Circle map
Named by 7 essays across one field — 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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Rotation numberOrbitBifurcationPeriodic orbitSelf-similarityStabilityConjugacyContinued fractionsIrrational rotationMeasureParameterRenormalisation