A twist that cannot avoid two points

Brouwer’s theorem says a continuous map of a disc into itself leaves a point where it is. On a ring — a disc with the middle removed — the theorem is false, and spectacularly so: rotating the ring by any angle moves every point.

So the question becomes what has to be added. Rotation moves the two boundary circles in the same direction, and the answer is that turning them in opposite directions is nearly enough, provided the map also leaves areas alone.

That is Poincaré’s last geometric theorem: an area-preserving homeomorphism of the annulus that fixes each boundary circle and rotates them in opposite senses has at least two fixed points. He conjectured it in 1912, a few months before his death, having been unable to prove it and having judged it too important to leave unpublished. Birkhoff proved it the following year.

What the hypotheses are doing #

Three conditions, and each is indispensable. The map must send the annulus to itself and leave each boundary circle where it is. It must preserve area. And it must turn the two boundaries in opposite directions — the twist condition.

The last one deserves a comment about what is really needed. The theorem as stated asks only about the two boundaries, not about the interior; nothing requires the rotation to vary monotonically with the radius, though the maps the theorem is usually applied to do have that property and are called twist maps because of it. What the boundary condition supplies is a place where the angular displacement is negative and a place where it is positive, and hence — by continuity along each radius — a whole curve where it is zero.

The map drawn here is the time-one flow of a Hamiltonian,

$$H(\theta, r) = \tfrac12 (r - c)^2 + \varepsilon\,(r-a)(b-r)\cos\theta ,$$

on the annulus $a \le r \le b$. That choice is not decoration. A Hamiltonian flow preserves area exactly, and a Hamiltonian constant along each boundary circle leaves those circles alone, so both hypotheses hold by construction — and both are then measured anyway: the Jacobian determinant on a grid, and the angular displacement at each boundary.

The proof in one picture #

Here is the argument, and it is the reason the theorem belongs in a collection of pictures.

Because the inner boundary turns one way and the outer the other, every radius contains a point whose angle does not change under the map. Those points form a loop $\Gamma$ running all the way round the annulus. Now apply the map to it, giving a second loop $f(\Gamma)$.

Every point of $\Gamma$ moves radially and only radially, since its angle is unchanged. So $f(\Gamma)$ is another loop going round the annulus — and the two loops enclose the same area, because the map preserves area and the region inside $\Gamma$ maps onto the region inside $f(\Gamma)$.

Two loops going round the annulus, enclosing equal areas: neither can lie strictly inside the other, so they cross. And a crossing is a point that neither turns nor moves in or out — a fixed point. Since two closed curves crossing transversally cross an even number of times, and they cross at least once, they cross at least twice.

The figure computes each step. The loop is found by bisecting on the angular displacement at every angle; the two enclosed areas are integrated and compared; the crossings are located and each is checked to be genuinely fixed, to within a millionth. The two that turn up are of different kinds — one a centre, around which nearby points circulate, one a saddle, which points approach along one direction and leave along another — and the classification is computed from the map’s derivative rather than read off the picture.

Birkhoff’s actual proof is harder than this sketch, because $\Gamma$ need not be a nice curve for a general continuous map: the set where the angular displacement vanishes can be a horrible closed set rather than a loop. The area argument survives in a form that does not need $\Gamma$ to be a curve, and supplying that form is what took a year and what makes the theorem a real one rather than a picture.

Drop area preservation and it all goes #

The hypothesis that looks most like a technicality is the one that does the work.

Push every interior point outward by a fixed rule that vanishes on the two boundaries. The boundaries stay put and still turn opposite ways; every point strictly inside moves outward, so nothing inside can be fixed; and nothing on the boundaries is fixed either, since they are turning. The map has no fixed point at all, and the only hypothesis it fails is area preservation, whose determinant the figure measures running down to less than three quarters.

That is why the theorem is a statement about mechanics rather than about topology alone. Area preservation is not a technical convenience; it is the property that the maps arising from Hamiltonian systems have and that arbitrary continuous maps do not — Liouville’s theorem, which says that the flow of a Hamiltonian system carries any region to a region of the same volume in phase space. The Poincaré–Birkhoff theorem is a topological theorem with a physical hypothesis, and the hypothesis is the whole content.

Where the fixed points sit #

A fixed point of a return map is a periodic orbit of the system it came from, and that is the reason anybody wanted the theorem.

