Counting pairs of saddle connections
Jayadev Athreya, Samantha Fairchild, and Howard Masur
Contact the author: Please use for correspondence this email.
Submission date: 24. Jan. 2022
MSC-Numbers: 32G15, 52C23, 30F30, 28C10
Keywords and phrases: translation surfaces, ergodic theory
Link to arXiv: See the arXiv entry of this preprint.
We show that for almost every translation surface the number of pairs of saddle connections with bounded virtual area has asymptotic growth like cR2 where the constant c depends only on the area and the connected component of the stratum. The proof techniques combine classical results for counting saddle connections with the crucial result that the Siegel-Veech transform is in L2. In order to capture information about pairs of saddle connections, we consider pairs with bounded virtual area since the set of such pairs can be approximated by a ﬁbered set which is equivariant under geodesic ﬂow. In the case of lattice surfaces, small virtual area is equivalent to counting parallel pairs of saddle connections, which also have a quadratic growth of cR2 where c depends in this case on the given lattice surface.