Individual Horocyclic Orbits on $H \backslash \Gamma(1)$, Closed and Otherwise

TL;DR

This paper investigates the detailed trajectories of individual horocycles on the modular surface, introducing algorithms for locating points with specific angles, analyzing homotopy classes during descent, and exploring the transitivity of an infinite horocycle path.

Contribution

It provides new algorithms based on continued fractions for locating points on horocycles and analyzes the stability and changes in homotopy classes during descent, along with initial insights into the transitivity of a specific infinite horocycle path.

Findings

Algorithms for locating points with specific angles using continued fractions.

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

The infinite horocycle path at radius 1 is suggested to be transitive.

Abstract

This paper endeavors to track the trajectories of individual horocycles on \modsurf. It is far more common to study \emph{sets} of such trajectories, seeking some asymptotic behavior using an averaging process (see section \ref{previous}). Our work is only marginally related to these efforts. We begin by examining horocycles defined using the pencil of circles whose common point (in the words of the Nielsen-Fenchel manuscript \cite{wF}) 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. Using Ford circles of Farey sequences we find their lifts to the Standard Fundamental Region (SFR) and find points of these lifts making given angles with a horizontal. Next, we offer two algorithms, both involving continued fractions, of locating points whose angle with the horizontal is near any target angle and whose lifts are near any given point in the SFR. 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 horocycle 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 single doubly infinite path is transitive.