Iterates of holomorphic maps, fixed points, and domains of univalence

V. V. Goryainov   ; O. S. Kudryavtseva   ; A. P. Solodov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 77 (2022) no. 6, pp. 959-1020

Résumé

Fixed points play an important part in the dynamics of a holomorphic map. Given a holomorphic self-map of a unit disc, all of its fixed points, with the exception of at most one of them, lie on the boundary of the disc. Furthermore, it turns out that the existence of an angular derivative and its value at a boundary fixed point affect significantly the behaviour of the map itself and its iterates. In addition, some classical problems in geometric function theory acquire new settings and statements in this context. These questions are considered in this paper. The presentation focuses on the problem of fractional iterations, domains of univalence, and the influence of the angular derivative at a boundary fixed point on the regions of values of Taylor coefficients. Bibliography: 90 titles.

Détail

Keywords: holomorphic map, fixed point, angular derivative, univalent covering domain, fractional iterates, one-parameter semigroup, infinitesimal generator, Koenigs function.
Mots-clés : domain of univalence, coefficient regions

Bibliographie

[1] J. Dieudonné, “Recherches sur quelques problèmes relatifs aux polynômes et aux fonctions bornées d'une variable complexe”, Ann. Sci. École Norm. Sup. (3), 48 (1931), 247–358 | DOI | MR | Zbl

[2] L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York–Düsseldorf–Johannesburg, 1973, ix+157 pp. | MR | Zbl

[3] H. Unkelbach, “Über die Randverzerrung bei konformer Abbildung”, Math. Z., 43:1 (1938), 739–742 | DOI | MR | Zbl

[4] R. Osserman, “A sharp Schwarz inequality on the boundary”, Proc. Amer. Math. Soc., 128:12 (2000), 3513–3517 | DOI | MR | Zbl

[5] O. Kudryavtseva and A. Solodov, “On the boundary Dieudonné–Pick lemma”, Mathematics, 9:10 (2021), 1108, 9 pp. | DOI

[6] V. V. Goryainov, “Holomorphic mappings of the unit disc into itself with two fixed points”, Mat. Sb., 208:3 (2017), 54–71 ; English transl. in Sb. Math., 208:3 (2017), 360–376 | DOI | MR | Zbl | DOI

[7] H. Yanagihara, “On the locally univalent Bloch constant”, J. Anal. Math., 65 (1995), 1–17 | DOI | MR | Zbl

[8] P. R. Mercer, “An improved Schwarz lemma at the boundary”, Open Math., 16:1 (2018), 1140–1144 | DOI | MR | Zbl

[9] F. G. Avkhadiev and L. A. Aksent'ev, “The main results on sufficient conditions for an analytic function to be schlicht”, Uspekhi Mat. Nauk, 30:4(184) (1975), 3–60 ; Enlish transl. in Russian Math. Surveys, 30:4 (1975), 1–63 | MR | Zbl | DOI

[10] G. V. Kuz'mina, “Numerical determination of radii of univalence of analytic functions”, Papers on approximate analysis, Tr. Mat. Ist. Steklova, 53, Publishing house of the USSR Academy of Sciences, Moscow–Leningrad, 1959, 192–235 (Russian) | MR | Zbl

[11] K. Noshiro, “On the theory of schlicht functions”, J. Fac. Sci. Hokkaido Imp. Univ. Ser. I, 2:3 (1934), 129–155 | DOI | Zbl

[12] S. E. Warschawski, “On the higher derivatives at the boundary in conformal mapping”, Trans. Amer. Math. Soc., 38:2 (1935), 310–340 | DOI | MR | Zbl

[13] E. Study, Vorlesungen über ausgewählte Gegenstände der Geometrie, v. II, Konforme Abbildung einfach zusammenhängender Bereiche, B. G. Teubner, Leipzig–Berlin, 1913, iv+142 pp. | Zbl

[14] J. W. Alexander, “Functions which map the interior of the unit circle upon simple regions”, Ann. of Math. (2), 17:1 (1915), 12–22 | DOI | MR | Zbl

