Chords that never cross

Two rays for every fraction followed the external rays of the Mandelbrot set into the roots of the bulbs on its main cardioid. At the root of the $p/q$ bulb exactly two rays land, their angles are the two members of one cycle of the doubling map that turn round the circle $p$ places at a time, and the binary expansions of those angles are written by the staircase under a line of slope $p/q$. That account is complete for the cardioid, and it covers a small part of the set. The bulbs have bulbs of their own, the filaments carry small copies of the whole set, and each of those pieces has a root where a pair of rays lands.

Which pairs? The question has a startlingly economical answer, found by Pierre Lavaurs in 1986. List the angles of each period in increasing order and join them in pairs by one rule: each angle takes the next angle of its period whose chord, drawn across the circle, crosses none of the chords drawn so far. The rule never looks at the set. It is a statement about chords in a disc, and it produces, period by period, exactly the pairs of angles whose rays land together. This essay runs the rule, counts what it makes against the components it should be naming, and traces the rays into the plane to see the pairs land.

Periodic angles and why they come in pairs #

An external angle is a fraction of a turn, and the doubling map $\theta \mapsto 2\theta \bmod 1$ moves it the way squaring moves a point far from the set. An angle has period $n$ when $n$ doublings bring it back for the first time, and those angles are exactly the fractions $m/(2^n - 1)$ that are not already of some smaller period. In binary they are the expansions that repeat a block of $n$ digits, and doubling is the shift that drops the first digit, as the orbit written as a word set out — $1/7 = 0.\overline{001}$, $3/7 = 0.\overline{011}$ — and the count of them is the count of binary blocks of length $n$ that are not powers of shorter blocks: 2 at period 2, 6 at period 3, 12 at period 4, 30 at period 5. Windows counted like necklaces met the same count dressed as necklaces, and it is computed the same way, by Möbius inversion of $2^n$ over the divisors of $n$.

Douady and Hubbard proved that every parameter ray with a periodic angle lands, and lands at the root of a hyperbolic component — a region of the interior where the critical orbit is attracted to a cycle — whose period equals the angle’s. They also proved that every such root receives exactly two rays, with the one exception of the main cardioid’s root at $c = 1/4$, which receives only the ray of angle $0$. So the periodic angles of period $n$ fall into pairs, and the number of pairs is the number of components of period $n$. What their theorem does not hand over directly is which angle is paired with which.

The pairing is not a matter of neighbouring angles, nor of angles from the same cycle of doubling, although both are true for the cardioid’s bulbs. For the bulb at $1/2$ the pair is $1/3$ and $2/3$, neighbours of period 2. For the component of period 3 at $c \approx -1.7549$, on the real axis to the left of everything drawn so far, the pair is $3/7$ and $4/7$: neighbours again, but from two different cycles, $\{1/7, 2/7, 4/7\}$ and $\{3/7, 6/7, 5/7\}$. A rule that works for both has to be about something else.

The rule is about crossing, not about the set #

Lavaurs’s rule takes the periods in order. Period 2 has one pair, $1/3$ and $2/3$, drawn as a chord. Period 3 has six angles, $1/7$ to $6/7$; the smallest, $1/7$, takes $2/7$, since that chord crosses nothing; $3/7$ takes $4/7$; $5/7$ takes $6/7$. The next period is the first at which the rule has to refuse something.

The twelve angles of period four are $m/15$ for $m$ not a multiple of 5. The first two pairs are immediate: $1/15$ with $2/15$, $3/15$ with $4/15$. Then $6/15 = 0.4$ looks for a partner. Its nearest unpaired angle of the same period is $7/15 \approx 0.467$, but the chord from $0.4$ to $0.467$ would cross the chord from $3/7 \approx 0.429$ to $4/7 \approx 0.571$, since one end lies inside that chord’s arc and the other outside. So would $8/15 \approx 0.533$. The first candidate whose chord passes clear is $9/15 = 0.6$, which goes round the outside of $3/7$–$4/7$ altogether. The two refused angles then pair with each other, $7/15$ with $8/15$, inside the period-three chord, and the last two pairs, $11/15$–$12/15$ and $13/15$–$14/15$, are neighbours again.

All six pairs can be read against the set. The pair $6/15$ and $9/15$ lands at the root of the bulb of period four hanging from the period-two bulb at $c = -1$: one bulb further along the real axis, which the period-two chord fences off on the left of the disc. The pair $7/15$ and $8/15$ lands at the tip-side copy of period four near $c \approx -1.94$, inside the region the period-three pair fences off, which is where the copy of the set sits. The others are the cardioid’s $1/4$ and $3/4$ bulbs and two small copies on the filaments above and below.

The rule’s only use of geometry is that two pairs of rays landing at different points cannot cross, since rays are disjoint curves and each pair, together with its landing point, cuts the plane in two. A pair drawn as a chord must therefore never cross another. What Lavaurs proved is that this one necessary condition, applied greedily in the order of the angles, is also sufficient: it leaves no choice at all.

That is less obvious than it sounds, because non-crossing alone leaves a great deal of choice. Twelve points on a circle can be paired by non-crossing chords in 132 ways — the sixth Catalan number, the count every way to pair a polygon’s edges met as the gluings that make a sphere, and counting the paths that go wrong derived by reflection. The chords already drawn at periods two and three cut that number down, and a search through all 132 shows by how much: to two. One is the pairing in the figure. The other keeps four of its pairs and swaps the remaining two for chords that wrap round the angle $0$, joining $1/15$ to $14/15$ and $2/15$ to $13/15$ — non-crossing, consistent with every lower chord, and wrong. The greedy order rules it out, since $1/15$ takes its nearest admissible partner $2/15$ before anything wider is considered. That the smallest-first, nearest-first order picks the pairing the set actually uses, at every period, is the content of the theorem rather than a consequence of the picture.

One pair for every component #

The first test of the rule is arithmetic. The components of period $n$ have centres, the parameters at which $0$ returns to itself after exactly $n$ steps of $z \mapsto z^2 + c$. Those are the roots of the polynomial $f_c^n(0)$ in $c$, of degree $2^{n-1}$, with the roots of lower period removed; Gleason showed that all of them are simple, so the count of centres is a count of roots, and Möbius inversion gives $1, 1, 3, 6, 15, 27, 63, 120, \dots$ The rule’s pairs must match it.

They match at every period computed, to 2,010 pairs at period 12, and the agreement is not built in: the rule could in principle get stuck, with an angle left that has no partner it can reach without crossing, or it could pair angles that belong to different components. Neither happens. Every angle finds a partner, and there are always exactly half as many pairs as angles, as the count of centres requires.

The figure also splits the pairs into two kinds, and the split is the most useful thing the angles say about a component. In some pairs the two angles lie on the same cycle of doubling, as $1/3$ and $2/3$ do and as every pair on the cardioid did in the essay before this one. In the others they lie on two different cycles, as $3/7$ and $4/7$ do. The first kind are the satellite components: bulbs attached directly to a component of lower period, at a point where that component’s cycle has a multiplier $e^{2\pi i p/q}$. The second kind are the primitive components, each the main cardioid of a small copy of the whole set, sitting at the end of a filament.

The satellites can be counted independently. A component of period $d$ carries one bulb for each fraction $p/q$ in lowest terms, of period $dq$, so the satellites of period $n$ number the sum of $\varphi(n/d)$ over the components of each period $d$ dividing $n$, with $\varphi$ Euler’s totient that two rays for every fraction met in its identity $\sum \varphi(q)/(2^q - 1) = 1$. The same-cycle pairs match that sum at every period: 1, 2, 3, 4, 7, 6, 12, 12, 23, 10, 51. At period 11, which is prime, only the cardioid has a bulb of that period dividing it, and the count is $\varphi(11) = 10$. Everything else is primitive — 1,959 of the 2,010 pairs at period 12 — so almost every component of a large period is the heart of a small copy of the set, and only a thin fraction are bulbs on something larger.

The order of the angles matters less and less #

At period four the rule refused two candidates; at the cardioid’s bulbs it refused none. It is tempting to read the rule as “pair each angle with its neighbour, unless a chord is in the way”, with the chords a rare correction. The computation says the opposite.

