Limits of Teichmüller maps (Q2352234)

Let \(\Delta=\{ z \in \mathbb{C}: |z|<1 \}\) and \(\Delta^*=\{ z \in \mathbb{C}: |z|>1 \} \cup \{ \infty\}\). A Teichmüller map is a quasiconformal homeomorphism of \( \overline{\mathbb{C}} = \mathbb{C} \cup \{ \infty\}\, \) that is conformal outside \(\Delta^*\) and such that in \(\Delta\), the complex dilatation is of the form \(k \overline{\varphi}/\varphi,\) where \(0\leq k< 1\) and \(\varphi\) is holomorphic. The author studies sequences of Teichmüller maps whose complex dilatations are of the form \(k_j \overline{\varphi_j}/\varphi_j,\) where \( k_j \to 1\) and \(\varphi_j\) are holomorphic tending to a holomorphic function \(\varphi\) uniformly on compact subsets, introduces a new notion of convergence which is instrumental in the formulation of the main results.