Irrational rotation Rα: x↦x+α (mod 1) is uniquely ergodic when α∉ℚ — every orbit is dense and equidistributed. Birkhoff's theorem: time average → space average.
Arnold cat map [[2,1],[1,1]] on T² is Anosov ergodic — exponential mixing, positive entropy ln φ.
Weyl criterion: equidistribution iff all Fourier modes → 0.