Gaps of saddle connection directions for some branched covers of tori

Abstract: Consider the class of translation surfaces given by gluing two identical tori along a slit. Every such surface has genus two and two cone-type singularities of angle $4\pi$. There is a distinguished set of geodesics called saddle connections that are the geodesics between cone points. We can recover a vector in the plane representing the saddle connection by keeping track of the amount that the saddle connection moves in the vertical and horizontal direction. How random is the set of saddle connections? We motivate the gap distribution of slopes as a measure of randomness and compute the gap distribution of slopes of saddle connections for the class of translation surfaces given by gluing two identical tori along a slit.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar