are those curves which satisfy (or in the case of a half-translation surface ). The geodesic flow on with direction is the flow on where is the geodesic starting at with direction if is not singular. Dynamical properties On a flat torus the geodesic flow in a given direction has the property that it is either periodic or ergodic. In general this is not true: there may be directions in which the flow is minimal (meaning every orbit is dense in the surface) but not ergodic. On the other hand, on a compact translation surface the flow retains from the simplest case of the flat torus the property that it is ergodic in almost every direction. Another natural question is to establish asymptotic estimates for the number of closed geodesics or saddle connections of a given length. On a flat torus there are no saddle connections and the number of closed geodesics of length is equivalent to . In general one can only obtain bounds: if is a compact translation surface of genus then there exists constants (depending only on the genus) such that the both of closed geodesics and of saddle connections of length satisfy Restraining to a probabilistic results it is possible to get better estimates: given a genus , a partition of and a connected component of the stratum there exists constants such that for almost every the asymptotic equivalent holds: , The constants are called Siegel–Veech constants. Using the ergodicity of the -action on , it was shown that these constants can explicitly be computed as ratios of certain Masur-Veech volumes. Veech dichotomy The geodesic flow on a Veech surface is much better behaved than in general. This is expressed via the following result, called the Veech dichotomy: Let be a Veech surface and a direction. Then either all trajectories defied over are periodic or the flow in the direction is ergodic. Relation with billiards If is a polygon in the Euclidean plane and a direction there is a continuous dynamical system called a billiard. The trajectory of a point inside the polygon is defined as follows: as long as it does not touch the boundary it proceeds in a straight line at unit speed; when it touches the interior of an edge it bounces back (i.e. its direction changes with an orthogonal reflection in the perpendicular of the edge), and when it touches a vertex