Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the space of geodesics on $\widetilde\Sigma$. It is well-known that $\mu$ can only have atoms at closed geodesics. We can also look at 1-dimensional subsets, namely pencils $P(a)$, the set of geodesics with one endpoint at $a \in \partial\widetilde\Sigma$. If $a$ is not one of the limit points of a closed geodesic, then $\mu(P(a)) = 0$; see, e.g., Martelli, "An Introduction to Geometric Topology", Proposition 8.2.8.

Question. What is an example of a geodesic current without atoms so that $\mu(P(a)) \ne 0$ (where $a$ is necessarily the endpoint of a closed geodesic)?