[15] P. T. Mocanu, “Two simple sufficient conditions for starlikeness”, Mathematica (Cluj), 34(57):2 (1992), 175–181 | MR | Zbl

[16] S. Ponnusamy and V. Singh, “Criteria for strongly starlike functions”, Complex Variables Theory Appl., 34:3 (1997), 267–291 | DOI | MR | Zbl

[17] M. Obradović, “Simple sufficient conditions for univalence”, Mat. Vesnik, 49:3-4 (1997), 241–244 | MR | Zbl

[18] L. Špaček, “Prispevek k teorii funkci prostych [Contribution à la théorie des fonctions univalentes]”, Časopis Pěst. Mat. Fys., 62 (1932), 12–19 | Zbl

[19] W. Kaplan, “Close-to-convex schlicht functions”, Michigan Math. J., 1:2 (1952), 169–185 | DOI | MR | Zbl

[20] R. Nevanlinna, “Über die schlichten Abbildungen des Einheitskreises”, Översikt av Finska Vet. Soc. Förh., 62 (A) (1920), 7, 14 pp. | Zbl

[21] H. Grunsky, “Zwei Bemerkungen zur konformen Abbildung”, Jahresber. Deutsch. Math.-Verein., 43 (1934), 140–143 | Zbl

[22] V. V. Goryainov and V. Ya. Gutlyanskii, “The radius of starlikeness for conformal mappingsСЏ”, Dopovid. Akad. Nauk Ukrain RSR Ser. A, 1974 (1974), 100–102 (Ukrainian) | MR | Zbl

[23] V. V. Goryainov, Value domains of certain functionals and geometric properties of a conformal mapping, PhD Thesis, Kuban' State University, Krasnodar, 1975, 104 pp. (Russian)

[24] O. Lehto, Univalent functions and Teichmüller spaces, Grad. Texts in Math., 109, Springer-Verlag, New York, 1987, xii+257 pp. | DOI | MR | Zbl

[25] V. N. Dubinin, “Geometric estimates for the Schwarzian derivative”, Uspeki Mat. Nauk, 72:3(435) (2017), 97–130 ; English transl. in Russian Math. Surveys, 72:3 (2017), 479–511 | DOI | MR | Zbl | DOI

[26] V. N. Dubinin, “Schwarzian derivative and covering arcs of a pencil of circles by holomorphic functions”, Mat. Zametki, 98:6 (2015), 865–871 ; English transl. in Math. Notes, 98:6 (2015), 920–925 | DOI | MR | Zbl | DOI

[27] W. Kraus, “Über den Zusammenhang einiger Characteristiken eines einfach zusammenhängenden Bereiches mit der Kreisabbildung”, Mitt. Math. Sem. Univ. Giessen, 21 (1932), 1–28 | Zbl

[28] Z. Nehari, “The Schwarzian derivative and schlicht functions”, Bull. Amer. Math. Soc., 55:6 (1949), 545–551 | DOI | MR | Zbl

[29] E. Hille, “Remarks on a paper by Zeev Nehari”, Bull. Amer. Math. Soc., 55:6 (1949), 552–553 | DOI | MR | Zbl

[30] Z. Nehari, “Some criteria of univalence”, Proc. Amer. Math. Soc., 5 (1954), 700–704 | DOI | MR | Zbl

[31] V. V. Pokornyi, “Some sufficient conditions of univalence”, Dokl. Akad. Nauk SSSR, 79:5 (1951), 743–746 (Russian) | MR | Zbl

[32] G. M. Goluzin, Geometric theory of functions of a complex variable, 2nd ed., Nauka, Moscow, 1966, 628 pp. ; English transl. Transl. Math. Monogr., 26, Amer. Math. Soc., Providence, RI, 1969, vi+676 pp. | MR | Zbl | MR | Zbl

[33] Ch. Pommerenke, Univalent functions, Studia Math./Math. Lehrbücher, 25, Vandenhoeck Ruprecht, Göttingen, 1975, 376 pp. | MR | Zbl

[34] P. L. Duren, H. S. Shapiro, and A. L. Shields, “Singular measures and domains not of Smirnov type”, Duke Math. J., 33:2 (1966), 247–254 | DOI | MR | Zbl

