Quotients of Horocycle

In this paper we investigate the long term asymptotic properties of what might be called the induced horocycle ow on a compact quotient of the Poincar e upper half plane. We nd that this \\ow" exhibits chaotic properties in the sense that, in the long term, the area of the intersection of an open ball propagated forward by the \\ow," with the original ball, tends to what would be expected if the intersections were determined in a probabilistic way. In 1987 Doug McMahon wrote a paper on a phenomenon he named \Universal Observability," 7]. This property of ows on a manifold is deened in terms of the idea of \observability" of a dynamical system. Informally , we could say a ow is observable by a given function, F, from the manifold to the real or complex numbers, if the new function given by F composed with an orbit in the ow uniquely determines the orbit. That is, one can decide which orbit is being \observed" by F from the output of F along the trajectory of the orbit. Clearly, even with such an informal description of this property, it is easy to see, for example, that the constant functions do not observe any ow on any manifold with more than one point in it. For most ows familiar to us, it is possible to nd many non-constant functions which fail to observe the ow. On the other hand, McMahon found a class of ows which were observable for every non-constant function on the manifold. Such ows he calls universally observable. McMahon's examples draw heavily on the work of Marina Ratner who, in her papers, 8], shows that horocycle ows on certain quotients of the group SL(2; R) have very dramatic ergodic properties.