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

We construct a Poincar ́e map Ph 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 com- plete characterisation of the periodic orbits of Ph, and show that they are equidistributed with respect to the invariant measure of Ph and can be organised in a tree by using the Stern-Brocot tree of rational numbers. In addition we introduce a time-reparameterisation of Ph which gives an insight into the dynamics of the non-periodic orbits.