Billiards in right angle triangles. Find examples when it is ergodic.
According to , if the angles of the triangle are rational multiples of $\pi$, the flow is uniquely ergodic in almost every direction. It is also uniquely ergodic in almost every direction for other triangles whose angles are not rational multiples of $\pi$ (again, from ).
To expand on Rodrigo's comment, the KMS paper does not prove it for all irrational triangles but for ones which can be approximated well by rationals. Here we should also mention Vorobets' work where he gives an explicit degree of approximation for which you have ergodicity.
Some triangles have optimal dynamics- all orbits are either closed or uniformly distributed, which corresponds to the Veech property of the associated translation surface. The work of Kenyon-Smillie and Puchta classifies these.