Ergodicity of eigenfunctions for ergodic billiards

We give a simple proof of ergodicity of eigenfunctions of the Laplacian with Dirichlet boundary conditions on compact Riemannian manifolds with piecewise smooth boundaries and ergodic billiards. Examples include the Bunimovich stadium, the Sinai billiard and the generic polygonal billiard tables of Kerckhoff, Masur and Smillie.