# 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

## References

1. 1.
Abate, M.: Iteration theory of holomorphic maps on taut manifolds. Research and Lecture Notes in Mathematics. Complex Analysis and Geometry. Mediterranean, Rende (1989)Google Scholar
2. 2.
Aleksandrov, I.A.: Parametric continuations in the theory of univalent functions (Russian). Izdat. Nauka, Moscow (1976). MR0480952Google Scholar
3. 3.
Anderson, J.M., Vasil’ev, A.: Lower Schwarz-Pick estimates and angular derivatives. Ann. Acad. Sci. Fenn. Math. 33(1), 101–110 (2008). MR2386840Google Scholar
4. 4.
Arosio, L., Bracci, F., Wold, E.F.: Solving the Loewner PDE in complete hyperbolic starlike domains of $$\mathbb {C}^N$$. Adv. Math. 242, 209–216 (2013). MR3055993Google Scholar
5. 5.
Berkson, E., Porta, H.: Semigroups of holomorphic functions and composition operators. Michigan Math. J. 25, 101–115 (1978)Google Scholar
6. 6.
Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics, vol. 169. Springer, New York (1997). MR1477662Google Scholar
7. 7.
Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Aleksandrov-Clark measures and semigroups of analytic functions in the unit disc. Ann. Acad. Sci. Fenn. Math. 33(1), 231–240 (2008). MR2386848Google Scholar
8. 8.
Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Evolution families and the Loewner equation II: complex hyperbolic manifolds. Math. Ann. 344(4), 947–962 (2009)Google Scholar
9. 9.
Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disc. J. Reine Angew. Math. (Crelle’s J.) 672, 1–37 (2012)Google Scholar
10. 10.
Bracci, F., Contreras, M.D., Díaz-Madrigal, S., Vasil’ev, A.: Classical and stochastic Löwner-Kufarev equations. In: Harmonic and Complex Analysis and Its Applications, pp. 39–134. Trends in Mathematics, Birkhäuser, Cham (2014). MR3203100Google Scholar
11. 11.
Bracci, F., Contreras, M.D., Díaz-Madrigal, S., Gumenyuk, P.: Boundary regular fixed points in Loewner theory. Ann. Mat. Pura Appl. (4) 194(1), 221–245 (2015)Google Scholar
12. 12.
Contreras, M.D.: Geometry behind chordal Loewner chains. Complex Anal. Oper. Theory 4(3), 541–587 (2010). MR2719792Google Scholar
13. 13.
Contreras, M.D., Díaz-Madrigal, S.: Analytic flows in the unit disk: angular derivatives and boundary fixed points. Pac. J. Math. 222, 253–286 (2005)
14. 14.
Contreras, M.D., Díaz-Madrigal, S., Pommerenke, C.: Fixed points and boundary behaviour of the Koenigs function. Ann. Acad. Sci. Fenn. Math. 29(2), 471–488 (2004). MR2097244Google Scholar
15. 15.
Contreras, M.D., Díaz-Madrigal, S., Pommerenke, C.: On boundary critical points for semigroups of analytic functions. Math. Scand. 98(1), 125–142 (2006). MR2221548Google Scholar
16. 16.
Contreras, M.D., Díaz-Madrigal, S., Gumenyuk, P.: Loewner chains in the unit disk. Rev. Mat. Iberoam. 26, 975–1012 (2010)
17. 17.
Cowen, C.C., Pommerenke, C.: Inequalities for the angular derivative of an analytic function in the unit disk. J. Lond. Math. Soc. (2) 26(2), 271–289 (1982). MR0675170Google Scholar
18. 18.
de Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154(1-2), 137–152 (1985). MR0772434Google Scholar
19. 19.
Elin, M., Goryainov, V., Reich, S., Shoikhet, D.: Fractional iteration and functional equations for functions analytic in the unit disk. Comput. Methods Funct. Theory 2(2), 353–366 (2002), [On table of contents: 2004]. MR2038126Google Scholar
20. 20.
Frolova, A., Levenshtein, M., Shoikhet, D., Vasil’ev, A.: Boundary distortion estimates for holomorphic maps. Complex Anal. Oper. Theory 8(5), 1129–1149 (2014). MR3208806Google Scholar
21. 21.
Goryainov, V.V.: Semigroups of conformal mappings. Mat. Sb. (N.S.) 129(171)(4), 451–472, (1986) (Russian); Translation in Math. USSR Sbornik 57, 463–483 (1987)Google Scholar
22. 22.
