Concept

Itinerary

The sequence of labels an orbit collects, one for each region of the space it lands in, step by step. It turns the motion of a point into a string of symbols, so that questions about trajectories become questions about words.

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

Also named here as shift map — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

Shift map Symbolic dynamics Binary expansion Conjugacy Dense orbit Doubling map Fibonacci numbers Golden ratio Periodic orbit Topological entropy

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