On the asymptotic quadratic growth rate of saddle connections and periodic orbits on marked flat tori

Abstract

Asymptotic quadratic growth rates of saddle connections and families of periodic cylinders on translation tori with n marked points are studied. In particular, for any marking the existence of limits of the quadratic growth rate is shown using elementary methods (avoiding Ratner's theorem). We study the growth rate limit as function of the marking. Precise formulas for this function in the case of two marked points are given and the sets where the growth rate function is maximal and continuous are described. For rational two markings the index of the Veech group in $ {\rm SL}(2,{\Bbb Z}) $ is calculated in two different ways.