Two humps and one entropy

The entropy in a single orbit read the chaos of the logistic map $x \mapsto rx(1-x)$ off the orbit of its single turning point. Milnor and Thurston’s kneading theory turns that orbit’s itinerary — left of the turning point or right of it, step by step — into a power series whose smallest root gives the entropy, and the computation agreed with the growth of the map’s folds at every parameter tried. It also confirmed the property that makes the one-humped family orderly: as $r$ increases the entropy never decreases. Chaos, once switched on, does not switch off as the hump grows.

That essay ended with the question one step up. A map with two turning points — a hump and a dip — has two critical orbits, and its family needs two parameters, so the entropy is a function on a plane rather than on a line. “Never decreases” no longer has a meaning, since a plane has no direction of increase. What replaces it is a question about level sets: the isentropes, the sets of maps with a given entropy. On a line, monotonicity says that each level set is an interval. On a plane the analogue is that each level set is connected — one piece, so that any two maps of the same entropy can be joined by a path along which the entropy never changes. John Milnor conjectured that for cubic maps in 1992, and Milnor and Charles Tresser proved it in 2000.

What monotonicity meant for one hump #

The one-humped statement has a history that explains why the two-humped one was hard. That the logistic family’s entropy never decreases as $r$ grows was conjectured from computer experiments in the 1970s, when the period-doubling cascade and its universal constants were being discovered, and it was proved in the early 1980s by Milnor and William Thurston together with Adrien Douady, John Hubbard and Dennis Sullivan. Every proof went through the complex numbers. The real map $x \mapsto rx(1-x)$ is a slice of the complex quadratic family, and the rigidity of complex dynamics — two quadratic polynomials whose critical orbits behave the same way are the same polynomial — is what forbids the entropy from turning round.

The real line on its own offers no such argument. A family of real maps chosen differently, with the same one hump but a different shape, can have entropy that rises and falls as its parameter moves. Monotonicity is a property of the quadratic family, inherited from the complex plane, and not of one-humped maps as such. So when the family has two humps and two parameters, the question is not merely whether a plane can behave like a line, but whether the complex rigidity that saved the line survives the extra dimension. For cubics it does, and that is the content of Milnor and Tresser’s theorem.

A plane of cubic maps #

The family is every cubic polynomial $f$ with $f(0) = 0$ and $f(1) = 1$ that rises, falls and rises again on $[0, 1]$. Such a cubic has turning points $c_1 < c_2$ in the interval, and its derivative is $k(x - c_1)(x - c_2)$ for a constant $k$ fixed by the requirement $f(1) = 1$. Its hump is the value $M = f(c_1)$ and its dip is $m = f(c_2)$, and when $M$ is at most 1 and $m$ at least 0, the whole interval is carried into itself, so iterating makes sense. The cubic is determined by its critical values, so the family is a region of the plane with coordinates $(M, m)$: the hero draws the square $M \in [0.5, 1]$, $m \in [0, 0.5]$, and for every grid point of it the figures solve for the turning points by Newton’s method and compute the entropy.

At the corner $M = 1$, $m = 0$ is the full cubic, whose hump touches the top and whose dip touches the bottom; it maps each of its three monotone pieces onto the whole interval, exactly as the doubling map does with two, and its entropy is $\log 3$. At the far corner, with a low hump and a high dip, the map is nearly the identity with a small wiggle, and its orbits settle down without any chaos at all. The hero’s shading runs between the two, darkest at the full map.

Counting laps exactly #

The entropy of an interval map is the growth rate of its laps, the maximal pieces on which an iterate is monotone: if $f^n$ has $\ell_n$ laps, the entropy is $\lim \frac1n \log \ell_n$. The lap count is defined combinatorially, but counting it naively means finding every turning point of $f^n$, of which there can be $3^n$. The figures avoid that by following images instead of points.

Each lap of $f^n$ is mapped by $f^n$ monotonically onto some interval $J$. The laps of $f^{n+1}$ inside it correspond to the pieces of $J$ cut at whichever turning points $c_1, c_2$ lie inside $J$, each piece then mapped onwards by $f$. So the whole lap structure of $f^{n+1}$ is determined by the list of image intervals of the laps of $f^n$, and the crucial point is that this list has few distinct members: every endpoint of an image interval is an endpoint of $[0, 1]$ or a point on the orbit of one of the two critical values. Merging identical intervals and keeping their multiplicities, the figures follow a few hundred intervals rather than billions of laps, and the number of laps is the sum of the multiplicities. After 120 steps the entropy is read off as the average growth over the last twenty.

