The action of the mapping class group on the space of geodesic rays of a   punctured hyperbolic surface
  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. 
Brian H. Bowditch and Makoto Sakuma
math.GT