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 completely 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 axes. We also prove that such conditions are equivalent to the curve $[0,infty) i tmapsto phi_t(z)$ being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of $Omega$.