Geodesic
Holomorphic maps of the disk into itself with invariant diameter and bounded distortion
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2015), pp. 51-63

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We consider a problem of embeddability of holomorphic map of the unit disk into itself with invariant diameter and bounded distortion into one-parameter semigroup. Here we require that the elements of one-parameter semigroup have the same properties as the original map does. We obtain embeddability criteria, the solution is given in terms of the Königs function.
Keywords: one-parameter semigroup, fractional iterates, infinitesimal generator, Königs function, fixed points. 
O. S. Kudryavtseva. Holomorphic maps of the disk into itself with invariant diameter and bounded distortion. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2015), pp. 51-63. http://geodesic.mathdoc.fr/item/IVM_2015_8_a4/
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[1] Schröeder E., “Über iterierte Funktionen”, Math. Ann., 3:2 (1871), 296–322 | DOI | MR

[2] Königs G., “Recherches sur les intégrales des certaines equations fonctionelle”, Ann. Ecole Norm. Sup., 1:3 (1884), 3–41

[3] Baker I. N., “Fractional iteration near a fixpoint of multiplier 1”, J. Australian Math. Soc., 4:2 (1964), 143–148 | DOI | MR | Zbl

[4] Karlin S., McGregor J., “Embedding iterates of analytic functions with two fixed points into continuous groups”, Trans. Amer. Math. Soc., 132:1 (1968), 137–145 | DOI | MR | Zbl

[5] Berkson E., Porta H., “Semigroups of analytic functions and composition operators”, Michigan Math. J., 25:1 (1978), 101–115 | DOI | MR | Zbl

[6] Cowen C. C., “Iteration and the solution of functional equations for functions analytic in the unit disk”, Trans. Amer. Math. Soc., 265:1 (1981), 69–95 | DOI | MR | Zbl

[7] Goryainov V. V., “Drobnye iteratsii analiticheskikh v edinichnom kruge funktsii s zadannymi nepodvizhnymi tochkami”, Matem. sb., 182:9 (1991), 1281–1299 | MR | Zbl

[8] Goryainov V. V., Kudryavtseva O. S., “Odnoparametricheskie polugruppy analiticheskikh funktsii, nepodvizhnye tochki i funktsiya Kënigsa”, Matem. sb., 202:7 (2011), 43–74 | DOI | MR | Zbl

[9] Goryainov V. V., “Polugruppy analiticheskikh funktsii v analize i prilozheniyakh”, UMN, 67:6 (2012), 5–52 | DOI | MR | Zbl

[10] Kudryavtseva O. S., “Funktsiya Kënigsa i drobnoe iterirovanie analiticheskikh v edinichnom kruge funktsii s veschestvennymi koeffitsientami i nepodvizhnymi tochkami”, Izv. Sarat. gos. un-ta. Ser. Matem. Mekhan. Inform., 13:1 (2013), 67–71 | Zbl

[11] Contreras M. D., Díaz-Madrigal S., Pommerenke Ch., “Fixed points and boundary behaviour of the Koenigs function”, Ann. Acad. Sci. Fenn. Math., 29:2 (2004), 471–488 | MR | Zbl

[12] Valiron Zh., Analiticheskie funktsii, GITTL, M., 1957

[13] Ahlfors L. V., Conformal invariants: topics in geometric function theory, McGraw-Hill Book Company, New York, 1973 | MR | Zbl

[14] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[15] Krein M. G., Nudelman A. A., Problema momentov Markova i ekstremalnye zadachi, Nauka, M., 1973 | MR