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Symbolic dynamics — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · dynamics
  2. A map drawn through the cycle 0 → 1/3 → 1, and the graph its pieces make. The graph of a map made of two straight pieces through a cycle of three points, with the cycle drawn as a staircase, beside a two-node graph showing which piece may follow which and the matrix of that graph.

    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.

    part 2 · dynamics
  3. The logistic map applied 6 times at 3.5, 3.83, 4, and its folds. Side-by-side graphs of the logistic map composed with itself, one per parameter, each labelled with how many monotone pieces it has.

    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.

    part 3 · dynamics
  4. 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.

    part 4 · dynamics
  5. Where x ↦ βx mod 1 spends its time, the golden ratio φ. A histogram of 2000000 orbit points of the β-map for the golden ratio φ in 50 bins, with Parry's staircase density drawn over it; they agree, and the density steps down at the orbit of 1.

    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.

    part 5 · dynamics
  6. 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.

    part 6 · dynamics

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