Abstract
Let (ϕt) be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let Ω be the starlike at infinity domain image of the Koenigs function of (ϕt). In this paper we characterize the type of convergence of the orbits of (ϕt) to the Denjoy-Wolff point in terms of the shape of Ω. In particular we prove that the convergence is non-tangential if and only if the domain Ω is “quasi-symmetric with respect to vertical axis”. We also prove that such conditions are equivalent to the curve [0,∞)∋t↦ϕt(z) being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of Ω.
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