Semigroups versus evolution families in the Loewner theory

TL;DR

This paper characterizes when evolution families in the Loewner theory are commuting, linking this property to the associated Herglotz vector fields having separated variables and originating from semigroups of holomorphic self-maps.

Contribution

It establishes a precise equivalence between commuting evolution families, separated variables in Herglotz vector fields, and semigroup origins in the Loewner theory.

Findings

Commuting evolution families correspond to separated variables in Herglotz vector fields.

Such evolution families originate from semigroups of holomorphic self-maps.

The paper provides a characterization linking algebraic and analytic properties in Loewner theory.

Abstract

We show that an evolution family of the unit disc is commuting if and only if the associated Herglotz vector field has separated variables. This is the case if and only if the evolution family comes from a semigroup of holomorphic self-maps of the disc.