Irrational rotation
Named by 4 essays across one field — each of them below, with the objects they name alongside it.
Three gaps and no more
Turn a circle by the same irrational angle over and over. The points never repeat and never settle, and yet at every single stage the gaps they leave take at most three different lengths — never four, at any number of steps, for any angle.
The orbit that must come back
A system with finitely many states has to repeat itself. Poincaré showed the same thing holds when the states are a continuum — almost every starting point returns arbitrarily close to where it began, however complicated the rule, and the argument is the pigeonhole principle with volume in place of counting.
A bounce is a fold of the table
Reflect the room instead of the ball and every bounce disappears — the trajectory becomes a straight line through a tiled plane, and questions about what a ball does forever become questions about the slope of that line.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
OrbitEquidistributionIterationMeasurePeriodic orbitApproximationBifurcationBilliardsCausticCircle mapContinued fractionsEntropy