Itinerary
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
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.
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.
The entropy in a single orbit
Topological entropy measures how fast all the orbits of a map come apart, which sounds like a property of every orbit at once. For a map with one fold it is decided by just one: the orbit of the turning point. Read that orbit as a string of signs, make a power series of it, and the entropy is minus the logarithm of the series' first zero — ln 2 at the top of the logistic family, ln φ in the window of three, and nought where the signs become the Thue–Morse sequence.
Named alongside it
The objects these essays reach for when they reach for this one.
Shift mapSymbolic dynamicsTopological entropyBinary expansionConjugacyDense orbitDoubling mapEntropyExact arithmeticFibonacci numbersGolden ratioLogistic map