On the links between horocyclic and geodesic orbits on geometrically infinite surfaces

Abstract

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