Concept

Ergodic theorem

The theorem that, for a thoroughly mixing map, the share of time an orbit spends in a region equals the region's measure. It holds for almost every starting point, and it turns averages over time into averages over space.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Where one Hénon orbit spends a million steps. A density map of 1000000 steps of one Hénon orbit counted into 420 × 140 cells: 2542 cells visited, half the time spent in 608 of them.

Where the time goes on an attractor

A chaotic orbit's position is unpredictable within a few dozen steps. How it divides its time is not — start anywhere, follow long enough, and the fraction of time spent in each region comes out the same. That distribution lives on a set of no area, and among the infinitely many ways an orbit could spend its time, typical starts pick exactly one.

dynamics · Strange attractor $Base-φ digits of two fractions and two irrational numbers. Four rows of 72 base-φ digits each, for 1/3, 2/7, √2 − 1 and 1/π; the two fractions repeat with periods 8 and 16, the two irrational numbers show no period.$

Fractions repeat and roots look random

Almost every number between 0 and 1 has base-φ digits with the frequencies Parry's measure predicts. Which particular numbers do? Every fraction, provably, does not: its expansion repeats from the first digit, with a period equal to the period of the Fibonacci numbers modulo its denominator. And √2 − 1 and 1/π, computed exactly to twelve thousand digits, match every predicted frequency to within sampling error — which proves nothing about them at all.

dynamics · Symbolic dynamics

The cosines of the powers of two

Start at cos 1 and apply x ↦ 2x² − 1 over and over. The map is chaotic and exactly solvable: the k-th point is cos(2ᵏ), the cosine of two to the k radians. Iterate it in floating point and the computed orbit leaves the true one after 51 steps. Read the binary digits of 1/(2π) instead and the true orbit can be followed for as long as digits can be computed — here 262,080 steps — and at that length it is as fair as any orbit can look. Whether it stays fair is a question about those digits, and nobody can answer it.

dynamics · Iteration

Named alongside it

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

Invariant measure Normal number Beta expansion Binary expansion Chebyshev polynomial Fibonacci numbers Fractal Golden ratio Henon map Lorenz system Periodic orbit Rounding

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