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arXiv · 2207.03755

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

Claudio Bonanno·Alessio Del Vigna·Stefano Isola
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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.

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Claudio Bonanno, Alessio Del Vigna, Stefano Isola. 2022-07-08. A Poincar\'e map for the horocycle flow on $PSL(2,\mathbb{Z})\backslash \mathbb{H}$ and the Stern-Brocot tree. https://arxiv.org/abs/2207.03755

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