Asymptotic behavior of orbits of holomorphic semigroups

TL;DR

This paper characterizes the convergence behavior of orbits of holomorphic semigroups in the unit disc, linking it to the geometric shape of the associated domain and establishing conditions for non-tangential and tangential convergence.

Contribution

It provides a complete geometric characterization of orbit convergence types for holomorphic semigroups using domain shape and quasi-geodesic properties.

Findings

Convergence is non-tangential iff the domain is quasi-symmetric w.r.t. vertical axes.

Tangential convergence characterized by the shape of the domain.

Orbit curves are quasi-geodesics under certain geometric conditions.

Abstract

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 $[0,\infty)\ni t\mapsto \phi_t(z)$ being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of $\Omega$.