Generic infinite staircases and wind-tree billiards are uniquely ergodic in almost every direction, while periodic infinite translation surfaces are never uniquely ergodic.
S\'eminaire Bourbaki, Vol.\ 1987/88
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Unique Ergodicity For Infinite Area Translation Surfaces
Generic infinite staircases and wind-tree billiards are uniquely ergodic in almost every direction, while periodic infinite translation surfaces are never uniquely ergodic.