The action of the mapping class group on the space of geodesic rays of a punctured hyperbolic surface

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