Limits of Teichmuller geodesics in the Universal Teichmuller space

A Thurston boundary of the universal Teichmuller space T(D) is the set of projective bounded measured laminations PMLbdd(D) of D. We prove that each Teichmuller geodesic ray in T(D) converges to a unique limit point in the Thurston boundary of T(D) in the weak topology. In particular, there is an open and dense set of geodesic rays, which have unique (weak(*)-)limits in the Thurston boundary. We also show that the main result is sharp by providing an example of a Teichmuller geodesic, which does not converge in the uniform weak topology.