  Let $\Sigma$ be a complete finite-area orientable hyperbolic surface with one
cusp, and let $\mathcal{R}$ be the space of complete geodesic rays in $\Sigma$
emanating from the puncture. Then there is a natural action of the mapping
class group of $\Sigma$ on $\mathcal{R}$. We show that this action is "almost
everywhere" wandering.
