Analysis and Mathematical Physics

A translation surface is a closed surface that is obtained by gluing edges of a polygon in parallel. The group $GL_2(R)$ acts on the collection translation surfaces of a fixed genus g. For a fixed translation surface S and t>0, we obtain a probability measure on the collection of translation surfaces by rotating S with a uniform angle and then multiplying by $diag(e^t, e^-t)$. Alternatively, we can talk on expanding a piece of horospherical orbit. We prove equidistribution of this sequence of measures as $t -> ∞$. 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.