A classical translation surface is a closed surface that is equipped with a translation structure at all but finitely many cone points. In recent years, people have asked what happens when we allow non-closed surfaces. From that question the field of infinite translation surfaces has evolved.
It turns out that the behaviour of infinite translation surfaces is in many regards very different and more diverse than in the classical case. For instance, Kerckhoff, Masur, and Smillie showed that on a classical translation surface the geodesic flow is uniquely ergodic in almost every direction. This is not at all true for infinite translation surfaces in general.
In the talk, I will give some examples what can go wrong but will also state a version of the unique ergodicity result for a class of infinite translation surfaces. For this class, we get that almost every direction is uniquely ergodic or shows a degeneration behaviour under the Teichmüller flow.
The presented work is joint with Kasra Rafi.