Currents on cusped hyperbolic surfaces and denseness property

Abstract

The space $\mathrm{GC} (Σ)$ of geodesic currents on a hyperbolic surface $Σ$ can be considered as a completion of the set of weighted closed geodesics on $Σ$ when $Σ$ is compact, since the set of rational geodesic currents on $Σ$, which correspond to weighted closed geodesics, is a dense subset of $\mathrm{GC}(Σ)$. We prove that even when $Σ$ is a cusped hyperbolic surface with finite area, $\mathrm{GC}(Σ)$ has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on $Σ$ but also to weighted geodesics connecting two cusps. In addition, we present an example in which a sequence of weighted closed geodesics converges to a geodesic connecting two cusps, which is an obstruction for the intersection number to extend continuously to $\mathrm{GC}(Σ)$. To construct the example, we use the notion of subset currents. Finally, we prove that the space of subset currents on a cusped hyperbolic surface has the denseness property of rational subset currents.