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

Abstract

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