TITLE: Existence of nonergodic polygonal billiard
QUESTION [7 upvotes]: Let $P$ be a polygon in the plane. One can define the billiard flow on the unit tangent bundle of $P$, just following the trajectories of the billiard at speed one. 
A standard conjecture is that a typical billiard flow is ergodic. But are there examples of irrational polygonal billiards for which the flow is known not to be ergodic ? 
EDIT : It is obvious that in this sense every rational polygon is not ergodic. The good framework is irrational polygons, for which the question seems to be open.

REPLY [2 votes]: Numerical evidences for non-ergodic irrational triangles may be found in
Lima, Rodríguez-Pérez and de Aguiar, Phys. Rev. E 87, 062902 (2013).