Visual sphere and Thurston's boundary of the Universal Teichm\"uller space

Abstract

Thurston's boundary to the universal Teichm\"uller space $T(\mathbb{D})$ is the space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of $\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of Teichm\"uller type if it shrinks vertical foliation of an integrable holomorphic quadratic differential. In a prior work we established that each Teichm\"uller geodesic ray limits to a multiple (by the reciprocal of the length of the leaves) of vertical foliation of the quadratic differential. Certain non-integrable holomorphic quadratic differential induce geodesic rays and we consider their limit points in $PML_{bdd}(\mathbb{D})$. Somewhat surprisingly, the support of the limiting projective measured laminations might be a geodesic lamination whose leaves are not homotopic to leaves of either vertical or horizontal foliation of the non-integrable holomorphic quadratic differential.