Ergodic theory — the statistics of a single orbit

Parent: orbit

Setting: a probability space $(X, \mathcal{B}, \mu)$ ($\mathcal{B}$ the σ-algebra of measurable sets, $\mu$ a probability measure) and a measurable map $T: X \to X$.

Definition

• $T$ is measure-preserving if $\mu(T^{-1}A) = \mu(A)$ for every $A \in \mathcal{B}$. (Preimage, not image — $T$ need not be invertible.)
• $T$ is ergodic if every invariant set is trivial: $T^{-1}A = A \implies \mu(A) \in \{0, 1\}$. Intuition: the space does not split into two dynamically separate parts of positive size.

Poincaré recurrence

$T$ measure-preserving, $\mu(A) > 0$. Then almost every point of $A$ returns to $A$ infinitely often. (Needs only measure preservation — not ergodicity.)

Birkhoff ergodic theorem (1931)

$T$ measure-preserving and ergodic, $f \in L^1(\mu)$. Then for $\mu$-almost every $x$:

$$\frac{1}{N} \sum_{n=0}^{N-1} f\big(T^n x\big) \;\longrightarrow\; \int_X f \, d\mu \qquad (N \to \infty).$$

Time average along one orbit = space average over the whole space. This is the license to learn global facts by following a single typical trajectory.

Worked example: the doubling map and normal numbers

$T(x) = 2x \bmod 1$ on $X = [0,1)$ with Lebesgue measure $\mu$.

Measure-preserving: for an interval $A = [a,b)$, $T^{-1}A = \left[\tfrac{a}{2}, \tfrac{b}{2}\right) \cup \left[\tfrac{a+1}{2}, \tfrac{b+1}{2}\right)$ — two intervals of length $\tfrac{b-a}{2}$ each, total $b - a = \mu(A)$. ✓

What $T$ does to binary digits: writing $x = 0.d_1 d_2 d_3\ldots$ in base 2, $T$ deletes the first digit: $T(x) = 0.d_2 d_3 d_4\ldots$ — the orbit reads out the binary expansion. ($T$ is the two-sided face of the full 2-shift; the dictionary is symbolic-dynamics.) Ergodicity holds (e.g. via Fourier coefficients: an invariant $L^2$ function has $\hat{f}(k) = \hat{f}(2k)$ for all $k$, forcing all nonzero coefficients to vanish).

Birkhoff applied with $f = \mathbf{1}_{[0,1/2)}$ (indicator of "first digit is 0"): for almost every $x$,

Almost every real number has binary digits that are half zeros, half ones — Borel's normal number theorem, falling out of one orbit-average. The same scheme with the gauss-map gives the digit statistics of continued fractions.

The caveat that keeps ergodic theory honest

"Almost every" is with respect to $\mu$ — statements say nothing about your particular point. Every rational is an exception to the doubling-map statement above (their orbits are eventually periodic). This is precisely the wall in collatz-orbit-statistics: the natural numbers are a measure-zero set inside the space where the Collatz dynamics is understood.