Concept

Doubling map

The map sending x to 2x with the whole part discarded, on the numbers from 0 to 1. It shifts binary digits one place left, which makes its orbits exactly as varied as the digit sequences, and it is the simplest chaotic system that can be analysed completely.

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 $How long four orbits of the doubling map spend in each quarter. Grouped bars for four starting points of the doubling map, each group showing the fraction of time the orbit spends in each quarter of the unit interval, against a dashed line at one quarter.$

Almost every orbit is fair

Double a number and keep the fractional part, and do it again, and again. Where the orbit goes is written in the number's binary digits, and for almost every starting point it spends exactly a quarter of its time in each quarter of the interval. The proof is a short argument about Fourier coefficients being pushed to infinity — and it leaves room for a set of exceptions with no length at all, which includes every fraction and, for all anyone can prove, every number anyone has ever named.

dynamics · Iteration

Named alongside it

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

Binary expansion Conjugacy Dense orbit Ergodicity Fourier series Invariant measure Itinerary Measure zero Normal number Periodic orbit Shift map Symbolic dynamics

All concepts