# Parametric representations and boundary fixed points of univalent self-maps of the unit disk

## TL;DR

This paper extends Loewner's parametric representation to semigroups of conformal self-maps of the unit disk with specified boundary fixed points and Denjoy-Wolff points, broadening the understanding of their control systems.

## Contribution

It generalizes the classical Loewner representation to include maps with boundary fixed points and prescribed boundary Denjoy-Wolff points, completing previous interior point cases.

## Key findings

- Extended Loewner's representation to boundary fixed points
- Characterized semigroups with boundary Denjoy-Wolff points
- Connected boundary fixed points with control system frameworks

## Abstract

A classical result in the theory of Loewner's parametric representation states that the semigroup $\mathfrak U_*$ of all conformal self-maps $\phi$ of the unit disk $\mathbb{D}$ normalized by $\phi(0) = 0$ and $\phi'(0) > 0$ can be obtained as the reachable set of the Loewner - Kufarev control system $$ \frac{\mathrm{d} w_t}{\mathrm{d} t}=G_t\circ w_t,\quad t\geqslant0,\qquad w_0=\mathsf{id}_{\mathbb{D}}, $$ where the control functions $t\mapsto G_t\in\mathsf{Hol}(\mathbb{D},\mathbb{C})$ form a certain convex cone. Here we extend this result to the semigroup $\mathfrak U[F]$ consisting of all conformal $\phi:\mathbb{D}\to\mathbb{D}$ whose set of boundary regular fixed points contains a given finite set $F\subset\partial\mathbb{D}$ and to its subsemigroup $\mathfrak U_\tau[F]$ formed by $\mathsf{id}_{\mathbb{D}}$ and all $\phi\in\mathfrak U[F]\setminus\{\mathsf{id}_{\mathbb{D}}\}$ with the prescribed boundary Denjoy - Wolff point $\tau\in\partial\mathbb{D}\setminus F$. This completes the study launched in [P. Gumenyuk, Preprint 2016, ArXiv:1603.04043], where the case of interior Denjoy - Wolff point $\tau\in\mathbb{D}$ was considered.