Ratner’s Theorem on Horocyclic Flows

Let X be a complete hyperbolic surface, perhaps the hyperbolic plane H , and let X denote the unit tangent bundle T (X) to X (and H = T H). There are three flows on X which will concern us here. They are realized by three cars, as represented in Figure 1. The cars all have their steering wheels locked in position: the first car drives straight ahead, the second one steers to the left so as to follow a path of geodesic curvature 1, and the third steers to the right, also following a path of geodesic curvature 1. All three cars have an arrow painted on the roof, centered at the rear axle; for the first the arrow points straight ahead, and for the other two it points sideways – in the direction towards which the car is steering for the second car and in the opposite direction for the third. The flows at time t ∈ R starting at a point x = (x, ξ) ∈ X are defined as follows: