Parametric Representations and Boundary Fixed Points of Univalent Self-Maps of the Unit Disk

Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 26)

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.

Keywords

Parametric representation Univalent function Conformal mapping Boundary fixed point Loewner equation Loewner-Kufarev equation Infinitesimal generator Evolution family Lie semigroup 

2010 Mathematics Subject Classification

Primary: 30C35 30C75; Secondary: 30D05 30C80 34H05 37C25 

Notes

Acknowledgements

The author was partially supported by Ministerio de Economía y Competitividad (Spain) project MTM2015-63699-P.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Natural SciencesUniversity of StavangerStavangerNorway

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