Concept

Binary expansion

A number written as a sum of powers of two, so that it is a string of noughts and ones. Doubling shifts the string one place, which is why maps that double are exactly readable from the expansion and why finite arithmetic loses them after fifty-odd steps.

Named by 2 essays across one field — 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 $The orbit a computer draws, and the orbit. Two orbits of the tent map from the same starting fraction plotted against the step number — one computed exactly in whole-number arithmetic and periodic, one computed in double precision and reaching zero.$

The orbit a computer draws

A chaotic orbit computed in floating point is not the orbit of the point it started from. Sometimes it is the true orbit of a nearby point, which is enough; sometimes the arithmetic simply runs out, and the picture is of the rounding.

dynamics · iteration

Named alongside it

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

Periodic orbit Conjugacy Dense orbit Doubling map Exact arithmetic Floating point Iteration Itinerary Numerical artefact Rounding error Sensitive dependence Shadowing

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