Asymptotic behavior of orbits of holomorphic semigroups

Let (phi(t)) 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 Omega be the starlike at infinity domain image of the Koenigs function of (phi(t)). In this paper we characterize the type of convergence of the orbits of (phi(t)) to the Denjoy-Wolff point in terms of the shape of Omega. In particular we prove that the convergence is non-tangential if and only if the domain Omega is "quasi-symmetric with respect to vertical axis". We also prove that such conditions are equivalent to the curve [0, infinity) (sic) t bar right arrow phi(t) (z) being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of Omega. (C) 2019 Elsevier Masson SAS. All rights reserved.