Classification of horocycle orbit closures in $ \mathbb{Z} $-covers

TL;DR

This paper classifies all horocycle orbit closures in $ ext{Z}$-covers of compact hyperbolic surfaces, revealing their fractal structure and Hausdorff dimension through analysis of geodesic rays.

Contribution

It provides a complete description of horocycle orbit closures in $ ext{Z}$-covers, a previously unresolved classification in hyperbolic geometry.

Findings

All non-maximal horocycle orbit closures are fractal with integer Hausdorff dimension.

The analysis of geodesic rays is key to understanding orbit closure structures.

The results extend understanding of dynamical systems on hyperbolic surfaces.

Abstract

We fully describe all horocycle orbit closures in $ \mathbb{Z} $-covers of compact hyperbolic surfaces. Our results rely on a careful analysis of the efficiency of all distance minimizing geodesic rays in the cover. As a corollary we obtain in this setting that all non-maximal horocycle orbit closures, while fractal, have integer Hausdorff dimension.