Convergence of some horocyclic deformations to the Gardiner-Masur boundary

We introduce a deformation of Riemann surfaces and we are interested in the convergence of this deformation to a point of the Gardiner-masur boundary of Teichmueller space. This deformation, which we call the horocyclic deformation, is directed by a projective measured foliation and belongs to a certain horocycle in a Teichmueller disc. Using works of Marden and Masur and works of Miyachi, we show that the horocyclic deformation converges if its direction is given by a simple closed curve or a uniquely ergodic measured foliation.