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Partially rectangular domains are compact two-dimensional Riemannianmanifolds $X$, either closed or with boundary, that contain a flat rectangle orcylinder. In this paper we are interested in partially rectangular domains withergodic billiard flow; examples are the Bunimovich stadium, the Sinai billiardor Donnelly surfaces.  We consider a one-parameter family $X_t$ of such domains parametrized by theaspect ratio $t$ of their rectangular part. There is convincing theoretical andnumerical evidence that the Laplacian on $X_t$ with Dirichlet or Neumannboundary conditions is not quantum unique ergodic (QUE). We prove that this istrue for all $t \in [1,2]$ excluding, possibly, a set of Lebesgue measure zero.This yields the first examples of ergodic billiard systems proven to benon-QUE.

Ergodic billiards that are not quantum unique ergodic