A spectrum that is a Cantor set

The Fibonacci chain is this collection’s simplest aperiodic object: two tile lengths in the order the substitution A → AB, B → A produces, which is also the order in which a line of irrational slope cuts a square lattice. The essays on it so far have asked what the chain is — how many windows of each length it contains, where its diffraction peaks sit, which periodic approximants come closest.

This asks what happens to a wave in it, and the answer is a set nobody would design: the values at which a wave neither grows nor decays form a Cantor set. There are no bands. Between any two allowed values there is a gap, and inside the neighbours of every gap there are more gaps.

Transfer matrices, and why a trace decides #

A wave on a chain of sites obeys a three-term recurrence: the amplitude at one site is determined by the two before it and by whichever of the two site types it is. Written as a matrix carrying a pair of consecutive amplitudes to the next pair, that is a transfer matrix — one per site, depending on the value being tested and on the site’s type.

The matrix over a whole period is the product of them, and it has determinant one. A matrix of determinant one is either a rotation-like map, whose repeated application keeps a vector bounded, or a stretch, which sends almost everything to infinity. The two cases are separated by the trace: bounded when |tr| ≤ 2, unbounded otherwise.

So for a periodic approximant of the chain — the chain truncated at a Fibonacci number of sites and repeated — the allowed values are exactly those where the trace of the period’s transfer matrix lies in the interval from −2 to 2. That is a classical criterion and it needs no more theory than the determinant.

The recursion the traces obey #

The chain’s substitution structure gives the traces a life of their own, and this is the part particular to the Fibonacci chain rather than to chains in general.

Writing xₙ for half the trace at level n, the words at successive levels satisfy wₙ₊₁ = wₙwₙ₋₁, and the corresponding matrices multiply. Taking traces and using the two-by-two identity tr(AB) + tr(AB⁻¹) = tr(A)tr(B) gives

$$x_{n+1} = 2\,x_n x_{n-1} - x_{n-2}$$

which is the trace map. It is a recursion on three numbers with no matrices in it, and it computes every level from the three before.

Here the machinery does both and compares. The first three values come from actual matrix products; everything after them comes from the recursion; and the two are required to agree at every level. They do, to the last few bits of the arithmetic — which is a test of the recursion rather than a restatement of it.

A quantity the recursion cannot change #

The trace map has a conserved quantity:

$$I = x_1^2 + x_2^2 + x_3^2 - 2x_1x_2x_3 - 1$$

and it takes the same value on every consecutive triple of the sequence. That is a polynomial identity rather than an approximation, and it is checked at every level of every run here.

Its value says something immediate. I is zero exactly when the two site types are the same — that is, when the chain is periodic — and positive otherwise. So a single number measures how far the chain is from being a crystal, and the size of the gaps in its spectrum is controlled by it.

That is a satisfying arrangement: the aperiodicity, which is a fact about an infinite word, has been reduced to a number computed from three traces, and the number decides the qualitative behaviour of the spectrum.

The drift, which is a measurement of something real #

Conserved quantities in a computation are usually reported as conserved and left there. This one drifts, and the drift is worth reporting because it is not a fault.

Over eight levels the invariant holds to the last bit of double-precision arithmetic — a drift of about 10⁻¹⁶. Over fourteen levels it has drifted by more than one. Nothing about the recursion has changed; the trace map is an expanding map, so a rounding error introduced at the third level is multiplied at every step after it, and by the fourteenth it is the size of the quantity itself.

The expansion is not an artefact either. It is the same expansion that makes the spectrum a Cantor set: the map’s sensitivity to its input is what drives bands apart at every scale. So the measured drift of a quantity that is exactly conserved in algebra is a measurement of the map’s instability, and reporting it as such is more informative than choosing eight levels and calling the invariant conserved.

Why the recursion exists at all #

A recursion on traces is a surprising object, and it is worth saying where it comes from, because the answer is the chain’s substitution and nothing else.

The Fibonacci word satisfies wₙ₊₁ = wₙ wₙ₋₁: the word at one level is the concatenation of the two before it. Transfer matrices multiply along a word, so the matrix of level n+1 is the product of the matrices of levels n and n−1, in that order.

