Concept

Invariant measure

A way of assigning sizes to sets that a map leaves unchanged: the preimage of every set has the same size as the set. It is what turns long-run questions about an orbit into questions about areas, and for a chaotic map it gives the share of time spent in each region.

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

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

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

The Chebyshev maps T₂, T₃, T₄ and T₅. Four small square plots of the Chebyshev polynomials of degrees two to five on the interval from minus one to one, each with the diagonal drawn; the graph of degree n sweeps between the bottom and top of the square n times.

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.

dynamics · Iteration

Named alongside it

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

Binary expansion Chaos Chebyshev polynomial Conjugacy Doubling map Ergodicity Fourier series Logistic map Measure zero Normal number Periodic orbit Topological entropy

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