Stretch, fold, and what is left

The previous rung measured that nearby trajectories separate exponentially. That measurement, taken together with one other fact, forces almost everything about the attractor’s shape.

The other fact is that the orbits stay in a bounded region. And the two together are in tension: a rule that multiplies every distance by more than one at every step, applied for ever, moves points arbitrarily far apart — unless something brings them back.

There is only one way to bring them back, and it is to fold. Not a metaphor: a continuous map of a bounded region that expands distances must send points that were far apart to points that are near, and the geometric operation that does that while staying continuous is a bend. Cutting would do it too and is not allowed; sliding would not do it at all.

The construction #

Take a square. Stretch it by a factor $\lambda > 2$ in one direction and squeeze it by more than the reciprocal in the other, so that areas shrink. The result is a long thin strip, far too long for the square.

Bend it into a horseshoe and lay it back down so that it crosses the original square twice.

That is Smale’s horseshoe map, and everything about it is chosen to be the simplest thing with the two properties: it stretches, and it maps the square into a bounded region.

The interesting question is which points stay in the square for ever. A point whose image leaves is not in the picture after one step; a point whose image stays but whose second image leaves survives one step. The set of points that stay for all time, forward and backward, is the invariant set, and it is what the dynamics is really about.

Counting what survives #

After one application, the part of the square still inside is the two strips where the horseshoe crosses it. Each has width $1/\lambda$ in the stretched direction.

After two, each of those strips is itself cut into two by the same construction applied again, so there are four strips of width $1/\lambda^2$. After $k$ steps there are $2^k$ strips of width $\lambda^{-k}$.

The total surviving width is $(2/\lambda)^k$, which goes to zero because $\lambda > 2$. So the surviving set has no width at all — and it is not empty, because it contains at least the fixed points and in fact uncountably many points.

A set with no length whose members can be listed by a binary expansion of infinite depth is a Cantor set, and that is exactly what this is. It is worth being explicit that “no length and not empty” is not a contradiction: the middle-thirds set has the same two properties for the same arithmetic reason, and it has as many points as the interval it sits in. The surviving set of the horseshoe is uncountable, has no width, and is totally disconnected — no two of its points are joined by a piece of it — which is the standard trio. Going backwards in time gives another Cantor set in the perpendicular direction, and the fully invariant set is the product of the two.

The dimension is immediate: $2^k$ pieces of size $\lambda^{-k}$ gives $\log 2 / \log \lambda$, which lies strictly between zero and one. The full invariant set, being a product of two such, has dimension twice that.

That last observation is worth a sentence, because it runs against intuition. Stretching harder makes the dynamics more sensitive — the exponent is $\log \lambda$ — and makes the invariant set smaller, because more of the square escapes. Sensitivity and size are separate quantities and they move in opposite directions, so “how chaotic” and “how large a set is involved” are two questions with two answers.

Why this is the general mechanism #

The horseshoe is an artificial map and its importance is that it is not a special case.

Smale’s theorem is that any map with a transverse homoclinic point — a point whose forward orbit and backward orbit both approach the same fixed point, with the two approaches crossing rather than touching — contains a horseshoe. Not resembles one: contains one, as an invariant subset on which the dynamics is exactly the horseshoe’s.

Transverse homoclinic points are generic. Poincaré found them in the three-body problem in 1890 and wrote that the resulting tangle was too complicated to draw — which was true, and it is what a horseshoe is. So the horseshoe is what chaos looks like whenever it appears, and finding one is the standard way of proving a system is chaotic.

For the Lorenz system the relevant structure is a horseshoe in the return map, and its presence is why the one-dimensional reduction has slope everywhere above one with a fold in the middle. The tent shape of that map is the horseshoe, one dimension down.

The coding #

The horseshoe makes precise the correspondence between orbits and infinite words that this field’s earlier essays introduce.

Label the two strips $0$ and $1$. A point of the invariant set has, at each moment forward and backward, a strip it is in, so it determines a two-sided infinite sequence of zeros and ones. Two facts make this a genuine dictionary.

