A characterization of orthogonal convergence in simply connected domains

TL;DR

This paper characterizes when sequences in a simply connected domain converge orthogonally to the boundary in terms of hyperbolic geometry, with applications to semigroup slope problems.

Contribution

It provides a necessary and sufficient condition for orthogonal convergence in simply connected domains using hyperbolic distance and horocycles.

Findings

Characterization of orthogonal convergence in terms of hyperbolic distance and horocycles.

Application to the slope problem for continuous semigroups of holomorphic self-maps.

Conditions for sequences to converge orthogonally to boundary points.

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 $\mathbb D$.