[35] J. Becker, “Löwnersche Differentlialgleichung und quasikonform fortsetzbare schlichte Funktionen”, J. Reine Angew. Math., 1972:255 (1972), 23–43 | DOI | MR | Zbl

[36] J. Becker and Ch. Pommerenke, “Schlichtheitskriterien und Jordangebiete”, J. Reine Angew. Math., 1984:354 (1984), 74–94 | DOI | MR | Zbl

[37] O. A. Busovskaya and V. V. Goryainov, “Homeomorphic continuation of spiral functions”, Ukrain. Mat. Zh., 33:5 (1981), 656–660 ; English transl. in Ukrainian Math. J., 33:5 (1981), 501–504 | MR | Zbl | DOI

[38] P. Koebe, “Über die Uniformisierung der algebraischen Kurven. II”, Math. Ann., 69:1 (1910), 1–81 | DOI | MR | Zbl

[39] L. Bieberbach, “Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln”, Sitzungsber Preuß. Akad. Wiss., Phys.-Math. Kl., 138 (1916), 940–955 | Zbl

[40] A. Bloch, “Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation”, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3), 17 (1925), 1–22 | DOI | MR | Zbl

[41] L. V. Ahlfors, “An extension of Schwarz's lemma”, Trans. Amer. Math. Soc., 43:3 (1938), 359–364 | DOI | MR | Zbl

[42] M. Heins, “On a class of conformal metrics”, Nagoya Math. J., 21 (1962), 1–60 | DOI | MR | Zbl

[43] M. Bonk, “On Bloch's constant”, Proc. Amer. Math. Soc., 110:4 (1990), 889–894 | DOI | MR | Zbl

[44] H. Chen and P. M. Gauthier, “On Bloch's constant”, J. Anal. Math., 69 (1996), 275–291 | DOI | MR | Zbl

[45] L. V. Ahlfors and H. Grunsky, “Über die Blochsche Konstante”, Math. Z., 42:1 (1937), 671–673 | DOI | MR | Zbl

[46] E. Landau, “Der Picard–Schottkysche Satz und die Blochsche Konstante”, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl., 1926 (1926), 467–474 | Zbl

[47] M. Tsuji, Potential theory in modern function theory, Reprint of the 1959 original, Chelsea Publishing Co., New York, 1975, x+590 pp. | MR | Zbl

[48] P. Cacridis-Theodorakopulos, “Über die untere Grenze der Rundungsschranken der beschränkten Funktionen $f(z)$, deren $|f'(0)|$ Vorgegebenen ist”, Math. Ann., 113:1 (1937), 657–664 | DOI | MR | Zbl

[49] G. Pick, “Über den Koebeschen Verzerrungssatz”, Ber. Verh. sächs. Ges. Wiss. Leipzig, Math.-Phys. Kl., 68 (1916), 58–64 | Zbl

[50] O. S. Kudryavtseva and A. P. Solodov, “Generalization of the Landau and Becker–Pommerenke inequalities”, Dokl. Ross. Akad. Nauk. Mat. Inform. Proteassy Upr., 505:4 (2022), 46–49 ; English transl. in Dokl. Math., 106:1 (2022), 251–253 | DOI | Zbl | DOI | MR

[51] K. Löwner, “Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I”, Math. Ann., 89:1-2 (1923), 103–121 | DOI | MR | Zbl

[52] O. S. Kudryavtseva and A. P. Solodov, “Asymptotically sharp two-sided estimate for domains of univalence of holomorphic self-maps of a disc with an invariant diameter”, Mat. Sbornik, 211:11 (2020), 96–117 ; English transl. in Sb. Math., 211:11 (2020), 1592–1611 | DOI | MR | Zbl | DOI

[53] G. Valiron, Fonctions analytiques, Presses Univ. de France, Paris, 1954, 236 pp. | MR | Zbl

[54] Ch. Pommerenke, “On the iteration of analytic functions in a halfplane. I”, J. London Math. Soc. (2), 19:3 (1979), 439–447 | DOI | MR | Zbl

