be, because the map’s slope at $\tfrac12$ is nought and one factor of every product of slopes along the cycle is nought. They sit between successive forks, one per period, and each is a root of the equation $f^n(\tfrac12) = \tfrac12$ — an equation in $r$ that can be solved to any precision without iterating anything to convergence.

For the whole diagram they are $2$, $1 + \sqrt 5 \approx 3.236$, $3.4986$, $3.5546$, $3.5667$ and $3.5692$, and the ratios of successive gaps are $4.709$, $4.681$, $4.663$ and $4.668$ — Feigenbaum’s constant, $4.6692\ldots$, arriving. Inside the window the superstable parameters of periods $3$, $6$, $12$, $24$, $48$ and $96$ are $3.83187$, $3.84457$, $3.84834$, $3.84920$, $3.84938$ and $3.84942$, and the ratios of their gaps are $3.362$, $4.425$, $4.612$ and $4.657$. The window’s cascade converges on the same constant, a little more slowly, from further away.

The first gap inside the window is about ninety-seven times narrower than the first gap of the whole diagram. So the copy is a scaled copy, compressed by roughly a hundred along the parameter, and its doublings accumulate at about $3.8494$, where its own chaos begins, in the way the original’s accumulate at $3.5699$.

Why the third iterate is a hump #

The reason for the copies is a calculation about the map composed with itself, and it is short enough to follow in words.

A cycle of three for $f$ is a fixed point of the third iterate $f^3$. So what happens to the cycle as $r$ increases — whether it is stable, when it forks, when it becomes chaotic — is decided by what $f^3$ does near the cycle’s points. Look at the point near $\tfrac12$. The map $f$ has its maximum there, so $f(x)$ for $x$ near $\tfrac12$ is the top value, $r/4$, less a small multiple of $(x - \tfrac12)^2$. The next application of $f$ happens near $0.96$, where $f$ is steeply decreasing, so it turns that small dip into a small rise. The third application happens near $0.16$, where $f$ is increasing, and passes the rise on. So near $x = \tfrac12$,

$$f^3(x) \approx \tfrac12 + c\,\left(x - \tfrac12\right)^2$$

for a positive $c$: a parabola, opening upward, with its turning point at $\tfrac12$.

That is a one-humped map, upside down. On a small interval round $\tfrac12$ — the interval the middle strand lives in — $f^3$ looks like the logistic map itself, rescaled to fit the interval and turned over. And anything a one-humped map does, $f^3$ restricted to that interval does too: its fixed point is stable, then it forks, then its forks fork, then it goes chaotic, with the same universal proportions, because the proportions do not depend on which hump. The middle strand is the diagram of that restricted map, and the diagram of any hump is the same diagram. The inversion is the sign of $c$.

The same in every window #

Nothing in that argument used the number three. In a window of period $p$, the point of the cycle near $\tfrac12$ sees $f^p$ as a small hump, of one orientation or the other depending on how many decreasing stretches the orbit passes through, and the strand through it is a copy of the whole diagram.

The widest period-five window runs from about $3.7382$ to $3.7448$, a range fewer than seven thousandths wide, and its middle strand is again the whole diagram — a fork, a cascade, chaos, windows, the end. And the windows inside the copy are copies of windows, each with its own strands, each strand a copy of the whole. The diagram contains itself at infinitely many scales and infinitely many places.

This is also why the windows are dense in the chaotic half. Every copy of the diagram contains a copy of every window, so between any two parameters in the chaotic part — which are separated by some window’s copy, however small — there are windows of every larger scale. The picture’s grey is shot through with white at every magnification, and no magnification finds a stretch of chaos without a window in it.

Where a window ends #

The copy also explains the one feature of a window that looked most unlike the whole diagram: its abrupt end.

The whole diagram ends at $r = 4$, where the hump reaches the full height of the interval and the orbit wanders over all of it; beyond $4$ the orbit escapes the interval altogether. In a window, the strand’s copy is the diagram of a small hump, and that hump reaches the full height of its interval at some parameter inside the window. At that parameter the copy’s chaos fills the strand’s interval completely and touches its edges — and the edges of the strand’s interval are points of an unstable cycle, the pushing half of the pair that was born with the stable cycle at the window’s left edge. An orbit that reaches them is no longer confined to the strand. It escapes into the rest of the interval, and the grey floods back.

So the right edge of a window is the copy’s $r = 4$. The same event that ends the whole diagram ends every copy of it, and in the copy it looks abrupt because what lies beyond the copy’s interval is not empty space but the rest of the chaotic interval, waiting. The collision of a chaotic band with an unstable cycle is called a crisis, and it is to the window’s end what the tangent collision was to its beginning.

How narrow the copies are #

Each window’s copy is compressed by its own factor, and the factors grow quickly with the period.

In the period-three window the first gap of the copy’s cascade, from period three to period six, is about ninety-seven times narrower than the first gap of the whole cascade, from one to two. In the widest period-five window the corresponding gap, from its superstable parameter at $3.738915$ to the period-ten one at $3.741848$, is about $0.0029$ wide — some four hundred and twenty times narrower than the original. The factors grow roughly exponentially with the period, because a window of period $p$ has to fit its whole copy into a range of parameters over which $p$ steps of the map stretch small differences enormously. That is why the windows of high period are too narrow to see, and why there are so many of them: the folds that measure chaos count how fast the number of available cycles grows with the period, and the room for their windows shrinks at a comparable rate.

