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

We construct a Poincaré map \(\mathcal{P}_h\) for the positive horocycle flow on the modular surface \(PSL(2,\mathbb{Z})\backslash \mathbb{H}\), and begin a systematic study of its dynamical properties. In particular we give a complete characterisation of the periodic orbits of \(\mathcal{P}_h\), an...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2022-07
Main Authors: Bonanno, Claudio, Alessio Del Vigna, Isola, Stefano
Format: Article
Language:English
Subjects:
Flow mapping
Orbits
Poincare maps
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We construct a Poincaré 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.
ISSN:2331-8422