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 ${\mathfrak U\hskip .03em}_*$ of all conformal self-maps ϕ of the unit disk $\mathbb {D}$ normalized by ϕ(0) = 0 and ϕ′(0) > 0 can be obtained as the reachable set of the Loewner–Kufarev control system

$$\displaystyle \frac {\mathrm {d} w_t}{\mathrm {d} t}=G_t\circ w_t,\quad t\geqslant 0,\qquad w_0={\mathsf {id}}_{\mathbb {D}}, $$

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 ${\mathfrak U\hskip .08em}[\,F]$ 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 ${\mathfrak U\hskip .1em}_{\tau }\hskip -.09em[\,F]$ formed by ${\mathsf {id}}_{\mathbb {D}}$ and all $\phi \in {\mathfrak U\hskip .08em}[\,F]\setminus \{{\mathsf {id}}_{\mathbb {D}}\}$ with the prescribed boundary Denjoy–Wolff point $\tau \in {\partial \mathbb {D}}\setminus F$. This completes the study launched in Gumenyuk, P. (Constr. Approx. 46 (2017), 435–458, https://doi.org/10.1007/s00365-017-9376-4), where the case of interior Denjoy–Wolff point $\tau \in \mathbb {D}$ was considered.