The fork that flips, and a second constant #

The inversion has a counterpart inside a single cascade, and it names the second of Feigenbaum’s constants.

At each doubling of the main cascade, the two new points of the cycle open away from each other; the pair nearest $\tfrac12$ opens by a distance that shrinks, from one doubling to the next, by a factor of about $2.5029$. And the smaller pair opens on the opposite side from the larger one — the fork near the top of the hump is inverted relative to the fork before it. That factor, with its sign, is Feigenbaum’s second constant, $\alpha \approx -2.5029$: the scaling of the values, where $\delta$ is the scaling of the parameters.

The window’s copies are the same phenomenon at a larger scale. A cascade is a sequence of self-similar copies, each inside the last, rescaled by $\delta$ along the parameter and by $\alpha$ along the value, turned over at each step because $\alpha$ is negative. A window is one self-similar copy of the whole, rescaled by different factors that depend on the window, turned over or not depending on the orbit’s route. Both are the same fact: near the top of the hump, some iterate of the map is a hump again.

Renormalisation, and what it proves #

The step from “$f^3$ near $\tfrac12$ looks like a hump” to “its diagram is the same diagram, with the same constants” is the part that needs real mathematics, and it has a name.

Mitchell Feigenbaum, and independently Pierre Coullet and Charles Tresser, in 1978, turned the observation into an operation on maps: take a one-humped map, compose it with itself, restrict to the small interval round the top, rescale to full size and flip. Call the result the renormalised map. Doing this to the map at the accumulation point of the cascade gives a map at the accumulation point of a cascade again, so there is a map that renormalisation carries to itself — a fixed point of the operation — and $\delta$ is the rate at which renormalisation pushes nearby maps away from it. That is why $\delta$ is universal: it is a property of the operation, not of any particular hump.

Proving that the fixed point exists took until 1982, when Oscar Lanford gave a proof that used a computer to check inequalities with rigorous error bounds. Conceptual proofs followed from Dennis Sullivan, Curtis McMullen and Mikhail Lyubich in the following two decades, using the complex dynamics of the same maps. The copies in the windows come from the same operation applied with $f^p$ instead of $f^2$, and every window has its own renormalisation carrying its strand onto the whole diagram.

The same copies in the complex plane #

The logistic map is a real slice of a complex family, and the copies have famous relatives there.

Allow $x$ and the parameter to be complex, and the set of parameters for which the orbit of the critical point stays bounded is the Mandelbrot set, which the shape in every picture of itself drew. Its intersection with the real line is the range of the logistic family in which the orbit stays in the interval, and the period-three window is where a small copy of the whole Mandelbrot set sits on the real axis — the “midget” whose real slice is exactly the window’s range. Every real window is the real slice of such a copy, and the self-similarity of the bifurcation diagram is the one-dimensional shadow of the self-similarity of the Mandelbrot set.

What the magnifications cannot show #

The copies are drawn at two windows and one strand each. The claim that every window of every period contains a copy is the renormalisation argument’s, and the figures show two instances of it.

The copy is not exact. A window’s strand is the diagram of $f^p$ restricted to a small interval, which is a hump but not a parabola, so its proportions agree with the whole diagram’s only in the limit of deep magnification. The cascade inside the period-three window shows this directly: its first gap ratio is $3.362$, far from $4.669$, and its ratios approach the constant from further away than the main cascade’s do.

The cascade table stops at period ninety-six. Beyond that the superstable parameters are closer together than the arithmetic of a double can separate reliably, and the claim that the ratios converge to $\delta$ is Feigenbaum’s and Lanford’s, not the table’s.

Still open: how the windows are arranged in size #

Every window contains copies of every window, and the windows are dense. What is not known precisely is how much of the parameter line the windows take up. Mikhail Lyubich proved that almost every parameter either lies in a window, where an attracting cycle exists, or is stochastic, with an invariant density, and Michael Jakobson had shown that the stochastic parameters have positive total length. So the windows, dense as they are, leave out a set of positive measure.

How that measure is shared out — how the widths of the windows of period $p$ decrease with $p$, and what fraction of the chaotic half the stochastic parameters occupy — is known only through estimates and computations. The total length of the windows between the end of the cascade and $r = 4$ has been estimated by computation, window by window, and no formula for it, or proof of any value for it, is known — even whether the windows of period $p$ shrink with $p$ at a rate that a single exponent describes is a question the computations suggest and nothing settles.

A picture that contains itself #

The habit worth keeping is the reduction that explained the copies.

A window of period three looked like a new phenomenon, with its own stutter at the edge, its own forks and its own chaos. It was not new. Near the top of the hump, the map’s third iterate is a hump, so everything that happens in the window is something the whole diagram already does, rescaled and perhaps turned over. A system that contains a small copy of itself contains copies of the copy, without end, and every feature of the original — its constants, its windows, its dark lines — appears at every scale.

That is what the word universality means in this subject. It is not only that different maps share the constant $4.669$; it is that one map shares it with itself, in every window, because every window is the whole diagram again.