Divergent geodesics in the Universal Teichmüller space

Abstract

Thurston boundary of the universal Teichmüller space $T(\mathbb{D})$ is the space $PM{L}_{bdd}(\mathbb{D})$ of projective bounded measured laminations of $\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of generalized Teichmüller type if it shrinks the vertical foliation of a holomorphic quadratic differential. We provide the first examples of generalized Teichmüller rays which diverge near Thurston boundary $PL{M}_{bdd}(\mathbb{D})$. Moreover, for every $k\ge 1$ 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 ${\theta }_1,\dots ,{\theta }_k$ are rationally independent reals then the sequence $(\{{\theta }_{1}n\},\dots ,\{{\theta }_{k}n\})$ is dense in the $k$-torus ${\mathbb{T}}^k$.

32 pages, 5 figures