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Mathematics > Dynamical Systems

arXiv:2207.03755 (math)
[Submitted on 8 Jul 2022]

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

Authors:Claudio Bonanno, Alessio Del Vigna, Stefano Isola
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Abstract: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.
Comments: 33 pages, 7 figures
Subjects: Dynamical Systems (math.DS)
MSC classes: 37D40, 37A40, 37C25
Cite as: arXiv:2207.03755 [math.DS]
  (or arXiv:2207.03755v1 [math.DS] for this version)
  https://doi.org/10.48550/arXiv.2207.03755

Submission history

From: Claudio Bonanno [view email]
[v1] Fri, 8 Jul 2022 08:55:04 UTC (39 KB)
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