Rotation number
Named by 13 essays across 3 fields — each of them below, with the objects they name alongside it.
The staircase that is flat almost everywhere
A map of the circle advances by an average amount each step. Plot that average against the parameter driving it and the graph is flat over an interval at every rational, rises only on a set of measure zero, and still climbs from nothing to one.
How a lock comes apart
Inside a plateau the map has two periodic orbits, one attracting and one pushing away. Track them to the plateau's edge and they run into each other and vanish together — so the edge of a tongue is a collision, and the width of the plateau is how far the two can be pulled apart.
A rotation in different coordinates
At an unlocked parameter the map is not merely like a rigid rotation; it is one, after a change of coordinates built out of a single orbit. The theorem needs the map to be smooth enough, and the map that shows why is one whose orbit leaves a hole.
A whole interval of speeds
Below the critical line every orbit of the circle map goes round at the same average speed. Above it the map folds back on itself, and the speed depends on where the orbit starts — not a few different values but a whole interval of them, every fraction in it the speed of some periodic orbit, and almost none of them ever seen by an orbit started at random.
The last circle to break
Kick a spinning rotor once a turn and most of its motions stay on curves that wind round forever, walls no orbit can cross. As the kick grows the walls break one by one, and the last to go is the one whose winding is the golden ratio — at a kick of 0.9716, found by watching a sequence of periodic orbits approximate it and asking whether they are stable.
A triangle that fits once fits everywhere
Put one circle inside another and try to fit a triangle between them, its corners on the outer circle and its sides touching the inner. Usually no triangle fits. But if one does, then one fits starting from every point of the outer circle — and whether it does is decided by a single equation in the two radii and the distance between the centres.
How smooth the disguise is
An unlocked circle map is a rigid rotation in different coordinates, and whether those coordinates are smooth is decided by arithmetic. Solving for the change of coordinates means dividing, harmonic by harmonic, by numbers that come arbitrarily close to nought — and the golden rotation number keeps them far enough away that the result is not merely smooth but analytic, while a number lying close to a fraction makes it buckle.
The fraction written on every bulb
Round the edge of the Mandelbrot set's cardioid hangs one bulb for every fraction between nought and one. The fraction is the bulb's address, its period, and the turn its cycle makes round a fixed point, all at once, and it sets the bulb's size as well: sin(πp/q)/q², which 245 computed bulbs obey to within fourteen per cent.
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.
Two constants at the flat point
Turn up the circle map until its graph goes flat at one point, then hunt for the setting at which an orbit turns by the golden ratio. The settings that lock to 1/2, 2/3, 3/5, 5/8 close in on it, each gap 2.8336 times smaller than the last, and the orbit's returns to its start close in by 1.2886 at every step. Below the flat line the same two numbers are the golden ratio's own. Change the map to a different one with the same kind of flat point and both numbers come back unchanged: they belong to flatness, not to the sine.
The silver mean at the flat point
At the golden-mean rotation number, the circle map with a flat point has two universal constants, 2.8336 and 1.2886. Repeat the measurement at the silver mean, √2 − 1, and two different numbers appear — 6.7992 and 1.5868 — shared by every map with the same kind of flat point. Below the critical line the silver gaps shrink by (1 + √2)², the square of the silver ratio, exactly as a rigid rotation predicts; on it they do not. Every rotation number with a repeating continued fraction carries its own pair, and a number whose expansion ends in ones inherits the golden pair, whatever came first.
A disc that turns for ever
At each fraction on the Mandelbrot set's cardioid a bulb is born. At the irrational numbers between them something rarer can happen: a disc round the fixed point rotates rigidly for ever, with no point falling in or flying out. Whether it exists depends on nothing but the digits of one continued fraction, and on one sum of their logarithms.
An orbit stuck to an island
A chaotic orbit in the standard map stretches at 0.44 a step, and then for 19,000 steps it almost stops stretching at all. It has wandered into the fringe of an island of regular motion and been caught. So gaps between visits to a box do not die away exponentially as they would in a uniformly chaotic map; they fall as a power, and an orbit of 400 million steps still has not seen the longest.
Named alongside it
The objects these essays reach for when they reach for this one.
Circle mapPeriodic orbitOrbitStabilityBifurcationChaosSelf-similarityConformal mapConjugacyContinued fractionsConvergence rateDiophantine approximation