Poincaré’s motivation was the three-body problem. He had reduced the question of periodic orbits of a restricted three-body system to the question of fixed points of a map of an annulus — the return map to a surface cut across the flow, now called a Poincaré section — and the map was area-preserving with a twist. A fixed point of it is a closed orbit of the planets; a point of period $q$ is an orbit that closes after $q$ circuits.

That last observation is the one that multiplies the theorem’s value. Applying it to the $q$-th iterate of the map, restricted to a sub-annulus where the rotation number passes through $p/q$, gives at least two points of period $q$ — for every rational $p/q$ in the range of rotation numbers. So an area-preserving twist map has infinitely many periodic orbits, densely many rotation numbers’ worth, and the pattern of centres and saddles they form is the skeleton the whole dynamics hangs on.

The pattern visible in the figure is the generic one, and it has a name — the Poincaré–Birkhoff chain. Where an invariant circle of the unperturbed map carried a rational rotation number, a perturbation typically destroys it and leaves behind an even number of periodic points, alternating between centres and saddles: a chain of islands. The centres are surrounded by their own invariant curves; the saddles have streams entering and leaving; and what happens where those streams meet is the beginning of chaos, which is where a difference too small to draw becomes a difference too large to ignore.

The billiard connection #

Every billiard table is an instance, which is the cleanest source of examples.

Take the state of the ball to be the point of the cushion it hit together with the angle it left at. That is a point of an annulus: position round the boundary, angle between zero and a straight angle. The bounce map carries this annulus to itself, it fixes the two boundary circles — the grazing trajectories — and, with the right coordinates, it preserves area.

The twist condition holds because a shallower angle carries the ball further round the table before its next bounce. So Poincaré–Birkhoff applies, and applied to the $q$-th iterate it gives, for every $p/q$, at least two closed trajectories that wind $p$ times round the table in $q$ bounces. That is how a convex billiard table is known to have infinitely many periodic paths even though nobody can find them — an existence theorem with no construction anywhere in it, which is exactly what an obtuse triangle still lacks.

What the two fixed points are like #

The two points the theorem guarantees are not interchangeable, and their difference is visible in the derivative.

At a fixed point, the map’s derivative is a two-by-two matrix of determinant one, since area is preserved. Such a matrix has eigenvalues whose product is one, so either they are a conjugate pair on the unit circle — the elliptic case, with nearby points rotating around the fixed point — or they are real reciprocals — the hyperbolic case, with one direction contracted and one stretched. The trace decides: below two in size means elliptic, above means hyperbolic.

The figures compute the trace at each crossing and report which is which; the two the theorem produces here are one of each, and that pairing is typical rather than accidental. An index argument explains it: the sum of the indices of the fixed points inside a region is determined by the behaviour on its boundary, and the boundary here forces the sum to be zero, so the points come in pairs of opposite index — a centre and a saddle.

The elliptic point behaves like an attracting fixed point without the attraction: nearby orbits neither approach nor escape, they circulate, because area preservation forbids both. That is a genuine difference from the dissipative case, and it is why the pictures in this essay have no basins and no attractors — nothing in an area-preserving system converges to anything.

The count, and what it is really counting #

There is a way of seeing why two is the right number, and it is worth having because the number looks arbitrary until it does not.

Attach to each fixed point an index: walk a small loop round it and count how many times the displacement vector — the arrow from a point to its image — turns while doing so. A centre has index $+1$, since the displacement rotates once with the loop; a saddle has index $-1$, since it rotates once the other way. The index is a whole number, it is unchanged by small perturbations, and the total index inside a region is decided entirely by the displacement on the region’s boundary.

On the annulus the boundary contributes nothing: the displacement on the inner circle points one way round and on the outer circle the other, and the two contributions cancel. So the indices of the fixed points inside must add to zero — which means either there are none, or there are at least two of opposite sign.

The theorem rules out the first case, and the index argument then supplies the pairing: a centre for every saddle. That is the same accounting that makes a vector field on a sphere have zeros of total index two, and the same accounting that makes the winding number of a polynomial’s image count its roots. Every one of those statements is a boundary determines a total, and a total that is not zero forces an interior.

Every rotation number, not only the rational ones #

The theorem applied to iterates gives periodic orbits at every rational rotation number in the twist range, which leaves an obvious gap: the irrationals, which are almost all of them. That gap was filled in the early 1980s, independently by Aubry and by Mather, and the answer is stranger than the rational case.

