Comparing Iterated Grafting and Grafting Rays

LetX be a closed oriented hyperbolic surface and γ a simple closed geodesic, which is short on X. We show that the iterated grafting grtγgrsγX is close to the grafting ray gr(s+T )γX in the Teichmüller metric, where the bounds depend on the length of γ only. Using this result, we show that the holonomy lifts gr2πγgrtγX of a grafting ray all lie in a r-tube around the ray grtγX and the rays grtγgr n 2πγX accumulate exponentially fast in the Hausdorff topology. Finally we study the asymptotic behaviour of grafting rays and iterated grafting sequences and show that both grtγX and gr n 2πγX converge geometrically to a cusped surface for any X.