On links between horocyclic and geodesic orbits on geometrically infinite surfaces

Abstract

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].