On links between horocyclic and geodesic orbits on geometrically infinite surfaces

We study the topological dynamics of the horocycle flow $h_\mathbb{R}$ on a geometrically infinite hyperbolic surface S. Let u be a non-periodic vector for $h_\mathbb{R}$ in T^1 S. Suppose that the half-geodesic $u(\mathbb{R}^+)$ is almost minimizing and that the injectivity radius along $u(\mathbb{R}^+)$ has a finite inferior limit $Inj(u(\mathbb{R}^+))$. We prove that the closure of $h_\mathbb{R} u$ meets the geodesic orbit along un unbounded sequence of points $g_{t_n} u$. Moreover, if $Inj(u(\mathbb{R}^+)) = 0$, the whole half-orbit $g_{\mathbb{R}^+} u$ is contained in $h_\mathbb{R} u$. When $Inj(u(\mathbb{R}^+)) > 0$, it is known that in general $g_{\mathbb{R}^+} u \subset h_\mathbb{R} u$. Yet, we give a construction where $Inj(u(\mathbb{R}^+)) > 0$ and $g_{\mathbb{R}^+} u \subset h_\mathbb{R} u$, which also constitutes a counterexample to Proposition 3 of [Led97].