Angular extents and trajectory slopes in the theory of holomorphic semigroups in the unit disk

TL;DR

This paper investigates the asymptotic behavior of holomorphic semigroups in the unit disk, linking trajectory slopes to the geometry of their domains, and provides new conditions for convergence to the Denjoy-Wolff point.

Contribution

It introduces a new sufficient condition for trajectories to converge with a definite slope, connecting geometric properties of the domain to dynamical behavior.

Findings

Established a new condition for trajectory convergence with slope

Connected domain geometry to asymptotic behavior of semigroups

Reproduced known conditions as corollaries

Abstract

We study relationships between the asymptotic behaviour of a non-elliptic semigroup of holomorphic self-maps of the unit disk and the geometry of its planar domain (the image of the Koenigs function). We establish a sufficient condition for the trajectories of the semigroup to converge to its Denjoy-Wolff point with a definite slope. We obtain as a corollary two previously known sufficient conditions.