Characterizations of convergence by a given set of angles in simply connected domains

TL;DR

This paper characterizes the convergence of sequences in simply connected domains using hyperbolic geometry, providing necessary and sufficient conditions for boundary convergence at specific angles.

Contribution

It introduces a geometric framework to determine boundary convergence of sequences via angle sets in simply connected domains.

Findings

Provides criteria for convergence by a given set of angles

Uses hyperbolic geometry to characterize boundary behavior

Establishes necessary and sufficient conditions for angular convergence

Abstract

Let $\Delta$ be a simply connected domain and $f:\mathbb{D} \to \Delta$, where $\mathbb{D}$ is the unit disk, be a corresponding Riemann map. Let ${z_n}\subset \Delta$ be a sequence with no accumulation points inside $\Delta$. In the present article, we give necessary and sufficient conditions in terms of hyperbolic geometry which certify that ${f^{-1}(z_n)}$ converges to a point of $\partial \mathbb{D}$ by a certain angle $\theta$ or by a certain set of angles $[\theta_1, \theta_2]$.