Horocyclic Orbits on $\Gamma(1)\frontslash\mathcal{H}$, \ Closed and Otherwise

TL;DR

This paper investigates the behavior of horocyclic orbits on the modular surface, using continued fractions and Ford circles to analyze their paths, stability, and transitivity properties.

Contribution

It introduces new methods involving continued fractions to locate horocycle points and studies their homotopy classes and transitivity on the modular surface.

Findings

Horocyclic orbits can be effectively tracked using Ford circles and continued fractions.

Homotopy classes of horizontal horocycles are stable between encounters with elliptic fixed points.

The open horocycle path at the Golden Mean exhibits transitive behavior.

Abstract

This paper studies certain horocyclic orbits on $\Gamma(1)\frontslash\mathcal{H}$. In the first instance we examine horocycles defined using the pencil of circles whose common point (in the words of the Nielsen-Fenchel manuscript is $\infty$. The orbits involved in this case are closed and long - judged by arc length between two points compared to the hyperbolic distance between them. We are concerned with tracking the paths of individual horocycles. Using Ford circles of Farey sequences we find lifts to the Standard Fundamental Region (SFR) and find points of these lifts making given angles with a horizontal. Next, we offer two methods, both involving continued fractions, of locating points with such angles whose lifts are near any given point in the SFR. This establishes in an effective manner a sort of transitivity, which necessarily involves infinitely many such horocycles. Next, we study the homotopy classes of horizontal horocycles as we descend to the real axis. We find these are stable during descent between encounters of the horizontal with elliptic fixed points. Such encounters change - complicate - the homotopy classes. We give these explicitly down to height $1/(2\sqrt{3})$. Finally we do an initial study of the open (infinite length) horocycle path with unit euclidean radius anchored at $\phi -1$, where $\phi$ is the Golden Mean. Enough information is adduced to suggest that this path is itself transitive. The methods resemble the Hardy-Littlewood Circle Method in a certain regard, albeit without the exponential sums.