Topological entropy
Named by 7 essays across one field — each of them below, with the objects they name alongside it.
A matrix that counts the returns
Draw a map straight through the cycle 0 → 1/3 → 1 and its two pieces carry each other in a fixed pattern: the left piece only across the right, the right across both. The orbits' words are then walks on a two-node graph, and the number of points that come back after n steps is the trace of that graph's matrix to the nth power — 1, 3, 4, 7, 11, 18 — each one checked by solving for the points exactly.
The folds that measure chaos
Apply the logistic map six times and its graph goes up and down 38 times at r = 3.5 and 64 times at r = 4. How fast that number of folds multiplies with each further step is the map's topological entropy: zero through the whole cascade of period doublings, log of the golden ratio in the window of three, log 2 at the top — and it never decreases as r rises.
A solvable chaos of every degree
The logistic map at four is chaotic and, through a change of coordinates, completely solvable: its orbits are cosines of doubling angles. The trick is not a one-off. For every whole number n there is a polynomial of degree n that multiplies angles by n instead of 2, and every one of them is exactly as solvable, has exactly nᵏ points of period k, and preserves the same distribution — and any two of them commute, which almost no two polynomials do.
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.
Where a base-β orbit spends its time
Follow x ↦ βx mod 1 for a million steps and count how often the orbit visits each part of the interval. For the doubling map the answer is evenly; for the golden ratio it is a staircase with one step, spending 1.17 times the average near 0 and 0.72 times it near 1. The step sits exactly where the orbit of the number 1 lands, and in a base whose 1 never has a finite expansion the staircase has infinitely many steps.
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.
Two humps and one entropy
A map of an interval with one hump has its chaos measured by one number, the entropy, and that number never decreases as the hump rises. Give the map two turning points — a hump and a dip — and the parameter space becomes a plane. The entropy is no longer a function of one setting, and the question becomes whether the maps of a given entropy form one connected curve or several. John Milnor conjectured one. Counting the laps of every cubic's iterates exactly, over the whole plane of hump heights and dip depths, draws the curves — and each comes out whole.
Named alongside it
The objects these essays reach for when they reach for this one.
Golden ratioLogistic mapPeriodic orbitSymbolic dynamicsChaosInvariant measureItineraryMarkov partitionCharacteristic polynomialChebyshev polynomialConjugacyConnectedness