A characterization of orthogonal convergence in simply connected domains

Abstract

Let $\mathbb{D}$ be the unit disc in $\mathbb{C}$ and let $f:\mathbb{D}\to \mathbb{C}$ be a Riemann map, $\Delta =f(\mathbb{D})$. We give a necessary and sufficient condition in terms of hyperbolic distance and horocycles which assures that a compactly divergent sequence $\{z_n\}\subset \Delta$ has the property that $\{f^{-1}(z_n)\}$ converges orthogonally to a point of $\partial \mathbb{D}$. We also give some applications of this to the slope problem for continuous semigroups of holomorphic self-maps of $\mathbb{D}$.

arXiv comment15 pages