Goryainov, V.V.: Fractional iterates of functions that are analytic in the unit disk with given fixed points. Mat. Sb. 182(9), 1281–1299 (1991); Translation in Math. USSR-Sb. 74(1), 29–46 (1993). MR1133569Google Scholar
23. 23.
Goryainov, V.V., Evolution families of analytic functions and time-inhomogeneous Markov branching processes. Dokl. Akad. Nauk 347(6), 729–731 (1996); Translation in Dokl. Math. 53(2), 256–258 (1996)Google Scholar
24. 24.
Goryainov, V.V.: Semigroups of analytic functions in analysis and applications. Uspekhi Mat. Nauk 676(408), 5–52 (2012); Translation in Russian Math. Surveys 67(6), 975–1021 (2012)Google Scholar
25. 25.
Goryainov, V.V.: Evolution families of conformal mappings with fixed points. (Russian. English summary). Zírnik Prats’ Instytutu Matematyky NAN Ukrayiny. National Academy of Sciences of Ukraine (ISSN 1815-2910) 10(4-5), 424–431 (2013). Zbl 1289.30024Google Scholar
26. 26.
Goryainov, V.V.: Evolution families of conformal mappings with fixed points and the Löwner-Kufarev equation. Mat. Sb. 206(1), 39–68 (2015); Translation in Sb. Math. 206(1-2), 33–60 (2015)Google Scholar
27. 27.
Goryainov, V.V., Ba, I.: Semigroups of conformal mappings of the upper half-plane into itself with hydrodynamic normalization at infinity. Ukr. Math. J. 44, 1209–1217 (1992)
28. 28.
Goryainov, V.V., Kudryavtseva, O.S.: One-parameter semigroups of analytic functions, fixed points and the Koenigs function. Mat. Sb. 202(7), 43–74 (2011) (Russian); Translation in Sbornik: Mathematics 202(7-8), 971–1000 (2011)Google Scholar
29. 29.
Gumenyuk, P.: Parametric representation of univalent functions with boundary regular fixed points. Constr. Approx. 46, 435–458 (2017). https://doi.org/10.1007/s00365-017-9376-4
30. 30.
Koch, J., Schleißinger, S.: Value ranges of univalent self-mappings of the unit disc. J. Math. Anal. Appl. 433(2), 1772–1789 (2016). MR3398791Google Scholar
31. 31.
Löwner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. Math. Ann. 89, 103–121 (1923)
32. 32.
Loewner, C.: On totally positive matrices. Math. Z. 63, 338–340 (1955). MR0073657
33. 33.
Loewner, C.: Seminars on Analytic Functions, vol. 1. Institute for Advanced Study, Princeton, NJ (1957). Available at http://babel.hathitrust.org Google Scholar
34. 34.
Loewner, C.: On generation of monotonic transformations of higher order by infinitesimal transformations. J. Anal. Math. 11, 189–206 (1963). MR0214711
35. 35.
Poggi-Corradini, P.: Canonical conjugations at fixed points other than the Denjoy-Wolff point. Ann. Acad. Sci. Fenn. Math. 25(2), 487–499 (2000). MR1762433Google Scholar
36. 36.
Pommerenke, Ch.: Univalent functions. With a chapter on quadratic differentials by Gerd Jensen. Vandenhoeck & Ruprecht, Göttingen (1975)
37. 37.
Pommerenke, Ch.: Boundary Behaviour of Conformal Mappings. Springer, New York (1992)
38. 38.
Pommerenke, C., Vasil’ev, A.: Angular derivatives of bounded univalent functions and extremal partitions of the unit disk. Pac. J. Math. 206(2), 425–450 (2002). MR1926785 (2003i:30024)Google Scholar
39. 39.
Prokhorov, D., Samsonova, K.: Value range of solutions to the chordal Loewner equation. J. Math. Anal. Appl. 428(2) (2015). MR3334955
40. 40.
Roth, O., Schleißinger, S.: Rogosinski’s lemma for univalent functions, hyperbolic Archimedean spirals and the Loewner equation. Bull. Lond. Math. Soc. 46(5), 1099–1109 (2014). MR3262210Google Scholar
41. 41.
Sarason, D.: Sub-Hardy Hilbert Spaces in the Unit Disk. University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10. Wiley, New York (1994). MR1289670Google Scholar
42. 42.
Shoikhet, D.: Semigroups in Geometrical Function Theory. Kluwer, Dordrecht (2001) 