Ergodic Theory: Time vs Space Averages
When orbits sample the whole space — and Birkhoff's theorem links dynamics to measure.
A measure-preserving transformation $T: (X, \mu) \to (X, \mu)$ is ergodic if every $T$-invariant set has measure $0$ or $1$. Equivalently, the only $T$-invariant $L^2$ functions are constants.
Birkhoff's ergodic theorem: for $f \in L^1(\mu)$,
$$\lim_{N \to \infty} \frac{1}{N} \sum_{n=0}^{N-1} f(T^n x) = \int f\, d\mu \quad \text{a.e. } x,$$provided $T$ is ergodic. Time averages along almost every orbit equal the space average — the foundational link between dynamics and statistical mechanics. Boltzmann's ergodic hypothesis: typical Hamiltonian flows are ergodic on their energy surface, so thermodynamic averages can be computed as time averages.
Examples:
- Irrational rotation $T_\alpha(x) = x + \alpha \pmod 1$ on $[0, 1)$ with $\alpha \notin \mathbb Q$ is uniquely ergodic; orbits equidistribute (Weyl).
- Bernoulli shift: mixing, strongest form of ergodicity, $K$-property entropy positive.
- Geodesic flow on negatively curved manifolds: ergodic, mixing, hyperbolic.
The hierarchy: ergodic ⊋ weakly mixing ⊋ strongly mixing ⊋ Bernoulli. Kolmogorov–Sinai entropy measures dynamical complexity; positive entropy ↔ chaos.