Products of two-by-two matrices of determinant one satisfy an identity — tr(AB) + tr(AB⁻¹) = tr(A) tr(B) — which is elementary and is the entire mechanism. Applying it to the concatenation gives the three-term recursion, with the third term being the trace of a product involving an inverse, which the identity converts into the level two steps back.

So the trace map is the substitution rule, read through a matrix identity. A different substitution gives a different map; a periodic chain gives a map with a degenerate invariant; and the family of inflation factors this collection has enumerated is, from this point of view, a family of trace maps.

Bands that split rather than grow #

The picture of the approximants is the essay’s main evidence and it is worth reading carefully.

The approximant of N sites has exactly N bands — one per site, which is a theorem about periodic chains and is the check the machinery makes on its own scan. At the next level there are N′ bands, with N′ the next Fibonacci number, and each of them lies inside a band of the previous level.

So the sequence of allowed sets is nested and shrinking. Every gap opened at one level is present at every level after it, and new gaps open inside the surviving bands. The limit of a nested sequence of that kind, if the total width goes to zero, is a Cantor set: closed, containing no interval, and of measure zero.

The gaps have labels #

A gap in a spectrum is not merely an absence: each one can be given an integer, and the integers are what make the picture more than a sequence of pretty rows.

The gap labelling theorem says that the fraction of the spectrum lying below a given gap takes values in a discrete set determined by the chain’s own arithmetic — for the Fibonacci chain, numbers of the form m + nτ reduced modulo one, with m and n integers. So each gap carries a pair of integers, and the pair is stable: the same gap in a finer approximant carries the same label.

That is why the gaps opened at one level survive at every level after it. A gap is not a feature of the approximant that might close when the approximation improves; it is indexed by a pair of integers, and the integers do not change.

The labelling is a theorem of Bellissard’s and it is not computed here. What is worth carrying from it is the reason a nested picture is the right picture: the levels are not successive approximations that might converge to an interval, they are successive refinements of a structure whose gaps are already labelled.

What is proved and what is measured #

The distinction matters here more than usual, because the headline claim is about an infinite object and everything computed is finite.

Measured: that the band count equals the site count at each level; that the total width falls, monotonically, by a roughly constant factor; that the invariant is conserved to the precision of the arithmetic and drifts thereafter.

Not proved here: that the limiting spectrum has measure zero. That is a theorem of Sütő’s from 1989, and the measurements above are consistent with it and do not establish it. A sequence of shrinking sets need not shrink to nothing, and no finite number of levels can settle the question.

The collection’s standing practice applies: a bounded computation reports what it found inside its bound, and the theorem is cited as a theorem rather than implied by a graph.

Where the chain came from, and what else it is #

The chain’s two site types are the two tile lengths of the Fibonacci tiling, and that tiling has three descriptions this collection has already established as equivalent: a substitution applied repeatedly, a cut through a square lattice at the golden slope, and a sequence of approximants converging to it.

Every one of those descriptions produces the same word, and so every one produces the same spectrum. That is worth stating because the trace map looks like a fact about the substitution and is really a fact about the chain: cut-and-project would reach the same recursion by a longer road.

The equivalence also says which parts generalise. A substitution with a different rule gives a different recursion; a cut at a different irrational slope gives a different chain with a different invariant; and the qualitative conclusion — a Cantor spectrum — survives both.

The refusal that keeps the picture honest #

A band is found here by scanning the range and bisecting at the crossings, and a scan can miss a band. As the level rises the bands narrow, so the same scan that resolved every band at level five will miss several at level nine.

A missed band and a gap look identical in the output, which is exactly the sort of silent failure this collection builds refusals against. The refusal is available because the answer is known in advance: an approximant of N sites has N bands, so a scan that finds fewer has failed rather than discovered.

The machinery therefore raises an error naming the sample count rather than reporting the smaller number. A picture of a Cantor set built from a scan too coarse to resolve it would be a picture of the scan.

Comparing with the periodic case #

The clearest way to see what the chain is doing is to set the two site types equal and watch everything collapse.

At V = 0 the chain is periodic — every site is the same — and the transfer matrices are all equal. The trace map’s invariant is exactly zero, every level’s allowed set is a single interval, and there are no gaps at all. That is the ordinary band of a uniform chain.

