Existence of nonergodic polygonal billiard 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. 
 A: E. Gutkin, Billiard dynamics: An updated survey with the emphasis on open problems Chaos 22, 026116 (2012) gives as open problem 8: Give an example of an irrational but non-ergodic polygon.  Some recent numerical results are given in J. Wang et al, Non-ergodicity and localization of invariant measure for two colliding masses arxiv:1309.7617.
For rational polygons, there is a nice example of an L-shaped billiard and a non-ergodic direction in Fig 3 of L. Demarco The conformal geometry of billiards Bull. AMS 48 33-52 (2011).
Update: There is also literature on the somewhat related question of non-dense aperiodic orbits.  If a positive measure of orbits is not dense, the billiard is not ergodic.  G. A. Galpern Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons Commun. Math. Phys. 91 187-211 (1983) gives examples of polygons with $n\geq 3$, including with irrational angles, which have non-dense aperiodic orbits.  G. W. Tokarsky Galpern's triangle example Commun. Math. Phys. 335 1211–1213 (2015) shows that the $n=3$ example orbit is in fact periodic, so the existence of a non-dense aperiodic orbit in a triangular billiard appears to be still open.
A: The dynamics of billiard flows in rational polygons (J. Smillie, 2000): The billiard in a rational polygon is ergodic in "almost all" directions, more precisely, the Hausdorff dimension of the set of non-ergodic directions is at most 1/2.
A: There are examples in the slit torus. 
Prof. Smillie explains this in:
http://www2.warwick.ac.uk/fac/sci/maths/people/staff/john_smillie/course/notes14february.pdf
A: Numerical evidences for non-ergodic irrational triangles may be found in
Lima, Rodríguez-Pérez and de Aguiar, Phys. Rev. E 87, 062902 (2013).
