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
Pages: 32
Bibtex
MSC-Numbers: 32G15, 52C23, 30F30, 28C10
Keywords and phrases: translation surfaces, ergodic theory
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
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.