A characterization of orthogonal convergence in simply connected domains

2018/06/18 by Bracci, Filippo, Contreras, Manuel D., Díaz-Madrigal, Santiago +1
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.06582

Abstract

Let \mathbb D be the unit disc in \mathbb C and let f:\mathbb D → \mathbb C be a Riemann map, Δ=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\⊂ Δ has the property that \f^-1(z_n)\ converges orthogonally to a point of ∂ \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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