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Concept

Dense orbit

A sequence of points under a repeated map that comes arbitrarily close to every point of the space. An irrational rotation of a circle has one from every starting point, and density is the first step towards the stronger property of equidistribution.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The orbit of 13/32 under doubling, and its word. A cobweb of the doubling map with one orbit drawn, the interval split in half beneath it, and the letter each step contributes written out in order.

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.

dynamics · Symbolic dynamics Lissajous figures for every coprime pair of frequencies up to 4. A 4 by 4 grid of Lissajous figures x = sin(pt + 0.3), y = sin(qt), with the crossing count 2pq − p − q under each and the non-coprime pairs left blank.

When two circular motions come home

Drive a point across with one sine wave and up and down with another. If the two frequencies are in a whole-number ratio the point retraces a closed figure whose crossings can be counted in advance — 2pq − p − q of them — and if they are not, it never comes back and fills the square, spending twenty times longer in the corners than in the middle.

analysis · Circular functions

Named alongside it

The objects these essays reach for when they reach for this one.

Arcsine distributionBinary expansionConjugacyCounting argumentDoubling mapEquidistributionIrrational rotationItineraryPeriodic orbitPeriodicityShift mapSine

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