Recurrent and irregular orbits of the horocyclic flow on the unit tangent bundle of the untwisted flute

TL;DR

This paper investigates the behavior of certain infinite quasi-minimizing rays in the unit tangent bundle of an untwisted flute surface, showing they do not intersect closed geodesics and analyzing their orbit structure under the horocyclic flow.

Contribution

It establishes conditions under which infinite quasi-minimizing rays have trivial horocyclic orbits on the untwisted flute surface.

Findings

Infinite quasi-minimizing rays do not intersect closed geodesics.

Such rays have trivial horocyclic orbits.

The orbit set T_u is only zero for these rays.

Abstract

The aim of this article is to show that if there exists $u \in \Omega_{h} \subset T^{1}S$ an infinite quasi-minimizing ray which do not intersect any closed geodesic on the surface $S$ (untwisted flute), then $T_{u}=\{ t \in \mathbb{R} \; ; \; g_{t}u \in \overline{h_{\mathbb{R}}u} \}=\{0\}$.