Convergence of some horocyclic deformations in the Gardiner-Masur boundary

Vincent Alberge (Universite de Strasbourg)

It is well known that a point of the Teichmueller space and a measured foliation determine an isometric embedding of the hyperbolic disc to the Teichmüller space equipped with the so-called Teichmueller metric. In this talk, we will consider the mage by this embedding of a particular horocycle whose points will be called an horocyclic deformation. To be more precise, we will be interested in the closure of this subset in the GM compactification. As the embedding of the disc does not admit a continuous extension to boundaries, we cannot say that the boundary of the set of horocyclic deformations consists of one point. However, according to Miyachi's results we will see that it is the case if the given foliation is either a simple closed curve or a uniquely ergodic foliation.