Individual Closed Horocyclic Orbits on the Modular Surface

TL;DR

This paper provides a constructive analysis of individual horocycle trajectories on the modular surface, establishing their topological properties and describing homotopy class changes during deformation.

Contribution

It offers an effective, constructive approach to understanding horocycle flows, including topological transitivity and homotopy class jumps, on the modular surface.

Findings

Establishes topological transitivity of horocycle flow

Describes homotopy class jumps under deformation

Provides an effective method for tracking horocycle trajectories

Abstract

We track the trajectories of individual horocycles on the modular surface. Our tracking is constructive, and we thus \emph{effectively} establish topological transitivity and even line-transitivity for the horocyclic flow. We also describe homotopy class jumps that occur under continuous deformation of horocycles.