K\oe nigs type linearization models and asymptotic behavior of one-parameter semigroups

D. Shoikhet

Contemporary Mathematics. Fundamental Directions, Proceedings of the Seminar on Differential and Functional Differential Equations supervised by A. L. Skubachevskii (Peoples' Friendship University of Russia), Tome 21 (2007), pp. 149-167.

Résumé

In this paper we study linear-fractional models for one-parameter semigroups of holomorphic mappings via Schröder's and Abel's functional equation. By using some limit schemes in the spirit of Kœnigs to solve those equation, we obtain new results on the asymptotic behavior of one-parameter semigroups having a boundary Denjoy–Wolff fixed point. In addition, we establish infinitisimal versions of the Burns-Krantz rigidity theorem for semigroups and their generators.

Bibliographie

[1] Abate M., “The infinitesimal generators of semigroups of holomorphic maps”, Ann. Mat. Pura Appl. (4), 161 (1992), 167–180 | DOI | MR | Zbl

[2] Aharonov D., Elin M., Reich S., Shoikhet D., “Parametric representations of semi-complete vector fields on the unit balls in $\mathbb C^n$ and Hilbert space”, Rend. Mat. Acc. Lincei, 10 (1999), 229–253 | MR | Zbl

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

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

[5] Bourdon P. S., Shapiro J. H., Cyclic phenomena for composition operators, Mem. Amer. Math. Soc., 125, no. 596, 1997 | MR | Zbl

[6] Bracci F., Tauraso R., Vlacci F., “Identity principles for commuting holomorphic self-maps of the unit disc”, J. Math. Anal. Appl., 270 (2002), 451–473 | DOI | MR | Zbl

[7] Brickman L., “$\Phi$-like analytic functions, I”, Bull. Amer. Math. Soc., 79 (1973), 555–558 | DOI | MR | Zbl

[8] Burns M., Krantz S. G., “Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary”, J. Amer. Math. Soc., 7 (1994), 661–676 | DOI | MR | Zbl

[9] Contreras M. D., Díaz-Madrigal S., “Analytic flows on the unit disk: angular derivatives and boundary fixed points”, Pacific J. Math., 222 (2005), 253–286 | DOI | MR | Zbl

[10] Contreras M. D., Díaz-Madrigal S., Pommerenke Ch., Second Angular Derivatives and Parabolic Iteration in the Unit Disk, Preprint, 2005

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

[12] Cowen C., MacCluer C., MacCluer B. D., Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995 | MR | Zbl

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

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

[15] Elin M., Levinshtein M., Reich S., Shoikhet D., Commuting Semigroups of Holomorphic Mappings, Preprint

[16] Elin M., Reich S., Shoikhet D., “Complex dynamical systems and the geometry of domains in banach spaces”, Dissertationes Math. (Rozprawy Mat.), 427 (2004) | MR

[17] Elin M., Shoikhet D., “Dynamic extension of the Julia–Wolff–Carathéodory Theorem”, Dynam. Systems Appl., 10 (2001), 421–438 | MR

[18] Elin M., Shoikhet D., “Semigroups with boundary fixed points on the unit Hilbert ball and spirallike mappings”, Geometric Function Theory in Several Complex Variables, World Sci. Publishing, River Edge, NJ, 2004, 82–117 | MR | Zbl

[19] Elin M., Shoikhet D., “An angle distortion theorem for starlike and spirallike functions with respect to a boundary point”, IJMMS, 2006 (2006), 1–13 | DOI | MR

[20] Goebel K., Reich S., Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York–Basel, 1984 | MR

[21] Gurganus K. R., “$\Phi$-like holomorphic functions in $\mathbb C^n$ and Banach space”, Trans. Amer. Math. Soc., 205 (1975), 389–406 | DOI | MR | Zbl

[22] Hadamard J., “Two works on iteration and related questions”, Bull. Amer. Math. Soc., 50 (1944), 67–75 | DOI | MR | Zbl

[23] Harris T. E., The Theory of Branching Processes, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1963 | MR | Zbl

[24] Kœœnigs G., “Recherches sur les intégrales de certaines équations fonctionnelles”, Ann. Sci. École Norm. Sup. 1, 1884, 2–41

[25] Kuczma M., “On the Schröder equation”, Dissertationes Math. (Rozprawy Mat.), 34 (1963) | MR | Zbl

[26] Lecko A., “On the class of functions convex in the negative direction of the imaginary axis”, J. Aust. Math. Soc., 73 (2002), 1–10 | DOI | MR | Zbl

[27] Migliorini S., Vlacci F., “A new rigidity result for holomorphic maps”, Indag. Mathem. (N.S.), 13:4 (2002), 537–549 | DOI | MR | Zbl

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

[29] Pommerenke Ch., Boundary behavior of conformal maps, Springer-Verlag, New York–Berlin–Heidelberg, 1992 | MR | Zbl

[30] Poreda T., “On generalized differential equations in Banach space”, Dissertationes Math. (Rozprawy Mat.), 310 (1991) | MR | Zbl

[31] Reich S., Shoikhet D., “Generation theory for semigroups of holomorphic mappings in Banach spaces”, Abstr. Appl. Anal., 1 (1996), 1–44 | DOI | MR | Zbl

[32] Reich S., Shoikhet D., “Semigroups and generators on convex domains with the hyperbolic metric”, Atti. Acad. Naz. Lincei (9), 8 (1997), 231–250 | MR | Zbl

[33] Reich S., Shoikhet D., “The Denjoy–Wolff theorem”, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 51 (1997), 219–240 | MR | Zbl

[34] Reich S., Shoikhet D., “Metric domains, holomorphic mappings and nonlinear semigroups”, Abstr. Appl. Anal., 3 (1998), 203–228 | DOI | MR | Zbl

[35] Schrœder E., “Über itierte Funktionen”, Math. Ann., 3 (1871), 296–322 | DOI

[36] Shapiro J. H., Composition Operators and Classical Function Theory, Springer-Verlag, Berlin, 1993 | MR

[37] Shoikhet D., Semigroups in Geometrical Function Theory, Kluwer, Dordrecht, 2001 | MR

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

[39] Siskakis A., “Semigroups of composition operators on spaces of analytic functions, a review”, Studies on composition operators (Laramie, WY, 1996), Contemp. Math., 213, Amer. Math. Soc., Providence, RI, 1998, 229–252 | MR | Zbl

[40] Tauraso R., “Commuting holomorphic maps of the unit disc”, Ergodic Theory Dynam. Systems, 24 (2004), 945–953 | DOI | MR | Zbl

[41] Tauraso R., Vlacci F., “Rigidity at the boundary for self maps of the disk”, Complex Var. Theory Appl., 45:2 (2001), 151–165 | MR | Zbl

[42] Valiron G., Fonctions Analytiques, Presses Univ. France, Paris, 1954 | MR | Zbl

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

[44] Wolff J., “Sur l'iteration des fonctions bornees”, C. R. Acad. Sci., 182 (1926), 200–201 | Zbl

[45] Wolff J., “Sur une generalisation d'un theoreme de Schwarz”, C. R. Acad. Sci., 182 (1926), 918–920 | Zbl