Asymptotic behavior of orbits of holomorphic semigroups

2018/10/18 by Bracci, Filippo, Contreras, Manuel D., Díaz-Madrigal, Santiago +2 · 1 citation
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.07947

Abstract

Let (ϕ_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 Ω be the starlike at infinity domain image of the Koenigs function of (ϕ_t). In this paper we completely characterize the type of convergence of the orbits of (ϕ_t) to the Denjoy-Wolff point in terms of the shape of Ω. In particular we prove that the convergence is non-tangential if and only if the domain Ω is `quasi-symmetric with respect to vertical axes'. We also prove that such conditions are equivalent to the curve [0,∞)\ni t↦ ϕ_t(z) being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of Ω.

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• From discrete iteration in the unit disc to continuous semigroups of holomorphic functions

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