On links between horocyclic and geodesic orbits on geometrically infinite surfaces

TL;DR

This paper investigates the relationship between horocycle and geodesic flows on geometrically infinite hyperbolic surfaces, revealing conditions under which their orbits intersect and providing counterexamples to previous propositions.

Contribution

It establishes new links between horocycle and geodesic orbits under specific geometric conditions and constructs counterexamples to prior assumptions.

Findings

Closure of horocycle orbit meets geodesic orbit along an unbounded sequence

If injectivity radius tends to zero, the entire half-geodesic is in the horocycle orbit

Counterexample shows positive injectivity radius does not guarantee orbit inclusion

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].