On the K\"onigs function of semigroups of holomorphic self-maps of the unit disc

TL;DR

This paper introduces an abstract approach to defining the K"onigs function for semigroups of holomorphic self-maps of the unit disc, linking it to holomorphic models and the infinitesimal generator.

Contribution

It presents a novel abstract framework for the K"onigs function and holomorphic models, enabling deduction of properties of the infinitesimal generator.

Findings

Established a new method to define the K"onigs function abstractly.

Connected the K"onigs function to the infinitesimal generator.

Provided insights into the structure of semigroups of holomorphic self-maps.

Abstract

Let $(\phi_t)$ be a semigroup of holomorphic self-maps of~$\mathbb D$. In this note, we use an abstract approach to define the K\"onigs function of $(\phi_t)$ and "holomorphic models" and show how to deduce the existence and properties of the infinitesimal generator of $(\phi_t)$ from this construction.