For the full map the count is exactly $3^n$: its line rises by $\log_{10} 3$ per step and reaches $10^{28.6}$ at sixty. The map with hump 0.95 and dip 0.08 grows by a smaller factor, with entropy 0.744. The map with hump 0.8 and dip 0.3 reaches only 3,664 laps after sixty steps, about $n^2$, which is the polynomial growth of a map with zero entropy; its average growth over steps 100 to 120 reads 0.018, the small positive value that a quadratic count produces over a finite window. That floor is visible as the palest shade in the hero, and everything darker is genuinely chaotic.

The full map in the corner is worth a closer look, because its entropy can be checked without counting anything. A cubic is fixed by its endpoints and its critical values, and the cubic $T_3(x) = 4x^3 - 3x$ on $[-1, 1]$ has fixed endpoints and critical values $\pm 1$; so the full map of the hero is $T_3$ in different coordinates. Writing $x = \cos\theta$ turns $T_3$ into $\theta \mapsto 3\theta$ — a solvable chaos whose every orbit can be written down — and tripling an angle has entropy exactly $\log 3$. The lap count’s $3^n$ is the same fact seen from the interval. Every other map in the plane is a deformation of this one, and the entropy measures how much of tripling’s mixing survives the deformation.

Two critical orbits #

A one-humped map’s dynamics are organised by one orbit, its turning point’s. A two-humped map has two, and they interact.

The orbit of the hump and the orbit of the dip wander over the interval separately, and each visits the regions left of the hump, between the turning points, and right of the dip in its own sequence. Recording those visits gives two kneading sequences, and Milnor and Thurston’s theory, extended to maps with several turning points, computes the entropy from the pair.

The two sequences are as irregular as chaos makes them, and that is the point of the plane of critical values. A one-humped family is a line because one number — the height of the hump — controls one kneading sequence. A two-humped family needs two numbers to control two sequences, and the plane of $(M, m)$ is a coordinate system for pairs of kneading sequences. A level set of entropy is the set of pairs whose combined growth rate is the same.

When the laps follow a pattern #

For most maps in the hero the critical orbits wander for ever, and the image intervals the lap count follows keep changing. For special maps they do not. If both critical orbits are eventually periodic — the hump’s orbit lands on a cycle and so does the dip’s — then the image intervals have endpoints drawn from a finite set, and the lap structure of every iterate is a rearrangement of finitely many pieces. The interval then splits into finitely many subintervals that the map carries across one another in a fixed pattern, and the number of laps grows like the powers of a fixed matrix. The entropy is exactly the logarithm of that matrix’s largest eigenvalue, as the matrix that counts the returns showed for maps of one hump, and it is an algebraic number.

That is how the entropy can be computed exactly at special parameters, and it is part of what makes the picture a staircase rather than a smooth slope: wherever both critical orbits are attracted to stable cycles, the entropy is constant on a whole open region of the plane, and at the centre of such a region the critical orbits are themselves periodic, so the constant value is the logarithm of an algebraic number. A single-humped map with entropy $\log \beta$ is, in the right coordinates, counting in base $\beta$, with its kneading sequence playing the part of the expansion of 1. Two humps make it two expansions at once, interleaved, and the pair is what the plane’s two coordinates choose.

The level curves are connected #

The hero answers the question on the computed grid. For each of the levels 0.2, 0.4, 0.6, 0.8 and 1.0, the figures trace the level curve through the grid cells by linear interpolation and then check how many separate pieces the traced segments form, joining segments that share an endpoint. Every level comes out as one piece. The curves run from the hero’s top edge, where the dip is at the floor, round to its right edge, where the hump is at the ceiling, and they nest one inside another towards the full map in the corner.

