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.