A Characterization of Orthogonal Convergence in Simply Connected Domains

Abstract

Let $${\mathbb {D}}$$ be the unit disc in $${\mathbb {C}}$$ and let $$f:{\mathbb {D}} \rightarrow {\mathbb {C}}$$ 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 $${\mathbb {D}}$$.