Divergent geodesics in the Universal Teichm\"uller space

TL;DR

This paper constructs examples of divergent geodesic rays in the universal Teichmüller space, including those with limit sets homeomorphic to higher-dimensional cubes, using number theory techniques.

Contribution

It provides the first examples of generalized Teichmüller rays that diverge near the Thurston boundary and constructs rays with complex limit sets.

Findings

First examples of diverging generalized Teichmüller rays

Construction of rays with limit sets homeomorphic to k-dimensional cubes

Application of Kronecker approximation theorem in Teichmüller theory

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 $\mathbb{T}^k$.