Contact points and fractional singularities for semigroups of holomorphic self-maps in the unit disc

TL;DR

This paper investigates boundary singularities in semigroups of holomorphic self-maps in the unit disc, introducing fractional singularities, and characterizes them through geometric and functional criteria involving Koenigs functions.

Contribution

It introduces regular fractional singularities for semigroup generators and provides geometric and functional characterizations of these singularities.

Findings

Characterization of regular fractional singularities via Koenigs functions

Geometric criteria for the shape of the Koenigs function image

Correspondence between contact points and maximal boundary arcs

Abstract

We study boundary singularities which can appear for infinitesimal generators of one-parameter semigroups of holomorphic self-maps in the unit disc. We introduce "regular" fractional singularities and characterize them in terms of the behavior of the associated semigroups and Koenigs functions. We also provide necessary and sufficient geometric criteria on the shape of the image of the Koenigs function for having such singularities. In order to do this, we study contact points of semigroups and prove that any contact (not fixed) point of a one-parameter semigroup corresponds to a maximal arc on the boundary to which the associated infinitesimal generator extends holomorphically as a vector field tangent to this arc.