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 \({\mathbb {D}}\).
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M. D. Contreras and S. Díaz-Madrigal: Partially supported by the Ministerio de Economía y Competitividad and the European Union (FEDER) MTM2015-63699-P and by La Consejería de Educación y Ciencia de la Junta de Andalucía.
F. Bracci: Partially supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
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Bracci, F., Contreras, M.D., Díaz-Madrigal, S. et al. A Characterization of Orthogonal Convergence in Simply Connected Domains. J Geom Anal 29, 3160–3175 (2019). https://doi.org/10.1007/s12220-018-00109-8
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DOI: https://doi.org/10.1007/s12220-018-00109-8