Turn V up slightly and gaps open at every level, narrow at first. The invariant is small and positive, and the total width falls slowly.

Turn V up further and the gaps widen, the invariant grows, and the total width falls faster. Nothing qualitative changes: the spectrum is a Cantor set for every non-zero V, and V decides how quickly the width falls rather than whether it does.

So the invariant is not merely a check on the recursion. It is the parameter the whole picture depends on, and its vanishing is exactly the boundary between a crystal and a quasicrystal in this one-dimensional setting.

Where the chain’s aperiodicity shows up #

It is worth connecting the spectrum back to the chain’s own properties, because the two are usually presented separately.

The chain never repeats and repeats everything in it: every finite block occurs again within a bounded distance. Its diffraction is pure point — sharp peaks, indexed by two integers rather than one. And its spectrum, by this essay, is a Cantor set.

Those three facts are not independent, and the middle one is the odd member. Sharp diffraction peaks are the signature of long-range order, which the chain has; a Cantor spectrum with no bands is the signature of something closer to disorder. The chain manages both because it is neither periodic nor random, and the whole interest of quasicrystals is in that gap.

What the theorem says about the limit #

The figures show a nested sequence of finite sets shrinking, and the claim about the limit is quoted rather than derived. It is worth stating what has actually been proved, because it is stronger than the pictures suggest and it took a long time.

For every non-zero contrast between the two site types, the spectrum of the infinite Fibonacci chain is a Cantor set of zero Lebesgue measure. That is Sütő’s result of 1989, and each half of it is doing work: Cantor says the set is closed, has empty interior and no isolated points; measure zero says the total width the figures show falling really does fall to nothing.

Zero measure is the surprising half. A nested sequence of sets whose widths fall can perfectly well converge to something of positive width — the widths would have to fall geometrically to a positive limit, which nothing forbids in general. The measurement in the figure shows them falling by a roughly constant factor at each level, which is the shape a zero limit takes, and the theorem is what turns that observation into a statement about the limit.

How big the set is in a finer sense is a later and harder question. The Hausdorff dimension of the spectrum is strictly between zero and one, it depends on the contrast, and it tends to one as the two site types become alike and to zero as they become very different. So the set is small in measure and not small in dimension, and how small is a continuous function of a parameter — which is a much more informative statement than the measure alone.

The same phenomenon, in the model that made it famous #

The Fibonacci chain is not where a Cantor spectrum was first met, and its better-known relative is worth naming because the two are the same phenomenon under different arithmetic.

Put a periodic chain in a magnetic field and the flux through each cell enters the recurrence as a phase. When the flux per cell is an irrational fraction of the flux quantum, the problem is quasiperiodic in exactly the sense this essay is about — one irrational number generating a sequence — and the spectrum is again a Cantor set. Plotting the allowed values against the flux gives the Hofstadter butterfly, a figure of self-similar bands within bands that is one of the most reproduced pictures in the subject.

That the spectrum is a Cantor set for every irrational flux was conjectured in the 1980s and became known as the Ten Martini Problem, after an offer of ten martinis for a proof. It was settled by Avila and Jitomirskaya in 2009.

The connection to this page is the mechanism rather than an analogy. Both problems reduce to iterating a map on transfer matrices, both have a substitution or a continued fraction organising the iteration, and in both the gaps are labelled by the same kind of arithmetic — a discrete set of values determined by the irrational number itself. The Fibonacci chain is the case where the substitution is explicit and the arithmetic is the golden ratio’s, which is why it is where the machinery is easiest to build and check.

Where this goes #

The natural extension is the other one-dimensional quasiperiodic chains — the ones built from other inflation factors — where the trace map is different and the invariant is different and the qualitative conclusion is the same. Comparing the invariants across the family would say what the conserved quantity is measuring in general.

The nearer neighbour is the crystal obtained by rounding τ off, whose periodic approximants are exactly the chains whose spectra are drawn here. Each row of the hero figure is one of those approximants, so the two essays are looking at the same objects from opposite ends: one asks how well an approximant imitates the quasicrystal, and this one asks what happens to its spectrum as the imitation improves.