Ergodic Billiard — Bunimovich Stadium vs Rectangle

The Bunimovich stadium (rectangle with semicircular caps) is a proven ergodic billiard — a single orbit uniformly fills all of phase space (Sinai 1970, Bunimovich 1979). A rectangular billiard is non-ergodic: orbits with rational slope are periodic; irrational-slope orbits fill only a subset (invariant tori).