Limits of Teichm\"uller geodesics in the Universal Teichm\"uller space

Thurston's boundary to the universal Teichm\"uller space $T(\mathbb{H})$ is the set of asymptotic rays to the embedding of $T(\mathbb{H})$ in the space of geodesic currents; the boundary is identified with the projective bounded measured laminations $PML_{bdd}(\mathbb{H})$ of $\mathbb{H}$. We prove that each Teichm\"uller geodesic ray in $T(\mathbb{H})$ has a unique limit point in Thurston's boundary to $T(\mathbb{H})$ unlike in the case of closed surfaces.