Periodic geodesics on translation surfaces

Let M be a compact connected oriented surface. The surface M is called a translation surface if it is equipped with a translation structure, that is, an atlas of charts such that all transition functions are translations in R2. It is assumed that the chart domains cover all surface M except for finitely many points called singular. The translation structure induces the structure of a smooth manifold, a flat Riemannian metric, and a Borel measure on the surface M punctured at the singular points. We require that the metric has a cone type singularity at each singular point; then the area of the surface is finite. The cone angle is of the form 2πm, where m is an integer called the multiplicity of the singular point. A singular point of multiplicity 1 is called removable; it is rather a marked point than a true singularity of the metric. Furthermore, the translation structure allows us to identify the tangent space at any nonsingular point x ∈ M with the Euclidean space R2. In particular, the unit tangent space at any point is identified with the unit circle S1 = {v ∈ R2 : |v| = 1}. The velocity is an integral of the geodesic flow with respect to this identification. Thus each oriented geodesic has a direction, which is a uniquely determined vector in S1. The direction of an unoriented geodesic is determined up to multiplying by ±1. SupposeX is a Riemann surface (one-dimensional complex manifold) homeomorphic to the surface M . Any nonzero Abelian differential on X defines a translation structure onM . The zeroes of the differential are singular points of the translation structure, namely, a zero of order k is a singular point of multiplicity k + 1. Every translation structure without removable singular points can be obtained this way. Any geodesic joining a nonsingular point to itself is periodic (or closed). We regard periodic geodesics as simple closed unoriented curves. Any periodic geodesic is included in a family of freely homotopic periodic geodesics of the same length and direction. The geodesics of the family fill an open connected domain. Unless the translation surface is a torus without singular points, this domain is an annulus. We call it a cylinder of periodic geodesics (or simply a periodic cylinder). A periodic cylinder is bounded by geodesic segments of