Limits of Teichm\"uller geodesics in the Universal Teichm\"uller space

TL;DR

This paper investigates the behavior of Teichmüller geodesic rays in the universal Teichmüller space, showing they have unique boundary limits, contrasting with the case of closed surfaces.

Contribution

It proves the uniqueness of boundary limit points for Teichmüller geodesic rays in the universal Teichmüller space, a result not true for closed surfaces.

Findings

Each Teichmüller geodesic ray has a unique limit point in Thurston's boundary.

The boundary is identified with projective bounded measured laminations.

Contrast with closed surfaces where uniqueness does not hold.

Abstract

Thurston's boundary to the universal Teichm\"uller space $T(\mathbb{H})$ is the set of asymptotic rays to the embedding of $T(\mathbb{H})$ in the space of geodesic currents; the boundary is identified with the projective bounded measured laminations $PML_{bdd}(\mathbb{H})$ of $\mathbb{H}$. We prove that each Teichm\"uller geodesic ray in $T(\mathbb{H})$ has a unique limit point in Thurston's boundary to $T(\mathbb{H})$ unlike in the case of closed surfaces.