Where the fixed point escapes

A theorem with three hypotheses invites the same question three times: what happens without this one? For the fixed-point theorem the answers are unusually clean, because each hypothesis has a counterexample that can be drawn in a single panel, and in each the escaping point is visible.

Each of the first three maps is continuous, and each sends its set into itself. What each lacks is one hypothesis, and the fourth panel is there because a search that reports nothing found on every input is not a search: the same sweep that finds no fixed point in the ring finds one on the disc, to three decimal places.

The hole #

Turn a ring about its centre and every point moves. The only point a rotation leaves alone is the centre, and the centre has been removed.

This is the counterexample that feels most like cheating and is the most instructive. Nothing has been done to the map — the same rotation on the full disc has a fixed point, and it is in the hole. What the ring lacks is not size or closedness but the ability to shrink a loop.

The connection is exact. A loop around the hole cannot be contracted, and the standard proof of the fixed-point theorem works by contracting a loop: assume no fixed point, use the direction from each point to its image to build a map onto the boundary circle, and derive a loop that must be both contractible and not. Remove the hypothesis that the set has no holes and the contradiction evaporates, because the loop was never contractible in the first place.

So the correct statement of the hypothesis is not no holes but contractible — a set that can be shrunk continuously to a point within itself. A disc is; a ring is not; a disc with a spike, a star, a comb and every convex set are.

The missing boundary #

The second panel takes the open disc — every point strictly inside the unit circle — and maps each point halfway toward a fixed point of the rim. Every point moves toward the rim without reaching it. The map’s fixed point exists and is on the rim, at distance exactly one from the centre, and the open disc does not contain it.

This is a failure of closedness, and it has a distinctive feel: nothing goes wrong anywhere, and the sequence of iterates converges perfectly well — to a point that is not in the space. The smallest distance any point moves is not zero, but it is arbitrarily small for points near the rim, which is what the figure reports.

The same failure in the simplest possible dress: on the half-open interval $(0, 1]$, the map $x \mapsto x/2$ moves every point, and its fixed point is $0$. Nothing exotic is required, and the missing point is always exactly the one that was wanted.

The room to move #

The third panel shifts the entire plane one unit to the right. Every point moves by exactly the same amount, so there is no fixed point and nothing approaches being one: the smallest movement is not merely positive, it is a half everywhere.

This is a failure of boundedness, and it is the crudest of the three. There is always somewhere further to go.

What makes it interesting is the repair. Add a single point at infinity to the plane — the construction that turns a sphere minus a point back into a sphere — and the shift extends to a continuous map of the sphere that fixes that added point. The counterexample does not survive compactification; it was never a map with no fixed point, only a map whose fixed point had been left out of the space.

That is worth putting beside the second panel, because the two failures now look identical: in both cases the fixed point exists as a limit and has been excluded by the choice of set. Closedness and boundedness are the two halves of compactness, and compactness is the hypothesis that says limits stay inside. One condition, two ways of breaking it.

What is actually being assumed #

The three hypotheses are therefore two, and stated properly they are:

Compact — closed and bounded, in Euclidean space; more precisely, every sequence has a convergent subsequence with its limit in the set. This is what makes the limiting step of every proof legitimate, and it is what both the open disc and the shifted plane violate.

Contractible — shrinkable to a point inside itself. This is what the ring violates, and it is the hypothesis that carries all the topology.

Convexity, which is how the theorem is usually stated, is neither of these: it is a convenient condition implying contractibility, and the theorem holds for many non-convex sets. A comb, a spiral arm and a solid letter S all satisfy it. What matters is the absence of a loop that cannot be shrunk, and convexity is simply the easiest sufficient condition to check.

The whole table, in one dimension #

Everything above happens on the line as well, where it can be checked by hand.

On $[0, 1]$, a continuous map into itself must cross the diagonal, because $f(0) - 0 \ge 0$ and $f(1) - 1 \le 0$ and a continuous function with a sign change has a zero. That is the whole proof in this dimension, and every hypothesis is visible in it.

Remove closedness — work on $(0, 1)$ — and the sign change may happen at an endpoint that is not there: $x \mapsto x/2$ has no fixed point in the open interval. Remove boundedness — work on $[0, \infty)$ — and $x \mapsto x + 1$ has none. Remove connectedness, which in one dimension is what plays the role of the missing hole: on the set $[0, 1] \cup [2, 3]$ the map sending the first piece to the second and the second to the first is continuous on its domain and moves everything.

Four sets, four verdicts, and the same three failures as in the plane. The one-dimensional case is where the hypotheses are easiest to feel, because the proof is one sign change and each counterexample removes exactly the part of the argument that needed the hypothesis.

What a turned ring does have #

The rotation of the ring is the sharpest counterexample above, and it is worth saying what remains true, because the answer is a theorem rather than nothing.

An area-preserving map of the annulus that turns the two boundary circles in opposite directions has at least two fixed points. That is the Poincaré–Birkhoff theorem, conjectured by Poincaré in 1912 in the last paper of his life and proved by Birkhoff the following year, and its hypotheses are exactly what a rigid rotation fails: a rigid rotation turns both circles the same way.

The theorem matters because it is what supplies periodic orbits in mechanics, where area preservation is not an assumption but a consequence of the equations of motion, and where a map that twists differently at different radii is the normal case rather than a special one. So the ring is not a place where fixed points are absent; it is a place where their existence needs a hypothesis about how the boundary is turned, rather than following from continuity alone.

