A Poincaré map for the horocycle flow on $PSL(2,\mathbb{Z})\mathrm{\backslash }\mathbb{H}$ and the Stern-Brocot tree

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