[55] I. N. Baker and Ch. Pommerenke, “On the iteration of analytic functions in a halfplane. II”, J. London Math. Soc. (2), 20:2 (1979), 255–258 | DOI | MR | Zbl

[56] J. Becker and Ch. Pommerenke, “Angular derivatives for holomorphic self-maps of the disk”, Comput. Methods Funct. Theory, 17:3 (2017), 487–497 | DOI | MR | Zbl

[57] O. S. Kudryavtseva and A. P. Solodov, “Two-sided estimates for domains of univalence for classes of holomorphic self-maps of a disc with two fixed points”, Mat. Sb., 210:7 (2019), 120–144 ; English transl. in Sb. Math., 210:7 (2019), 1019–1042 | DOI | MR | Zbl | DOI

[58] A. P. Solodov, “The exact domain of univalence on the class of holomorphic maps of a disc into itself with an interior and a boundary fixed points”, Izv. Ross. Akad. Nauk Ser. Mat., 85:5 (2021), 190–218 ; English transl. in Izv. Math., 85:5 (2021), 1008–1035 | DOI | MR | Zbl | DOI

[59] O. S. Kudryavtseva and A. P. Solodov, “Inverse function theorem on the class of holomorphic self-maps of a disc with two fixed points”, Uspekhi Mat. Nauk, 77:1(463) (2022), 187–188 ; English transl. in Russian Math. Surveys, 77:1 (2022), 177–179 | DOI | MR | Zbl | DOI

[60] V. V. Goryainov, “Holomorphic mappings of a strip into itself with bounded distortion at infinity”, Complex analysis and its applcations, Tr. Mat. Inst. Steklova, 298, MAIK Nauka/Interperiodika, Moscow, 2017, 101–111 ; English transl. in Proc. Steklov Inst. Math., 298 (2017), 94–103 | DOI | MR | Zbl | DOI

[61] E. Schröder, “Ueber iterirte Functionen”, Math. Ann., 3:2 (1870), 296–322 | DOI | MR | Zbl

[62] G. Koenigs, “Recherches sur les intégrales de certaines équations fonctionelles”, Ann. Sci. École Norm. Sup. (3), 1 (1884), 3–41 | DOI | MR | Zbl

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

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

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

[66] V. V. Goryainov, “One-parameter semigroups of analytic functions and a compositional analogue of infinite divisibility”, Tr. Inst. Prikl. Mat. Mekh., 5, Institute of Applied Mathematics and Mechanics. National Academy of Sciences of Ukraine, Donetsk, 2000, 44–57 (Russian) | MR | Zbl

[67] A. Denjoy, “Sur l'itération des fonctions analytiques”, C. R. Acad. Sci. Paris, 182 (1926), 255–257 | Zbl

[68] J. Wolff, “Sur l'itération des fonctions holomorphes dans une région, et dont les valeurs appartiennent à cette région”, C. R. Acad. Sci. Paris, 182 (1926), 42–43 | Zbl

[69] J. Wolff, “Sur l'itération des fonctions bornées”, C. R. Acad. Sci. Paris, 182 (1926), 200–201 | Zbl

[70] V. V. Goryaĭnov, “Fractional iterates of functions analytic in the unit disk, with given fixed points”, Mat. Sb., 182:9 (1991), 1281–1299 ; English transl. in Sb. Math., 74:1 (1993), 29–46 | MR | Zbl | DOI

[71] D. Shoikhet, “Representations of holomorphic generators and distortion theorems for spirallike functions with respect to a boundary point”, Int. J. Pure Appl. Math., 5:3 (2003), 335–361 | MR | Zbl

[72] M. D. Contreras, S. Díaz-Madrigal, and Ch. Pommerenke, “On boundary critical points for semigroups of analytic functions”, Math. Scand., 98:1 (2006), 125–142 | DOI | MR | Zbl

[73] F. Bracci, M. D. Contreras, and S. Díaz-Madrigal, “Aleksandrov–Clark measures and semigroups of analytic functions in the unit disc”, Ann. Acad. Sci. Fenn. Math., 33:1 (2008), 231–240 | MR | Zbl

