Concept

Shift map

The map that deletes the first symbol of an infinite string and moves the rest one place along. Many maps of an interval are this map in disguise once each point is replaced by its itinerary, which is what makes symbolic dynamics work.

Named by 4 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

Writing 0.3 in base φ. The graph of x ↦ βx mod 1 for the golden ratio φ, the diagonal, and the cobweb staircase of the orbit of 0.3, beside the 8 digits it writes.

Counting in a base that is not a whole number

Multiply by β and keep the fractional part, over and over, and the whole parts you throw away are the digits of the starting number in base β — even when β is the golden ratio. Which digit strings can ever appear is decided by one string alone: the way the number 1 is written in that base. In base φ it is .11, so 11 is the only thing forbidden; in base 1.8 it never ends, and no finite list of rules describes what is allowed.

dynamics · Symbolic dynamics

The Collatz map on the 2-adic integers, before and after changing to parity coordinates. Two scatter plots of 2048 points: the Collatz map on 2-adic integers in binary-digit coordinates, a scattered cloud, and the same map in parity-vector coordinates, where every point lies on the two lines of the doubling map.

Where the Collatz map is a coin

Extend the Collatz map from the whole numbers to the 2-adic integers — binary strings that run on for ever to the left — and it stops being mysterious. It becomes, after a change of coordinates, the simplest chaotic system there is: shifting a string of coin tosses one place. Everything about it is then known, and none of it says anything about the whole numbers, which is the most instructive failure in the whole story.

dynamics · Collatz

One number, fair in base 2 and gappy in base 6. α binary: 000010101011100011100011100011110110100001001011…; base 6: 013014043000333425113050213000000124355504543223…; zero runs after 3^k: 81:16, 243:42, 729:121.

Fair in base two and not in base six

Almost every number has digits that look like fair coin tosses in every base, and almost no number anyone can name is known to. Stoneham's number, 1/(3·2³) + 1/(9·2⁹) + 1/(27·2²⁷) + …, is one of the few: its binary digits are provably fair. Written in base 6 the same number has a run of 121 zeros starting at the 730th digit, and a longer one at every power of three after — because in base 6 each of its terms is a terminating fraction, and the gaps between them never fill.

dynamics · Symbolic dynamics

Named alongside it

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

Symbolic dynamics Conjugacy Doubling map Itinerary Binary expansion Collatz Decimal expansion Dense orbit Equidistribution Fibonacci numbers Golden ratio Measure

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