That is what Milnor’s conjecture predicts, and what Milnor and Tresser proved for real cubic maps in 2000. Its content is not obvious. Entropy is a wild function — it is continuous on these families, but it climbs in a devil’s staircase, constant across every region where some orbit is stably periodic and rising only through a set of parameters of small measure — and wild functions on a plane can have level sets with many components, islands of high entropy cut off from others by valleys of lower. The theorem says this never happens for cubics. Any two cubic maps with the same entropy can be joined through cubic maps of that entropy.

The proof uses complex analysis. A real cubic is a cubic polynomial with real coefficients, and polynomials can be iterated on the complex plane, where their dynamics are rigid: two polynomials with the same combinatorics of their critical orbits are often the same polynomial up to a change of coordinates. Milnor and Tresser combined that rigidity with the kneading theory to show that each isentrope is a connected union of combinatorial classes. The general statement for polynomials of every degree — monotonicity of entropy in the many-parameter sense — was proved by Henk Bruin and Sebastian van Strien in 2015, using the density of hyperbolic maps established by Oleg Kozlovski, Weixiao Shen and van Strien.

Along a line through the plane #

Restrict the plane to a line and the old one-parameter picture returns, with a caveat.

Along the symmetric line, where $m = 1 - M$ and the map is unchanged by turning the square upside down, the entropy climbs from nothing to $\log 3$ in the same staircase of plateaus as the logistic family’s, each plateau a window in which some orbit is stably periodic. Along the line where the dip is held at 0.05 and the hump rises, it climbs from nothing to 0.967, short of $\log 3$ because the dip never reaches the floor. Both climbs look monotone at this resolution. The connectedness theorem does not require that — a line through the plane can cross an isentrope, leave it and cross it again, if the isentrope is curved — and nothing in the theorem forbids a line through the plane from seeing the entropy fall and rise again. These two lines happen not to, at the resolution drawn.

Entropy read at a hundred and twenty steps #

Entropy at 120 steps, not in the limit. The lap count is exact for each iterate, but the entropy is a limit, and the figures read it off the growth from step 100 to step 120. For chaotic maps that is accurate to several decimals; for maps of zero entropy it leaves a floor of about 0.02, as the laps-figure shows, and maps just beyond the onset of chaos, where entropy is small and positive, are not well separated from that floor.

Connected on the grid, not in the plane. The level curves are traced through an 80 × 80 grid by interpolation, and a component smaller than a grid cell, or two pieces joined only through a cell’s interior, would be invisible. The computation is evidence for connectedness at the grid’s resolution; the theorem is what makes it true at every resolution.

Coordinates found numerically. Each grid point is a pair of critical values, and the cubic with those values is found by Newton’s method from the nearest entry of a coarse table of turning points, then checked: its hump and dip must match the requested values to within $10^{-9}$, with the turning points in order inside the interval. Every one of the 6,561 grid points in the drawn square passed that check, so the picture has no holes; but the solution is found in floating point, and the critical values of the map whose entropy is reported differ from the grid’s by that small amount.

One normalisation. The family here fixes both endpoints of the interval, so every map has fixed points at 0 and 1. Other normalisations of cubic maps — maps that send the endpoints to each other, or that leave an interval invariant without fixing its ends — give different two-parameter families, and Milnor and Tresser’s theorem covers them as well; the figures draw only this one.

Still open: how the plane is laid out #

Connectedness answers the question it asks and leaves others. Where are the boundaries between zero entropy and positive entropy? On the logistic line it is a single point, the end of the period-doubling cascade at $r = 3.5699\ldots$. In the cubic plane it is a curve — the edge of the palest region in the hero — and it is built from infinitely many cascades, one for every way the two critical orbits can double their periods together. Its shape near the full map, its smoothness, and whether it has the universal scaling of Feigenbaum’s constants along it everywhere or only at special points, are known only in part.

A second question is about the words the orbits write. Every point of the plane has a pair of kneading sequences, and the pairs that actually occur are constrained by admissibility conditions that are known for one turning point and much more intricate for two. Which pairs of sequences are realised by some cubic, and how the realised pairs fill the plane, is the combinatorial skeleton of the hero’s picture, and it is only partly drawn. And for maps of degree four and higher, with three or more turning points, the plane becomes a space of three or more dimensions in which the same connectedness holds by Bruin and van Strien’s theorem — but where nobody has drawn the level sets, because a figure cannot.