Doubling map
Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.
The orbit written as a word
Cut the interval in two and record which half each step of an orbit lands in. The orbit becomes an infinite string of two letters, the map becomes the act of deleting the first letter, and questions about trajectories turn into questions about words.
Almost every orbit is fair
Double a number and keep the fractional part, and do it again, and again. Where the orbit goes is written in the number's binary digits, and for almost every starting point it spends exactly a quarter of its time in each quarter of the interval. The proof is a short argument about Fourier coefficients being pushed to infinity — and it leaves room for a set of exceptions with no length at all, which includes every fraction and, for all anyone can prove, every number anyone has ever named.
Sensitivity comes free
The standard definition of chaos asks for three things: an orbit that goes everywhere, periodic orbits everywhere, and sensitive dependence on the starting point. The third, the one the word chaos is usually taken to mean, turns out to follow from the other two. A periodic point and a wandering point that start side by side must eventually part, because the wanderer has to visit places the periodic orbit never goes.
Multiplying every number on the dial at once
Join every residue on a dial to twice itself and the chords draw a heart-shaped curve with one cusp; join each to three times itself and the curve has two. The picture is the whole multiplication map at once, and it holds three facts: the map splits the dial into cycles whose lengths are orders, those cycles on a dial of 2ⁿ − 1 are the binary necklaces of length n, and the curve is the caustic light draws inside a cup.
Chaos on the line between two roots
Newton's method for a quadratic is the one case solved completely: every starting point goes to the nearer root, and the plane splits into two half-planes. Cayley did it in 1879. What his solution also contains is a line on which the method never settles — and a change of variable shows that on that line Newton's method is exactly the doubling map, the simplest chaos there is. Run on the real line for x² + 1, which has no real root to find, it becomes a source of Cauchy-distributed randomness.
Two rays for every fraction
Seen from far away, every point of the Mandelbrot set's outside has an angle, and the curves of constant angle come in to land on the set. At the root of the p/q bulb exactly two of them land, and their angles are written by the staircase under a line of slope p/q. The limbs those pairs fence off share out the whole circle of angles between them, by the identity Σ φ(q)/(2^q − 1) = 1.
Fair in base two and not in base six
Almost every number has digits that look like fair coin tosses in every base, and almost no number anyone can name is known to. Stoneham's number, 1/(3·2³) + 1/(9·2⁹) + 1/(27·2²⁷) + …, is one of the few: its binary digits are provably fair. Written in base 6 the same number has a run of 121 zeros starting at the 730th digit, and a longer one at every power of three after — because in base 6 each of its terms is a terminating fraction, and the gaps between them never fill.
Chords that never cross
Every component of the Mandelbrot set's interior has a root, and exactly two external rays land there, with angles that repeat under doubling. Which two can be found without drawing the set at all: list the angles of each period in order, and join each to the next one whose chord crosses no chord already drawn. The rule makes 63 pairs at period 7 and 2,010 at period 12, one for every component; traced into the plane, every pair's two rays end together, beside a centre of the right period and no other.
Named alongside it
The objects these essays reach for when they reach for this one.
Periodic orbitBinary expansionConformal mapConjugacyDense orbitMandelbrot setNormal numberOrbitSensitive dependenceShift mapSymbolic dynamicsTotient