A Poincaré map for the horocycle flow on PSL(2,Z)\H and the Stern-Brocot tree

We construct a Poincaré map P_h for the positive horocycle flow on the modular surface PSL(2,Z)\H, and begin a systematic study of its dynamical properties. In particular, we give a complete characterisation of the periodic orbits of P_h, and show that they are equidistributed with respect to the invariant measure of P_h and can be organised in a tree by using the Stern-Brocot tree of rational numbers. In addition we introduce a time-reparameterisation of P_h which gives an insight into the dynamics of the non-periodic orbits.