Constructing Transversals to Horocycle Flow
Examining the distribution of the gaps between elements of a sequence can provide insight into how equidistributed these elements are. In the setting of translation surfaces, an important sequence is that of slopes of saddle connections. This distribution has been computed for specific examples, the square torus by Athreya and Cheung, and the double pentagon, by Athreya, Chaika and Lelievere. The strategy of proof involves reinterpreting the question in the setting of horocycle flow on the moduli space. Specifically, the gaps can be seen as return times under horocycle flow to a transversal. In joint work with Caglar Uyanik, we compute the distribution in the octagon and then provide a parametrization for the transversal for any lattice surface that depends on the parabolic elements of the Veech group. In the case of a generic surface, the situation becomes more complicated.