For every rotation number in the range, rational or not, there is an invariant set on which the map turns by exactly that amount. At a rational the set is a periodic orbit, which is the theorem above. At an irrational it is one of two things: an invariant circle running all the way round the annulus, or — when that circle has been destroyed by the perturbation — a Cantor set sitting where the circle used to be, with gaps in it, invariant under the map, and carrying the same rotation number.

Those remnants are called cantori, and they are the reason the phrase “the invariant circle is destroyed” is misleading. Nothing is destroyed; the circle develops gaps and becomes a set with no length and uncountably many points, which still constrains the dynamics, though far more weakly — a trajectory can now leak through the gaps, slowly, where before the circle was an impassable barrier.

The method that finds them all is variational, which pays off a remark the essay makes about the Fagnano orbit. Assign to each finite segment of a trajectory a quantity — an action — built from the map’s generating function, and look for the configurations that minimise it. Every Aubry–Mather set is a minimiser, the periodic orbits included: the saddle of a Poincaré–Birkhoff pair is the minimiser at that rotation number, and the centre is a saddle point of the same action rather than a minimum.

So the existence result of this essay, which produces a fixed point out of two loops crossing and hands over no construction, has a companion that produces the same orbits by minimising something. The variational route gives more and asks more: it needs the twist to vary monotonically with the radius, which the boundary condition alone does not supply, and in exchange it produces orbits at every rotation number rather than at the rationals.

The picture that results is the one this whole field has been assembling. Rational rotation numbers give chains of islands — centres and saddles, alternating, from Poincaré–Birkhoff. Irrational ones give circles while the perturbation is small and cantori once it is not, and which irrationals keep their circles longest is decided by how badly they are approximated by fractions, so the golden ones survive last. That is the same ordering that decides which rotations resist locking, arriving in a completely different setting for the same arithmetic reason.

Two theorems, then, covering the two kinds of number, and the boundary between them is a continued fraction.

Where it fails, and what it needs #

Both boundaries must be fixed. A map that carries the annulus into itself without fixing the boundaries can rotate everything, so the boundary condition is not decoration; there are generalisations to maps that only preserve the boundary circles setwise, and they need care.

Area preservation cannot be weakened to volume-like conditions in general. The theorem’s higher-dimensional analogues are largely open or false. The natural generalisation — a symplectic map of a higher-dimensional annulus has fixed points — became the Arnold conjecture, whose proof in the general case took Floer’s homology in the 1980s and is one of the origins of modern symplectic topology.

Two is the truth, not an artefact. The theorem gives at least two, and there are maps with exactly two, so the count cannot be improved without more hypotheses.

What the pictures cannot show #

The loop and its image are drawn for one particular map, and the theorem is about all of them. Worse, the drawing shows a case where the zero-displacement set is a nice curve, and Birkhoff’s difficulty was precisely that in general it is not — so the figure illustrates the easy case of the argument and quietly omits the hard one. That is stated here because a picture that hides the hard case is a picture that can mislead about what was proved.

Nor can a figure show a fixed point being forced. What is drawn is two loops that do cross; what the theorem says is that they must. The picture supports the argument by measuring the two enclosed areas and finding one number, which is the step from which everything else follows, but the conclusion is a piece of reasoning about all such loops and not an observation about these ones.

And the chain of islands is drawn at one perturbation size. How the chain appears as the perturbation grows from zero, which is the real phenomenon, would need a sequence of pictures and is the subject of a whole other ladder.

The ladder from here #

Below: Brouwer’s theorem, which is this statement on a disc with no hypothesis about area, and the contraction principle, which buys uniqueness and an algorithm at the price of a much stronger hypothesis. Sideways: the hairy ball theorem, another existence result whose proof is a counting argument about indices, and billiards, which is where the annulus comes from in practice. Above: KAM theory, which says which invariant circles survive a perturbation, and the symplectic fixed-point theorems that generalise this one to higher dimensions.

What is worth carrying away #

The theorem is a good example of a hypothesis that looks like bookkeeping and is the entire argument. Area-preserving sounds like a technical condition attached to make a proof work; the counterexample shows it is the condition, and the topology contributes only the observation that two loops of equal enclosed area must cross.

There is a second lesson in how the result was used. Poincaré did not want a theorem about annuli; he wanted periodic orbits of a three-body system, and he found them by reducing a question about a flow in a high-dimensional space to a question about a map of a two-dimensional one. The reduction — take a surface across the flow and study the return map — is worth more than the theorem, and it is the standard first move on any continuous dynamical system to this day.