At period five the rule refuses 6 candidates in making 15 pairs; at period seven, 120 in 63; at period nine, 1,372 in 252; at period twelve, 51,998 in 2,010 — twenty-six refusals for every pair. An angle of period twelve typically passes over dozens of nearer angles of its own period before it reaches one it can be joined to without cutting a chord drawn at some lower period. The chords of the lower periods form a nested system of fences, and by period twelve almost every pair is decided by which fenced region an angle sits in, not by which angle sits next to it.

That is the reason the rule must take the periods in order. The pairs of each period are only determined once every lower period’s chords are in place, and the whole construction is a recursion on the period, each level fixing the fences inside which the next level pairs up. The finished picture of the hero figure, with its nests of curves round $1/2$ and its fringes at the cardioid’s bulbs, is the record of that recursion. It is the circle seen as Thurston saw it: the quadratic minor lamination, the system of all chords between angles that land together, of which Lavaurs’s pairs are the periodic part.

Why two rays meet: the same angles in one picture #

The reason a pair lands together comes from the other plane. One c, one picture fixed a parameter and drew the filled Julia set of $z \mapsto z^2 + c$, the starting points whose orbits stay bounded, and the Julia set has rays of its own: curves of constant angle coming in from infinity to its boundary, on which squaring acts by doubling the angle. Douady’s principle for the subject is to plough in the dynamical plane and harvest in the parameter plane: a fact about rays in one Julia set becomes, by continuity in $c$, a fact about parameter rays.

Take $c$ at the centre of a component of period $n$. Its critical point $0$ lies in an attracting cycle of $n$ basins, and the basin containing $c$ itself has, on its boundary, a single point where two dynamical rays land — rays whose angles are periodic of period $n$ and which together separate that basin from the rest. Those two angles are the pair. For $c$ near $-1.7549$, the centre of the period-three copy, the basin round $c$ is fenced off by the dynamical rays $3/7$ and $4/7$, and as $c$ moves out of the component to its root, the cycle and the landing point of those two rays collide; at that parameter the rays of angle $3/7$ and $4/7$ in the parameter plane land as well. The pair is read off a picture of a single Julia set, and Lavaurs’s rule is a way of reading it off for every component at once, without computing any Julia set.

Tracing the rays into the plane #

The counts agree, but a count cannot tell whether $7/15$ really goes with $8/15$ rather than with some other angle of period four. For that the rays have to be drawn. Each parameter ray is traced in from far outside the set by Kawahira’s method, as in the essay before this one: at each depth, Newton’s method solves for the parameter $c$ whose critical orbit escapes at a prescribed radius and angle, starting from the previous point, with the angle doubled exactly in integers at each level.

Twenty-four pairs at periods three to five give forty-eight rays, and the picture shows them coming in from all sides in their pairs. The period-three pair $3/7$ and $4/7$ approaches along the real axis from either side and ends at the copy near $-1.75$. Period-five pairs reach the cardioid’s four fifth-bulbs, three more points along the negative real axis, and small copies of the set scattered above and below it. At the end of each pair a dot marks a centre: Newton’s method, started at the midpoint of the two rays’ ends and solving $f_c^n(0) = 0$, converges to a parameter whose critical orbit has exact period $n$.

The drawing shows the rays coming in together. To check that they land together, every pair to period seven was traced and measured.

Partners end together, and the two kinds end differently #

At a depth of 100 doublings a ray is very close to its landing point but has not reached it, and how close depends on the kind of point. The test is therefore relative: for each pair, compare the distance between the ends of its own two rays with the distance from either end to the nearest end of any other pair.

All 115 pairs pass. The nearest stranger is at least 5.1 times further away than a ray’s own partner, and for 78% of the pairs a thousand times further or more. From the ends of each pair, Newton’s method finds a centre of exact period $n$, and the centres found are all distinct: 1 at period 2, 3, 6, 15, 27 and 63 at period 7 — every centre of every period to seven, each reached from one pair and no centre reached twice. So the rule’s pairing, computed on a circle, is confirmed in the plane for 115 components.

