Equidistribution of Expanding Horospheres in the Space of Translation Surfaces

A translation surface is a closed surface that is obtained by gluing edges of a polygon in parallel. The group GL2(R) acts on the collection translation surfaces of a fixed genus g. For a fixed translation surface S and t greater than 0, we obtain a probability measure on the collection of translation surfaces by rotating S with a uniform angle and then multiplying by diag(et,e−t). Alternatively, we can talk on expanding a piece of horospherical orbit. We prove equidistribution of this sequence of measures as t− greater than ∞. This resolves a conjecture of Forni, and extends a result of Eskin and Mirzakhani that (in particular) showed our result with a Cesàro average. We will also discuss an application of this result to billiards with rational angles.