Boundary regular fixed points in Loewner theory

TL;DR

This paper characterizes boundary regular fixed points in Loewner theory through analytical and geometrical properties, providing examples and an embedding result for univalent self-maps with specified boundary fixed points.

Contribution

It offers a new characterization of regular fixed points in Loewner theory and establishes an embedding theorem for univalent maps with boundary fixed points.

Findings

Characterization of regular fixed points via Herglotz vector fields and Loewner chains

Examples illustrating the conditions for regular fixed points

An embedding theorem for univalent maps with boundary fixed points

Abstract

We characterize regular fixed points of evolution families in terms of analytical properties of the associated Herglotz vector fields and geometrical properties of the associated Loewner chains. We present several examples showing the r\^ole of the given conditions. Moreover, we study the relations between evolution families and Herglotz vector fields at regular contact points and prove an embedding result for univalent self-maps of the unit disc with a given boundary regular fixed point into an evolution family with prescribed boundary data.