Regular poles and $\beta$-numbers in the theory of holomorphic semigroups

TL;DR

This paper introduces the concept of regular boundary poles for infinitesimal generators of holomorphic semigroups in the unit disc, characterizing them via $eta$-points and duality, with applications to radial multi-slits.

Contribution

It defines regular poles in the context of holomorphic semigroup generators, characterizes them using $eta$-numbers, and explores duality and applications to complex slit structures.

Findings

Regular poles are characterized by $eta$-points of the semigroup.

A duality operation links regular poles to null poles of the dual generator.

An example shows a non-isolated radial slit with a tip lacking a positive $eta$-number.

Abstract

We introduce the notion of regular (boundary) poles for infinitesimal generators of semigroups of holomorphic self-maps of the unit disc. We characterize such regular poles in terms of $\beta$-points (i.e. pre-images of values with positive Carleson-Makarov $\beta$-numbers) of the associated semigroup and of the associated K\"onigs intertwining function. We also define a natural duality operation in the cone of infinitesimal generators and show that the regular poles of an infinitesimal generator correspond to the regular null poles of the dual generator. Finally we apply such a construction to study radial multi-slits and give an example of a non-isolated radial slit whose tip has not a positive Carleson-Makarov $\beta$-number.