Divergent geodesics in the Universal Teichm\"uller space

Abstract

Thurston boundary of 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 generalized Teichm\"uller type if it shrinks the vertical foliation of a holomorphic quadratic differential. We provide the first examples of generalized Teichm\"uller rays which diverge near Thurston boundary $PLM_{bdd}(\mathbb{D})$. Moreover, for every $k\geq 1$ we construct examples of rays with limit sets homeomorphic to $k$-dimensional cubes. For the latter result we utilize the classical Kronecker approximation theorem from number theory which states that if $\theta_1,\ldots,\theta_k$ are rationally independent reals then the sequence $(\{\theta_1 n\},\ldots,\{\theta_k n\})$ is dense in the $k$-torus $\mathbb{T}^k$.