I agree that it's probably ergodic, but to show every distribution will definitely come about (approximately) you'll need a slightly stronger property than just

I agree that it's probably ergodic, but to show every distribution will definitely come about (approximately) you'll need a slightly stronger property than just ergodicity. Ergodicity guarantees it is almost always the case, but if you want it to always be the case you'll need it to be uniquely ergodic. Since irrational rotations are uniquely ergodic I reckon this system will also be, but I think the proof won't be trivial.