Ergodic billiards that are not quantum unique ergodic

Partially rectangular domains are compact two-dimensional Riemannian manifolds X, either closed or with boundary, that contain a flat rectangle or cylinder.In this paper we are interested in partially rectangular domains with ergodic billiard flow; examples are the Bunimovich stadium, the Sinai billiard or Donnelly surfaces.We consider a one-parameter family X t of such domains parametrized by the aspect ratio t of their rectangular part.There is convincing theoretical and numerical evidence that the Laplacian on X t with Dirichlet, Neumann or Robin boundary conditions is not quantum unique ergodic (QUE).We prove that this is true for all t 2 Œ1; 2 excluding, possibly, a set of Lebesgue measure zero.This yields the first examples of ergodic billiard systems proven to be non-QUE.