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

Abstract

A classical result in the theory of Loewner's parametric representation states that the semigroup ${\mathfrak{U}}_{\ast }$ of all conformal self-maps $\varphi$ of the unit disk $\mathbb{D}$ normalized by $\varphi (0)=0$ and ${\varphi }^{\prime }(0)>0$ can be obtained as the reachable set of the Loewner - Kufarev control system $$\frac{\mathrm{d}{w}_t}{\mathrm{d}t}=G_t\circ w_t,t⩾0,w_0={\mathsf{id}}_{\mathbb{D}},$$ 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 $\varphi :\mathbb{D}\to \mathbb{D}$ 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 $\varphi \in \mathfrak{U}[F]\setminus \{{\mathsf{id}}_{\mathbb{D}}\}$ 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.