Two hypotheses replaced one, and the count went from zero to two. This is the usual shape of a repair: the failed theorem is not patched, it is replaced by a different theorem with different hypotheses that happens to cover the case of interest.

Where it came from, and what its author thought of it #

Brouwer proved the theorem for every dimension in 1910 and 1911, having proved the three-dimensional case a year earlier, and the proof was thoroughly non-constructive: it establishes that a map with no fixed point cannot exist, and produces no fixed point for any map.

Within a decade he had come to regard that style of argument as illegitimate. Brouwer became the founder of intuitionism, the position that a mathematical object exists only when it can be constructed, and under which a proof by contradiction of an existence statement establishes nothing. He never retracted the theorem, and he never used it again; the most cited result of his career sits on the wrong side of the line he spent the rest of it drawing.

The situation is not as paradoxical as it sounds, and Sperner’s lemma is why. The combinatorial half of the modern proof is entirely constructive — it points at a triangle and gives a walk that finds it — and only the final limit is not. What cannot be constructed is the limit point, and that is a defect of the real numbers rather than of the argument, which is a distinction Brouwer himself drew and which is why intuitionistic analysis has a fixed-point theorem of its own with a weaker conclusion.

Where the fixed point can be counted rather than found #

There is a stronger statement lurking, and it explains the whole table of cases at once.

For a reasonable space and a continuous map, an integer — the Lefschetz number — can be computed from what the map does to the space’s holes, and if it is not zero the map has a fixed point. For a contractible space the number is $1$ for every map, which is Brouwer’s theorem; for the ring, a rotation gives $0$, which is why a rotation may have none.

The sphere is the case worth having. A continuous map of the sphere to itself need not have a fixed point — the antipodal map, sending every point to the one opposite, has none — and the sphere is compact, so compactness is not the issue. It is not contractible, and the Lefschetz number of the antipodal map is exactly zero.

So the four panels of the first figure are four values of one invariant, and the theorem’s hypotheses are the conditions under which that invariant is guaranteed non-zero. Once that is the framing, which hypothesis is dropped stops being a list of special cases and becomes a computation.

What the counterexamples do not show #

None of them shows that the theorem is delicate. Each hypothesis is doing real work, and dropping it kills the conclusion entirely rather than weakening it — but the sets on which the theorem does hold are enormously varied, and the failures require exactly the structures drawn: a hole, a missing boundary, an unbounded direction.

Nor do the counterexamples show that fixed points are rare in the failing cases. The ring has plenty of maps with fixed points; a rotation is a particular map chosen because it has none. The theorem says every continuous map has one; a counterexample only has to exhibit one map that does not, and that asymmetry is the whole difference between a theorem and its refutation.

And the sweep is a check, not a proof. Two thousand sampled points that all move is evidence; the argument is the algebra beside it, which solves each map’s fixed-point equation and reads off where the solution lies. On the ring the equation’s only solution is the centre; on the open disc it is a point of the rim; on the plane it has none. That is what settles the matter, and the sweep is there to catch the case where the algebra was written down wrongly.

What the picture cannot show #

The panels are two-dimensional and the theorem is about every dimension. In three dimensions the ring becomes a solid torus and the argument is the same; the hairy ball theorem, by contrast, is genuinely dimension-dependent, holding on even-dimensional spheres and failing on odd ones for a reason no picture in three dimensions can display.

The open disc cannot be drawn honestly at all. Its boundary is not part of it, and the figure marks that with a dashed rim — a convention, not a depiction. A reader who did not know the convention would see a disc, and the whole counterexample lives in the difference.

Nor can a picture show every continuous map. Each panel draws one map, chosen to fail, and the theorem quantifies over all of them. The pictures refute; they do not establish.

The ladder from here #

Below: the theorem itself, the lemma that proves it and the contraction hypothesis that strengthens it. Sideways: loops that cannot be shrunk, which is the invariant behind the ring, and the sphere minus a point, which is the repair for the plane. Above: the fixed-point theorems that hold in settings this one does not reach — for maps that assign a set to each point, and for spaces of functions, where the contraction principle does the work instead.

What a hypothesis is doing #

It is worth asking, of each of the three, what would go wrong in the proof rather than in the conclusion.

The standard argument assumes no fixed point and builds a continuous map from the set to its own boundary that leaves the boundary alone. Compactness is what makes the construction continuous — the direction from a point to its image is well defined only while the two are a bounded distance apart and the point cannot escape. Contractibility is what makes the resulting map impossible, since a set that can be shrunk to a point admits no such retraction.

Reading the hypotheses off the proof rather than off the statement is generally the quicker route to knowing which counterexamples exist, and it predicts their shape: each of the three panels breaks exactly one step of that construction, and the panel that breaks none has the fixed point the construction rules out.

Three failures, one lesson #

The lasting point is that in all three cases the fixed point exists and the set does not contain it. The ring’s is at its centre, the open disc’s is on its rim, the plane’s is at infinity — and each is excluded by the same act of removing something.

That is the useful way to read the hypotheses of any theorem with a limit in its proof. The hypotheses are rarely about the map; they are about whether the space is closed under the operation the proof performs. Completeness in the contraction principle, compactness here and the extreme value theorem’s closed interval are three statements of one requirement: the thing the argument constructs must have somewhere to live.