Time average equals ensemble average — the foundation of statistical mechanics
The ergodic hypothesis (Boltzmann, Maxwell) states that over long times, a trajectory visits every accessible region of phase space with equal frequency. This means time averages equal ensemble (phase space) averages — justifying why we can use statistical ensembles to describe a single system.
A Lissajous figure with irrational frequency ratio is ergodic on the torus (Weyl's theorem) — it densely fills the surface. A rational ratio gives a closed loop — non-ergodic. Real Hamiltonian systems are generically non-ergodic (KAM theory), but chaotic regions are ergodic within their own accessible regions.