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

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