We construct a Poincar\'e map $\mathcal{ P}_h$ for the positive horocycle flow on the modular surface $PSL(2,\mathbb{ Z})\backslash \mathbb{ H}$, 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.
Categories: math.DS