The scatter separates into two clouds, and the separation is exactly the satellite–primitive split. The 23 same-cycle pairs end between 0.0034 and 0.063 apart; the 92 two-cycle pairs end within $3.7 \times 10^{-5}$ of each other, most within $10^{-6}$. The geometry explains it. A satellite’s two rays come down either side of a bulb, and at any finite depth they are still separated by the bulb’s width; they close only as the depth grows, and slowly, like one over the depth, which is how two rays for every fraction measured them reaching the roots on the cardioid. A primitive component’s rays come down either side of a filament that leads to the small copy, and a filament has almost no width, so its two rays run side by side for most of their length and pinch together long before they arrive.

What the chords cannot show #

The rule and the checks above settle the periodic angles completely, to the periods computed and, by Lavaurs’s theorem, at every period. They say nothing directly about the rest of the boundary.

The rays drawn here all land at roots, which are special points: parabolic parameters where a cycle has multiplier exactly one. Most of the boundary consists of other points. Rays with angles whose binary expansions are eventually periodic but not periodic — $1/6 = 0.0\overline{01}$, for instance — land at the tips of filaments, the Misiurewicz parameters where the critical orbit lands on a repelling cycle after a few steps; the rule here does not pair those, and they are joined in Thurston’s lamination by chords of their own. Rays with irrational angles are the bulk of all rays, and whether every one of them lands is not known.

The disc picture also loses the sizes. A chord’s length on the circle — the width of the arc it cuts off — is the share of external angles fenced off behind the component, which two rays for every fraction measured as harmonic measure, the chance that a random path from infinity first meets the set behind that pair. It is not the component’s size in the plane. A small copy of the set at period seven can be fenced off by an arc of width $10^{-6}$ and still be visible on a screen; a bulb can be large in the plane and fence off a modest arc. The lamination records how the pieces are joined, not how large they are.

And the figures stop at period seven for the traced rays, period twelve for the counts. The rays are traced numerically and their ends measured to a fixed depth; that they actually land, and land at the components found, is the theorem, and the computation is consistent with it at every point tested.

Still open: whether the chords are the whole set #

Is the Mandelbrot set exactly the closed disc with every lamination chord collapsed to a point? Thurston’s lamination contains Lavaurs’s chords, the chords of the pre-periodic angles, and their limits. If the set is locally connected — the conjecture, usually written MLC, that has stood since Douady and Hubbard’s work in the 1980s — then collapsing each chord of the lamination to a point turns the circle into the set’s boundary, and the combinatorics computed here is a complete model of its shape: two points of the boundary coincide exactly when their angles are joined. Without local connectivity, the model and the set could differ on parts of the boundary too intricate for any chord system to describe. The conjecture is known to hold at many points — at every parameter that is only finitely renormalisable, by Yoccoz’s work, and at many infinitely renormalisable ones, by Lyubich’s and others’ — and it is open in general, and it would also settle the density of hyperbolic components, the question the area of the Mandelbrot set, from two sides met as a number nobody can compute.

There is a smaller open question in the refusals. The count of refused candidates grows by a factor of about three a period, faster than the pairs themselves, which double. No formula for it is known, and none is needed for the rule to work, but the count measures how deeply the fences of all earlier periods nest round a typical angle — something like the depth of the lamination at period $n$ — and a formula would say how that depth grows.

Joined by a rule about crossing #

Every root in the Mandelbrot set receives exactly two periodic rays, and which two can be computed on a circle, without drawing the set. Take the periods in order; within a period, join each unpaired angle to the next one whose chord crosses no chord already drawn. The rule makes 1, 3, 6, 15, 27, 63, 120 pairs and on to 2,010 at period twelve, matching the count of centres every time; the pairs whose two angles share one cycle number exactly the satellite bulbs and the rest root the small copies of the whole set. Traced in to a depth of 100 doublings, all 115 pairs to period seven end closer to their partners than to any other ray, beside 115 distinct centres of the right periods — and the rule, which begins as a rule about neighbours, ends at period twelve refusing twenty-six candidates for every pair it makes.