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arXiv · 1806.06582

A characterization of orthogonal convergence in simply connected domains

Filippo Bracci·Manuel D. Contreras·Santiago Díaz-Madrigal·Hervé Gaussier
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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 $\mathbb D$.

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Filippo Bracci, Manuel D. Contreras, Santiago Díaz-Madrigal, Hervé Gaussier. 2018-06-18. A characterization of orthogonal convergence in simply connected domains. https://arxiv.org/abs/1806.06582

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