Currents on cusped hyperbolic surfaces and denseness property

Abstract: The space $\mathrm{GC} (\Sigma)$ of geodesic currents on a hyperbolic surface $\Sigma$ can be considered as a completion of the set of weighted closed geodesics on $\Sigma$ when $\Sigma$ is compact, since the set of rational geodesic currents on $\Sigma$, which correspond to weighted closed geodesics, is a dense subset of $\mathrm{GC}(\Sigma )$. We prove that even when $\Sigma$ is a cusped hyperbolic surface with finite area, $\mathrm{GC}(\Sigma )$ has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on $\Sigma$ 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 $\mathrm{GC}(\Sigma )$. 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.