Convergence of Teichmüller deformations in the universal Teichmüller space

Abstract

Let φ : D → C \varphi :\mathbb {D}\to \mathbb {C} be an integrable holomorphic function on the unit disk D \mathbb {D} and let D φ : D → T ( D ) D_{\varphi }:\mathbb {D}\to T(\mathbb {D}) be the corresponding Teichmüller disk in the universal Teichmüller space T ( D ) T(\mathbb {D}) . For a positive t t it is known that D φ ( t ) → [ μ φ ] ∈ P M L b ( D ) D_{\varphi }(t)\to [\mu _{\varphi }]\in PML_b(\mathbb {D}) as t → 1 t\to 1 , where μ φ \mu _{\varphi } is a bounded measured lamination representing a point on the Thurston boundary of T ( D ) T(\mathbb {D}) . We extend this result by showing that D φ : D → T ( D ) D_{\varphi }\colon \mathbb {D}\to T(\mathbb {D}) extends as a continuous map from the closed disk D ¯ \overline {\mathbb {D}} to the Thurston bordification. In addition, we prove that the rate of convergence of D φ ( λ ) D_{\varphi }(\lambda ) when λ → e i θ \lambda \to e^{i\theta } is independent of the type of the approach to e i θ ∈ ∂ D e^{i\theta }\in \partial \mathbb {D} .