[74] V. V. Goryainov and O. S. Kudryavtseva, “One-parameter semigroups of analytic functions, fixed points and the Koenigs function”, Mat. Sb., 202:7 (2011), 43–74 ; English transl. in Sb. Math., 202:7 (2011), 971–1000 | DOI | MR | Zbl | DOI

[75] V. V. Goryainov, “Semigroups of analytic functions in analysis and applications”, Uspekhi Mat. Nauk, 67:6(408) (2012), 5–52 ; English transl. in Russian Math. Surveys, 67:6 (2012), 975–1021 | DOI | MR | Zbl | DOI

[76] M. Elin, V. Goryainov, S. Reich, and D. Shoikhet, “Fractional iteration and functional equations for analytic in the unit disk”, Comput. Methods Funct. Theory, 2:2 (2002), 353–366 | DOI | MR | Zbl

[77] T. E. Harris, “Some mathematical models for branching processes”, Proceedings of the 2nd Berkeley symposium on mathematical statistics and probability (1950), Univ. of California Press, Berkeley–Los Angeles, CA, 1951, 305–328 | MR | Zbl

[78] T. E. Harris, The theory of branching processes, Grundlehren Math. Wiss., 119, Springer-Verlag, Berlin; Prentice-Hall, Inc., 1963, xiv+230 pp. | MR | Zbl

[79] S. Karlin and J. McGregor, “Embeddability of discrete time simple branching processes into continuous time branching processes”, Trans. Amer. Math. Soc., 132:1 (1968), 115–136 | DOI | MR | Zbl

[80] V. V. Goryaĭnov, “Fractional iteration of probability generating functions and imbedding discrete branching processes in continuous processes”, Mat. Sb., 184:5 (1993), 55–74 ; English transl. in Sb. Math., 79:1 (1994), 47–61 | MR | Zbl | DOI

[81] V. V. Goryaĭnov, “Semigroups of probability generating functions, and infinitely splittable random variables”, Theory Stoch. Process., 1(17):1 (1995), 2–9 | Zbl

[82] V. V. Goryainov, “Koenigs function and fractional iterates of probability generating functions”, Mat. Sb., 193:7 (2002), 69–86 ; English transl. in Sb. Math., 193:7 (2002), 1009–1025 | DOI | MR | Zbl | DOI

[83] P. R. Garabedian and M. Schiffer, “A proof of the Bieberbach conjecture for the fourh coefficient”, J. Rational Mech. Anal., 4:3 (1955), 427–465 | DOI | MR | Zbl

[84] L. de Branges, A proof of the Bieberbach conjecture, Preprint LOMI E-5-84, Leningrad Branch of the V. A. Steklov Math. Inst., Leningrad, 1984, 21 pp.

[85] L. de Branges, “A proof of the Bieberbach conjecture”, Acta Math., 154:1-2 (1985), 137–152 | DOI | MR | Zbl

[86] O. S. Kudryavtseva, “Fractional iterates of analytic functions with rational coefficients in the unit disc”, Vestn. Volgograd. Gos. Univ. Ser. 1 Mat. Mekh., 2011, no. 2(15), 50–62 (Russian)

[87] O. Tammi, Extremum problems for bounded univalent functions, Lecture Notes in Math., 646, Springer-Verlag, Berlin–New York, 1978, vii+313 pp. | DOI | MR | Zbl

[88] I. Schur, “Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind”, J. Reine Angew. Math., 1917:147 (1917), 205–232 | DOI | MR | Zbl

[89] O. S. Kudryavtseva, “Schwarz's lemma and estimates of coefficients in the case of an arbitrary set of boundary fixed points”, Mat. Zametki, 109:4 (2021), 636–640 ; English transl. in Math. Notes, 109:4 (2021), 653–657 | DOI | MR | Zbl | DOI

[90] C. C. Cowen and Ch. Pommerenke, “Inequalities for the angular derivative of an analytic function in the unit disk”, J. London Math. Soc. (2), 26:2 (1982), 271–289 | DOI | MR | Zbl