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

TL;DR

This paper develops a Poincaré map for the positive horocycle flow on the modular surface, characterizes its periodic orbits, and links them to the Stern-Brocot tree, advancing the dynamical understanding of horocycle flows.

Contribution

It introduces a Poincaré map for the horocycle flow, characterizes periodic orbits, and connects these orbits to the Stern-Brocot tree, providing new dynamical insights.

Findings

Periodic orbits are fully characterized.

Periodic orbits are equidistributed with respect to the invariant measure.

Periodic orbits can be organized in a Stern-Brocot tree structure.

Abstract

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.