Every sequence occurs. Given any sequence, there is exactly one point of the invariant set with that itinerary, because at each stage the surviving strips split in two and the choice at each stage is free. The uniqueness is the nesting: the strips specified by longer and longer prefixes are nested and their widths go to zero, so they close down on a single point.

And applying the map shifts the sequence. The dynamics is the operation of moving the reading position one place along, which is the simplest map anybody could write down.

So the horseshoe’s dynamics is the shift on binary sequences, and every question about it becomes a question about sequences. That is a complete solution of the system in a sense that almost no dynamical system admits: not an approximation, not a numerical study, but a relabelling after which every question is combinatorial. How many periodic orbits of period $k$? The number of sequences with period $k$, which is $2^k$. Is there a dense orbit? Yes — take the sequence listing every finite word in turn. Are there orbits doing anything one likes? Yes, by writing the sequence down. The whole of symbolic dynamics becomes available at once, and it is available because the folding is exactly a doubling.

What the attractor inherits #

The Lorenz attractor is not a horseshoe — it is an attractor, and the horseshoe’s invariant set attracts nothing, having no volume in either direction. But the mechanism is shared, and the attractor’s structure is what happens when the folding is combined with an attracting direction.

Across the sheet, the attractor is a Cantor set. Cut it with a small plane transverse to the flow and the cross-section is a Cantor set of leaves, exactly as in the horseshoe. Along the sheet it is a smooth two-dimensional surface. So it is locally a surface times a Cantor set, and its dimension is two plus a fraction.

It is worth putting a number on that. The Lorenz attractor’s dimension is close to $2.06$: two for the sheet, and $0.06$ for the Cantor set across it. That fraction is small, which is why the object looks like a surface in every picture and why establishing that it is not took decades. A dimension of $2.06$ is a surface with a dust’s worth of extra structure, and the extra structure is what makes the dynamics on it chaotic rather than a flow on a surface, which by the classification of surface flows could not be.

And the fold is visible in the picture nobody looks at it in. The famous two-lobed drawing of the Lorenz attractor is a projection, and the crossing that looks like a self-intersection is the fold seen edge-on. Trajectories do not cross, because the object is not flat; what the projection shows as a crossing is two sheets at different depths, and following them through is following the fold.

The baker’s version #

The horseshoe has a cousin that is easier to compute with and gives the same conclusions, and it is worth having because it makes the doubling completely explicit.

Take the unit square, stretch it to twice the width and half the height, cut it in half and stack the two halves. That is the baker’s map, and it is what kneading dough does.

In coordinates it is: $x \mapsto 2x \bmod 1$, and $y \mapsto (y + \lfloor 2x \rfloor)/2$. The first coordinate is the doubling map, whose action on a binary expansion is to delete the leading digit; the second collects the deleted digits in reverse. So a point of the square is a two-sided binary sequence with a marker showing where the present is, and the map moves the marker.

The correspondence with sequences is not a theorem here; it is the coordinates. That is the reason the baker’s map is the standard example, and it makes visible what the horseshoe’s version requires an argument for: the shift is not a model of the dynamics, it is the dynamics written in base two.

The difference between the two is that the baker’s map preserves area and the horseshoe does not. The baker’s map has no attractor and no escape — every point stays — while the horseshoe throws most points out and keeps a dust. Adding contraction to the baker’s map, so that the two halves are stacked with a gap, produces a Cantor set in the vertical direction and is the cleanest model of a strange attractor anybody has.

Putting that figure here rather than in the previous rung is deliberate. Read alone it is a measurement of a rate. Read after the horseshoe it is a measurement of the stretch factor, since $\lambda$ and the exponent are related by $\lambda = e^{\lambda_{\text{exp}} \tau}$ over a return time $\tau$ — so the geometry and the number are the same fact, and the figure that seemed to be about prediction is about a fold.

What it costs #

The horseshoe’s set is not an attractor. It has zero area and repels in one direction, so almost every point leaves. What is drawn in the figures is the set of survivors, and a randomly chosen starting point is not in it. The Lorenz attractor is genuinely attracting, and the difference is that it has a contracting direction which the horseshoe’s model omits.

