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A classical result in the theory of Loewner's parametric representationstates that the semigroup $\mathfrak U_*$ of all conformal self-maps $\phi$ ofthe unit disk $\mathbb{D}$ normalized by $\phi(0) = 0$ and $\phi'(0) > 0$ canbe obtained as the reachable set of the Loewner - Kufarev control system $$\frac{\mathrm{d} w_t}{\mathrm{d} t}=G_t\circ w_t,\quad t\geqslant0,\qquadw_0=\mathsf{id}_{\mathbb{D}}, $$ where the control functions $t\mapstoG_t\in\mathsf{Hol}(\mathbb{D},\mathbb{C})$ form a certain convex cone. Here weextend this result to the semigroup $\mathfrak U[F]$ consisting of allconformal $\phi:\mathbb{D}\to\mathbb{D}$ whose set of boundary regular fixedpoints contains a given finite set $F\subset\partial\mathbb{D}$ and to itssubsemigroup $\mathfrak U_\tau[F]$ formed by $\mathsf{id}_{\mathbb{D}}$ and all$\phi\in\mathfrak U[F]\setminus\{\mathsf{id}_{\mathbb{D}}\}$ with theprescribed boundary Denjoy - Wolff point $\tau\in\partial\mathbb{D}\setminusF$. 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.

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