Divergent geodesics in the universal Teichmüller space

Abstract:

Thurston boundary of the universal Teichmüller 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 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 $PML_{bdd}(\mathbb {D})$. 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 $\mathbb {T}^k$.