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

Let Δ C be a simply connected domain, let f : D → Δ be a Riemann map, and let {zk} ⊂ Δ be a compactly divergent sequence. Using Gromov’s hyperbolicity theory, we show that {f−1(zk)} converges nontangentially to a point of ∂D if and only if there exists a simply connected domain U C such that Δ ⊂ U and Δ contains a tubular hyperbolic neighborhood of a geodesic of U and {zk} is eventually contained in a smaller tubular hyperbolic neighborhood of the same geodesic. As a consequence we show that if (φt) is a non-elliptic semigroup of holomorphic self-maps of D with Koenigs function h and h(D) contains a vertical Euclidean sector, then φt(z) converges to the Denjoy-Wolff point non-tangentially for every z ∈ D as t → +∞. Using new localization results for the hyperbolic distance, we also construct an example of a parabolic semigroup which converges non-tangentially to the DenjoyWolff point but is oscillating, in the sense that the slope of the trajectories is not a single point.