The Return Map of the Cross Section of Horizontally Short Lattice Surfaces is Weakly Mixing

Abstract

We prove that the return map of the unstable horocycle flow on the space of horizontally short translation surfaces associated to a lattice surface $(X, \omega)$ is weakly mixing. This extends a result of Cheung-Quas for the square torus to all lattice surfaces. The proof adapts their criterion for weakly mixing and uses quantitative bounds for Siegel-Veech transforms restricted to the Poincar\'e section of horizontally short surfaces.