A Characterization of Orthogonal Convergence in Simply Connected Domains

Abstract

Let D be the unit disc in C and let f : D → C be a Riemann map, ∆ = f (D). We give a necessary and sufficient condition in terms of hyperbolic distance and horocycles which assures that a compactly divergent sequence {z n } ⊂ ∆ has the property that {f −1 (z n )} converges orthogonally to a point of ∂D. We also give some applications of this to the slope problem for continuous semigroups of holomorphic self-maps of D.

Angular Extents and Trajectory Slopes in the Theory of Holomorphic Semigroups in the Unit Disk

“…We also analyse the relationship between these functions and the non-tangential convergence of the orbits of the semigroup (see Proposition 5.1). Moreover, as a corollary, we recover results from [3] and [5].…”

“…As far as we know, apart from examples and some folklore results concerning strong symmetry of the planar domain, the unique three papers dealing with the above question are [3], [5] and [7]. In [3], it is shown that whenever the boundary of the planar domain is included in a vertical half-strip, the arrival slope set is equal to {0}.…”

“…In [3], it is shown that whenever the boundary of the planar domain is included in a vertical half-strip, the arrival slope set is equal to {0}. Likewise, in [5], it is shown that if the boundary of the planar domain is included in a horizontal strip, the arrival slope set is also equal to {0}. In [7], the authors introduce some "boundary distance" functions, which measure the distance of a vertical straight line to the boundary of the planar domain, and use them to characterize geometrically when the arrival slope set coincides with the singleton {π/2} or {−π/2}.…”

Non-tangential limits and the slope of trajectories of holomorphic semigroups of the unit disc

“…Note that η([v 0 , v 1 ]) ⊂ B by Proposition 5.2 (7). Moreover, since Im η(v 1 ) − Im η(v 0 ) > NR, by Proposition 5.2 (6)…”

“…Let ξ : [0, 1] → S R +a be the geodesic of S R +a such that ξ(0) = γ(u ′ 0 ) and ξ(1) = γ(t). By Proposition 5.2 (7), for all s ∈ [0, 1],…”