The two directions are not symmetric and the drawings suggest they are. Forward time gives a Cantor set of vertical strips; backward time gives a Cantor set of horizontal ones; and the invariant set is their intersection. In a genuine attractor the backward direction is the contracting one, so the backward Cantor set is what is seen and the forward one is not — which is why a picture of the Lorenz attractor shows layers rather than a dust.

And the correspondence with sequences is only exact for the model. In the Lorenz system the coding is approximate to the same accuracy as the return map’s thinness, and there are itineraries that no orbit realises because the map’s two branches do not have full range. Working out which sequences occur is the kneading theory of the map, and it is a genuine complication that the idealised horseshoe hides.

Where it came from #

Smale constructed the horseshoe in 1960, on a beach in Rio, while trying to prove something else.

He had conjectured that structurally stable systems — those whose qualitative behaviour survives a small perturbation — were essentially simple, with finitely many periodic orbits. Norman Levinson wrote to him about a forced oscillator with infinitely many periodic orbits that seemed nonetheless robust. The horseshoe is Smale’s attempt to see what Levinson’s example was doing, stripped to its simplest form, and it refuted his own conjecture completely: the horseshoe has infinitely many periodic orbits, of every period, and it is structurally stable.

A counterexample to one’s own conjecture, reduced until it is a picture of a square being folded, is close to the ideal form of a mathematical object. It explained Levinson’s example, explained Poincaré’s homoclinic tangle from seventy years earlier, and became the standard mechanism by which chaos is demonstrated in a given system.

The reduction is the part worth admiring. Levinson’s example is a differential equation; Poincaré’s is the three-body problem; the horseshoe is a square, a stretch and a fold, and the arithmetic of $2^k$ against $\lambda^{-k}$. Everything the complicated examples do, it does, and it can be checked by hand.

What the pictures cannot show #

The invariant set is the limit and the figure shows five stages. At the fifth stage there are thirty-two strips of finite width; the object being described has none. Every drawing of a Cantor set is a drawing of a finite stage, and the interesting properties — uncountable, no length, totally disconnected — belong only to the limit.

The fold is not drawn. The figures show the surviving strips at each stage, which is the result of stretching and folding, and not the motion. Drawing the bend requires showing the square being deformed, which is a sequence of frames rather than a picture, and the strips are what those frames leave behind.

The stages are drawn in one direction only, so the count on the page is the square root of the truth. At stage $k$ the figure shows $2^k$ strips; the full invariant set has $4^k$ small squares at that stage, and drawing them fills the panel with dust long before the pattern is visible.

And the second Cantor set is missing. The horseshoe’s invariant set is a Cantor set across the square and another along it, and the figures show only the first. The full set is a product of two dusts, and a drawing of it at any stage is a grid of small squares whose count doubles twice as fast.

Where the ladder goes next #

The last rung on this ladder asks what kind of set the folding leaves behind and what it means for a set to have a dimension between two and three: the attractor that is neither a surface nor a solid.

Named here as a debt: kneading theory, which decides exactly which itineraries a given fold realises and which the idealised horseshoe’s “every sequence occurs” glosses over.

Also left unwritten: the structural stability of the horseshoe — that a small perturbation of the map has a horseshoe conjugate to this one — which is the property that makes it a mechanism rather than an example, and which is stated above and not shown. It is what distinguishes an object that survives perturbation from one that is an artefact of exact parameters.

Sideways, the doubling-and-shrinking count is the same arithmetic as the middle-thirds construction, the coding of orbits by words is symbolic dynamics, and the fold in a one-dimensional map is what a bouncing ball’s table does to its phase space.

What is worth carrying away #

Two constraints that are individually harmless can force a structure when imposed together.

Stretching alone gives escape. Boundedness alone gives nothing. Both at once give folding, folding repeated gives a Cantor set, a Cantor set of itineraries gives a coding, and a coding gives every conclusion about periodic orbits, dense orbits and sensitivity at once. The whole of the subject’s structure follows from an incompatibility.

The habit worth taking is to ask what a pair of requirements makes impossible. A rule that expands and a region that confines cannot both hold in a simple way, and the geometry is the resolution — which is a more productive question than asking what either requirement implies alone.